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0705.3360.pdf | 0 | # The Road to Quantum Artificial Intelligence
Kyriakos N. Sgarbas
Wire Communications Lab., Dept. of Electrical and Computer Engineering,
University of Patras, GR-26500, Patras, Greece
E-mail: sgarbas $$@$$ upatras.gr
# Abstract
This paper overviews the basic principles and recent advances in the emerging field of
Quantum Computation (QC), highlighting its potential application to Artificial Intelligence
(AI). The paper provides a very brief introduction to basic QC issues like quantum registers,
quantum gates and quantum algorithms and then it presents references, ideas and research
guidelines on how QC can be used to deal with some basic AI problems, such as search and
pattern matching, as soon as quantum computers become widely available.
Keywords: Quantum Computation, Artificial Intelligence
# 1. Introduction
Quantum Computation (QC) is the scientific field that studies how the quantum
behavior of certain subatomic particles (i.e. photons, electrons, etc.) can be used to
perform computation and eventually large scale information processing.
Superposition and entanglement are two key-phenomena in the quantum domain that
provide a much more efficient way to perform certain kinds of computations than
classical algorithmic methods. In QC information is stored in quantum registers
composed of series of quantum bits (or qubits). QC defines a set of operators called
quantum gates that operate on quantum registers performing simple qubit-range
computations. Quantum algorithms are successive applications of several quantum
gates on a quantum register and perform more elaborate computations.
QC’s ability to perform parallel information processing and rapid search over
unordered sets of data promises significant advances to the whole scientific field of
information processing. This article focuses on the benefits QC has to offer in the area
of Artificial Intelligence (AI). In fact, several research papers have already reported
how QC relates to specific aspects of AI (e.g. quantum game theory [Miakisz et al.
(2006)], quantum evolutionary programming [Rylander et al. (2001)], etc). The
present article attempts a more global view on quantum methods for AI applications
addressing not only work already done but also some broad ideas for future work. But
first it presents a very brief (due to space limitation) introduction to QC basics and
algorithms, just the essentials to understand the subject. For a full introduction and
| <h1>The Road to Quantum Artificial Intelligence</h1>
<p>Kyriakos N. Sgarbas</p>
<p>Wire Communications Lab., Dept. of Electrical and Computer Engineering,
University of Patras, GR-26500, Patras, Greece
E-mail: sgarbas $$@$$ upatras.gr</p>
<h1>Abstract</h1>
<p>This paper overviews the basic principles and recent advances in the emerging field of
Quantum Computation (QC), highlighting its potential application to Artificial Intelligence
(AI). The paper provides a very brief introduction to basic QC issues like quantum registers,
quantum gates and quantum algorithms and then it presents references, ideas and research
guidelines on how QC can be used to deal with some basic AI problems, such as search and
pattern matching, as soon as quantum computers become widely available.</p>
<p>Keywords: Quantum Computation, Artificial Intelligence</p>
<h1>1. Introduction</h1>
<p>Quantum Computation (QC) is the scientific field that studies how the quantum
behavior of certain subatomic particles (i.e. photons, electrons, etc.) can be used to
perform computation and eventually large scale information processing.
Superposition and entanglement are two key-phenomena in the quantum domain that
provide a much more efficient way to perform certain kinds of computations than
classical algorithmic methods. In QC information is stored in quantum registers
composed of series of quantum bits (or qubits). QC defines a set of operators called
quantum gates that operate on quantum registers performing simple qubit-range
computations. Quantum algorithms are successive applications of several quantum
gates on a quantum register and perform more elaborate computations.</p>
<p>QC’s ability to perform parallel information processing and rapid search over
unordered sets of data promises significant advances to the whole scientific field of
information processing. This article focuses on the benefits QC has to offer in the area
of Artificial Intelligence (AI). In fact, several research papers have already reported
how QC relates to specific aspects of AI (e.g. quantum game theory [Miakisz et al.
(2006)], quantum evolutionary programming [Rylander et al. (2001)], etc). The
present article attempts a more global view on quantum methods for AI applications
addressing not only work already done but also some broad ideas for future work. But
first it presents a very brief (due to space limitation) introduction to QC basics and
algorithms, just the essentials to understand the subject. For a full introduction and</p>
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|
0705.3360.pdf | 1 | more details the reader is advised to read [Karafyllidis (2005a)], [Gruska (1999)] or
[Nielsen & Chuang (2000)].
# 2. Quantum Computation Basics
The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-
tion can be the energy state of an electron in an atom, the polarization of a photon, or
any other bi-state quantum system. When a qubit is measured (or observed), its state
is always found in one of two clearly distinct states, usually transcribed as $$|0>$$ and
$$|1>$$ . These are direct analogs of the 0 and 1 states of a classical bit but they are also
orthogonal states of a 2-dimensional Hilbert space and they are called basis states for
the qubit. Before the qubit is measured, its state can be in a composition of its basis
states denoted as:
In Eq.1 a and b are complex numbers called probability amplitudes; $$|\mathbf{a}|^{2}$$ is the
probability of the qubit to appear in state $$|0>$$ when observed, and $$|\mathfrak{b}|^{2}$$ is the probability
to appear in state $$|1>$$ . Equation 1 also presents the matrix notation of the qubit states.
A series of qubits is called a quantum register.
An n-qubit quantum register is denoted as:
It has $$2^{\mathtt{n}}$$ observable states, corresponding to the basis states of Eq.2, each one having
a probability of $$\left|\mathbf{c}_{\mathrm{i}}\right|^{2}$$ when measured. Again, this can be considered as a vector of an n-
dimensional Hilbert space with $$\sum_{i=0}^{2^{n}-1}\bigl|c_{i}\bigr|^{2}=1\,.$$
A single qubit can be considered as a trivial quantum register with $$\mathtt{n}{=}1$$ . When $$\mathfrak{n}^{>1}$$ the
quantum register can be considered as a series of qubits:
where $$\otimes$$ denotes the tensor product.
Quantum systems are able to simultaneously occupy different quantum states. This is
known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state
of Eq.2 for the quantum register represent superpositions of the basis states over the
same set of qubits. A quantum register can be in a superposition of two or more basis
states (with a maximum of $$2^{\ n}$$ , where n is the number of its qubits). The qubits of the
| <p>more details the reader is advised to read [Karafyllidis (2005a)], [Gruska (1999)] or
[Nielsen & Chuang (2000)].</p>
<h1>2. Quantum Computation Basics</h1>
<p>The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-
tion can be the energy state of an electron in an atom, the polarization of a photon, or
any other bi-state quantum system. When a qubit is measured (or observed), its state
is always found in one of two clearly distinct states, usually transcribed as $$|0>$$ and
$$|1>$$ . These are direct analogs of the 0 and 1 states of a classical bit but they are also
orthogonal states of a 2-dimensional Hilbert space and they are called basis states for
the qubit. Before the qubit is measured, its state can be in a composition of its basis
states denoted as:</p>
<p>In Eq.1 a and b are complex numbers called probability amplitudes; $$|\mathbf{a}|^{2}$$ is the
probability of the qubit to appear in state $$|0>$$ when observed, and $$|\mathfrak{b}|^{2}$$ is the probability
to appear in state $$|1>$$ . Equation 1 also presents the matrix notation of the qubit states.
A series of qubits is called a quantum register.</p>
<p>An n-qubit quantum register is denoted as:</p>
<p>It has $$2^{\mathtt{n}}$$ observable states, corresponding to the basis states of Eq.2, each one having
a probability of $$\left|\mathbf{c}_{\mathrm{i}}\right|^{2}$$ when measured. Again, this can be considered as a vector of an n-
dimensional Hilbert space with $$\sum_{i=0}^{2^{n}-1}\bigl|c_{i}\bigr|^{2}=1\,.$$</p>
<p>A single qubit can be considered as a trivial quantum register with $$\mathtt{n}{=}1$$ . When $$\mathfrak{n}^{>1}$$ the
quantum register can be considered as a series of qubits:</p>
<p>where $$\otimes$$ denotes the tensor product.</p>
<p>Quantum systems are able to simultaneously occupy different quantum states. This is
known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state
of Eq.2 for the quantum register represent superpositions of the basis states over the
same set of qubits. A quantum register can be in a superposition of two or more basis
states (with a maximum of $$2^{\ n}$$ , where n is the number of its qubits). The qubits of the</p>
| [{"type": "text", "coordinates": [49, 78, 432, 105], "content": "more details the reader is advised to read [Karafyllidis (2005a)], [Gruska (1999)] or\n[Nielsen & Chuang (2000)].", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [49, 117, 240, 132], "content": "2. Quantum Computation Basics", "block_type": "title", "index": 2}, {"type": "text", "coordinates": [49, 137, 432, 239], "content": "The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-\ntion can be the energy state of an electron in an atom, the polarization of a photon, or\nany other bi-state quantum system. When a qubit is measured (or observed), its state\nis always found in one of two clearly distinct states, usually transcribed as $$|0>$$ and\n$$|1>$$ . These are direct analogs of the 0 and 1 states of a classical bit but they are also\northogonal states of a 2-dimensional Hilbert space and they are called basis states for\nthe qubit. Before the qubit is measured, its state can be in a composition of its basis\nstates denoted as:", "block_type": "text", "index": 3}, {"type": "interline_equation", "coordinates": [166, 241, 321, 275], "content": "", "block_type": "interline_equation", "index": 4}, {"type": "text", "coordinates": [49, 282, 432, 335], "content": "In Eq.1 a and b are complex numbers called probability amplitudes; $$|\\mathbf{a}|^{2}$$ is the\nprobability of the qubit to appear in state $$|0>$$ when observed, and $$|\\mathfrak{b}|^{2}$$ is the probability\nto appear in state $$|1>$$ . Equation 1 also presents the matrix notation of the qubit states.\nA series of qubits is called a quantum register.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [50, 339, 238, 353], "content": "An n-qubit quantum register is denoted as:", "block_type": "text", "index": 6}, {"type": "interline_equation", "coordinates": [89, 359, 390, 394], "content": "", "block_type": "interline_equation", "index": 7}, {"type": "text", "coordinates": [50, 400, 431, 459], "content": "It has $$2^{\\mathtt{n}}$$ observable states, corresponding to the basis states of Eq.2, each one having\na probability of $$\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}$$ when measured. Again, this can be considered as a vector of an n-\ndimensional Hilbert space with $$\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.$$", "block_type": "text", "index": 8}, {"type": "text", "coordinates": [49, 467, 431, 493], "content": "A single qubit can be considered as a trivial quantum register with $$\\mathtt{n}{=}1$$ . When $$\\mathfrak{n}^{>1}$$ the\nquantum register can be considered as a series of qubits:", "block_type": "text", "index": 9}, {"type": "interline_equation", "coordinates": [84, 497, 398, 518], "content": "", "block_type": "interline_equation", "index": 10}, {"type": "text", "coordinates": [50, 526, 212, 538], "content": "where $$\\otimes$$ denotes the tensor product.", "block_type": "text", "index": 11}, {"type": "text", "coordinates": [49, 544, 432, 608], "content": "Quantum systems are able to simultaneously occupy different quantum states. This is\nknown as a superposition of states. In fact, the state of Eq.1 for the qubit and the state\nof Eq.2 for the quantum register represent superpositions of the basis states over the\nsame set of qubits. A quantum register can be in a superposition of two or more basis\nstates (with a maximum of $$2^{\\ n}$$ , where n is the number of its qubits). The qubits of the", "block_type": "text", "index": 12}] | [{"type": "text", "coordinates": [50, 79, 431, 94], "content": "more details the reader is advised to read [Karafyllidis (2005a)], [Gruska (1999)] or", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [50, 92, 175, 107], "content": "[Nielsen & Chuang (2000)].", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [50, 118, 239, 133], "content": "2. Quantum Computation Basics", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [50, 139, 430, 151], "content": "The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [50, 152, 431, 165], "content": "tion can be the energy state of an electron in an atom, the polarization of a photon, or", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [50, 165, 431, 177], "content": "any other bi-state quantum system. When a qubit is measured (or observed), its state", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [49, 176, 395, 191], "content": "is always found in one of two clearly distinct states, usually transcribed as ", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [395, 176, 411, 190], "content": "|0>", "score": 0.88, "index": 8}, {"type": "text", "coordinates": [411, 176, 432, 191], "content": " and", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [49, 189, 65, 202], "content": "|1>", "score": 0.59, "index": 10}, {"type": "text", "coordinates": [65, 189, 431, 203], "content": ". These are direct analogs of the 0 and 1 states of a classical bit but they are also", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [50, 203, 431, 215], "content": "orthogonal states of a 2-dimensional Hilbert space and they are called basis states for", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [50, 216, 431, 228], "content": "the qubit. Before the qubit is measured, its state can be in a composition of its basis", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [50, 229, 128, 239], "content": "states denoted as:", "score": 1.0, "index": 14}, {"type": "interline_equation", "coordinates": [166, 241, 321, 275], "content": "{\\big|}{\\mathbf{q}}{\\big\\rangle}={\\mathbf{a}}{\\big|}0{\\big\\rangle}+{\\mathbf{b}}{\\big|}1{\\big\\rangle}={\\mathbf{a}}{\\binom{\\!{\\mathsf{T}}{1}}{0}}+{\\mathbf{b}}{\\binom{\\!{\\mathsf{0}}}{1}}={\\left[\\!{\\mathbf{a}}\\!\\right]}", "score": 0.94, "index": 15}, {"type": "text", "coordinates": [50, 284, 383, 297], "content": "In Eq.1 a and b are complex numbers called probability amplitudes; ", "score": 1.0, "index": 16}, {"type": "inline_equation", "coordinates": [383, 282, 399, 296], "content": "|\\mathbf{a}|^{2}", "score": 0.89, "index": 17}, {"type": "text", "coordinates": [399, 284, 431, 297], "content": " is the", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [50, 296, 231, 310], "content": "probability of the qubit to appear in state", "score": 1.0, "index": 19}, {"type": "inline_equation", "coordinates": [232, 296, 247, 309], "content": "|0>", "score": 0.87, "index": 20}, {"type": "text", "coordinates": [247, 296, 338, 310], "content": " when observed, and", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [338, 295, 354, 309], "content": "|\\mathfrak{b}|^{2}", "score": 0.9, "index": 22}, {"type": "text", "coordinates": [354, 296, 430, 310], "content": " is the probability", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [49, 309, 128, 323], "content": "to appear in state", "score": 1.0, "index": 24}, {"type": "inline_equation", "coordinates": [128, 309, 144, 322], "content": "|1>", "score": 0.58, "index": 25}, {"type": "text", "coordinates": [144, 309, 431, 323], "content": ". Equation 1 also presents the matrix notation of the qubit states.", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [49, 322, 256, 336], "content": "A series of qubits is called a quantum register.", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [51, 340, 239, 354], "content": "An n-qubit quantum register is denoted as:", "score": 1.0, "index": 28}, {"type": "interline_equation", "coordinates": [89, 359, 390, 394], "content": "\\left|Q_{n}\\right\\rangle=c_{0}{\\big|}0\\cdots000{\\big\\rangle}+c_{1}{\\big|}0\\cdots001{\\big\\rangle}+\\cdots+c_{2^{n}-1}{\\big|}1\\cdots111{\\big\\rangle}=\\sum_{i=0}^{2^{n}-1}c_{i}{\\big|}i{\\big\\rangle}", "score": 0.91, "index": 29}, {"type": "text", "coordinates": [49, 401, 77, 414], "content": "It has", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [77, 401, 88, 411], "content": "2^{\\mathtt{n}}", "score": 0.84, "index": 31}, {"type": "text", "coordinates": [89, 401, 430, 414], "content": " observable states, corresponding to the basis states of Eq.2, each one having", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [50, 413, 120, 427], "content": "a probability of", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [120, 412, 137, 426], "content": "\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}", "score": 0.91, "index": 34}, {"type": "text", "coordinates": [137, 413, 430, 427], "content": " when measured. Again, this can be considered as a vector of an n-", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [51, 438, 190, 451], "content": "dimensional Hilbert space with ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [191, 426, 246, 461], "content": "\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.", "score": 0.74, "index": 37}, {"type": "text", "coordinates": [50, 469, 343, 482], "content": "A single qubit can be considered as a trivial quantum register with", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [344, 468, 362, 479], "content": "\\mathtt{n}{=}1", "score": 0.84, "index": 39}, {"type": "text", "coordinates": [363, 469, 395, 482], "content": ". When", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [395, 468, 414, 479], "content": "\\mathfrak{n}^{>1}", "score": 0.75, "index": 41}, {"type": "text", "coordinates": [414, 469, 430, 482], "content": " the", "score": 1.0, "index": 42}, {"type": "text", "coordinates": [50, 482, 298, 493], "content": "quantum register can be considered as a series of qubits:", "score": 1.0, "index": 43}, {"type": "interline_equation", "coordinates": [84, 497, 398, 518], "content": "\\left|\\!\\left|Q_{n}\\right\\rangle\\!\\right=\\!\\left|q_{n-1}\\right\\rangle\\otimes\\!\\left|q_{n-2}\\right\\rangle\\!\\cdots\\!\\left|q_{i}\\right\\rangle\\!\\cdots\\!\\left|q_{1}\\right\\rangle\\otimes\\!\\left|q_{0}\\right\\rangle\\!=\\!\\left|q_{n-1}q_{n-2}\\cdots q_{i}\\cdots q_{1}q_{0}\\right\\rangle", "score": 0.9, "index": 44}, {"type": "text", "coordinates": [50, 526, 79, 539], "content": "where", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [79, 526, 89, 536], "content": "\\otimes", "score": 0.83, "index": 46}, {"type": "text", "coordinates": [89, 526, 211, 539], "content": " denotes the tensor product.", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [50, 545, 431, 558], "content": "Quantum systems are able to simultaneously occupy different quantum states. This is", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [50, 558, 431, 572], "content": "known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [50, 570, 431, 584], "content": "of Eq.2 for the quantum register represent superpositions of the basis states over the", "score": 1.0, "index": 50}, {"type": "text", "coordinates": [50, 583, 431, 596], "content": "same set of qubits. A quantum register can be in a superposition of two or more basis", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [50, 596, 171, 608], "content": "states (with a maximum of", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [172, 595, 182, 606], "content": "2^{\\ n}", "score": 0.82, "index": 53}, {"type": "text", "coordinates": [183, 596, 431, 608], "content": ", where n is the number of its qubits). The qubits of the", "score": 1.0, "index": 54}] | [] | [{"type": "block", "coordinates": [166, 241, 321, 275], "content": "", "caption": ""}, {"type": "block", "coordinates": [89, 359, 390, 394], "content": "", "caption": ""}, {"type": "block", "coordinates": [84, 497, 398, 518], "content": "", "caption": ""}, {"type": "inline", "coordinates": [395, 176, 411, 190], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [49, 189, 65, 202], "content": "|1>", "caption": ""}, {"type": "inline", "coordinates": [383, 282, 399, 296], "content": "|\\mathbf{a}|^{2}", "caption": ""}, {"type": "inline", "coordinates": [232, 296, 247, 309], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [338, 295, 354, 309], "content": "|\\mathfrak{b}|^{2}", "caption": ""}, {"type": "inline", "coordinates": [128, 309, 144, 322], "content": "|1>", "caption": ""}, {"type": "inline", "coordinates": [77, 401, 88, 411], "content": "2^{\\mathtt{n}}", "caption": ""}, {"type": "inline", "coordinates": [120, 412, 137, 426], "content": "\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}", "caption": ""}, {"type": "inline", "coordinates": [191, 426, 246, 461], "content": "\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.", "caption": ""}, {"type": "inline", "coordinates": [344, 468, 362, 479], "content": "\\mathtt{n}{=}1", "caption": ""}, {"type": "inline", "coordinates": [395, 468, 414, 479], "content": "\\mathfrak{n}^{>1}", "caption": ""}, {"type": "inline", "coordinates": [79, 526, 89, 536], "content": "\\otimes", "caption": ""}, {"type": "inline", "coordinates": [172, 595, 182, 606], "content": "2^{\\ n}", "caption": ""}] | [] | [481.0, 680.0] | [{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "2. Quantum Computation Basics ", "text_level": 1, "page_idx": 1}, {"type": "text", "text": "The quantum analog of a bit is called a quantum bit or qubit. Its physical implementation can be the energy state of an electron in an atom, the polarization of a photon, or any other bi-state quantum system. When a qubit is measured (or observed), its state is always found in one of two clearly distinct states, usually transcribed as $|0>$ and $|1>$ . These are direct analogs of the 0 and 1 states of a classical bit but they are also orthogonal states of a 2-dimensional Hilbert space and they are called basis states for the qubit. Before the qubit is measured, its state can be in a composition of its basis states denoted as: ", "page_idx": 1}, {"type": "equation", "text": "$$\n{\\big|}{\\mathbf{q}}{\\big\\rangle}={\\mathbf{a}}{\\big|}0{\\big\\rangle}+{\\mathbf{b}}{\\big|}1{\\big\\rangle}={\\mathbf{a}}{\\binom{\\!{\\mathsf{T}}{1}}{0}}+{\\mathbf{b}}{\\binom{\\!{\\mathsf{0}}}{1}}={\\left[\\!{\\mathbf{a}}\\!\\right]}\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "In Eq.1 a and b are complex numbers called probability amplitudes; $|\\mathbf{a}|^{2}$ is the probability of the qubit to appear in state $|0>$ when observed, and $|\\mathfrak{b}|^{2}$ is the probability to appear in state $|1>$ . Equation 1 also presents the matrix notation of the qubit states. A series of qubits is called a quantum register. ", "page_idx": 1}, {"type": "text", "text": "An n-qubit quantum register is denoted as: ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\left|Q_{n}\\right\\rangle=c_{0}{\\big|}0\\cdots000{\\big\\rangle}+c_{1}{\\big|}0\\cdots001{\\big\\rangle}+\\cdots+c_{2^{n}-1}{\\big|}1\\cdots111{\\big\\rangle}=\\sum_{i=0}^{2^{n}-1}c_{i}{\\big|}i{\\big\\rangle}\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "It has $2^{\\mathtt{n}}$ observable states, corresponding to the basis states of Eq.2, each one having a probability of $\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}$ when measured. Again, this can be considered as a vector of an ndimensional Hilbert space with $\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.$ ", "page_idx": 1}, {"type": "text", "text": "A single qubit can be considered as a trivial quantum register with $\\mathtt{n}{=}1$ . When $\\mathfrak{n}^{>1}$ the quantum register can be considered as a series of qubits: ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\left|\\!\\left|Q_{n}\\right\\rangle\\!\\right=\\!\\left|q_{n-1}\\right\\rangle\\otimes\\!\\left|q_{n-2}\\right\\rangle\\!\\cdots\\!\\left|q_{i}\\right\\rangle\\!\\cdots\\!\\left|q_{1}\\right\\rangle\\otimes\\!\\left|q_{0}\\right\\rangle\\!=\\!\\left|q_{n-1}q_{n-2}\\cdots q_{i}\\cdots q_{1}q_{0}\\right\\rangle\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "where $\\otimes$ denotes the tensor product. ", "page_idx": 1}, {"type": "text", "text": "Quantum systems are able to simultaneously occupy different quantum states. This is known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state of Eq.2 for the quantum register represent superpositions of the basis states over the same set of qubits. A quantum register can be in a superposition of two or more basis states (with a maximum of $2^{\\ n}$ , where n is the number of its qubits). The qubits of the quantum register remain in superposition until they are measured (intentionally or not). At the time of measurement the state of the register collapses (or is resolved) to one of its basis states randomly, according to the probability assigned to that state. 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105], "lines": [{"bbox": [50, 79, 431, 94], "spans": [{"bbox": [50, 79, 431, 94], "score": 1.0, "content": "more details the reader is advised to read [Karafyllidis (2005a)], [Gruska (1999)] or", "type": "text"}], "index": 0}, {"bbox": [50, 92, 175, 107], "spans": [{"bbox": [50, 92, 175, 107], "score": 1.0, "content": "[Nielsen & Chuang (2000)].", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "title", "bbox": [49, 117, 240, 132], "lines": [{"bbox": [50, 118, 239, 133], "spans": [{"bbox": [50, 118, 239, 133], "score": 1.0, "content": "2. Quantum Computation Basics", "type": "text"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [49, 137, 432, 239], "lines": [{"bbox": [50, 139, 430, 151], "spans": [{"bbox": [50, 139, 430, 151], "score": 1.0, "content": "The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-", "type": "text"}], "index": 3}, {"bbox": [50, 152, 431, 165], "spans": [{"bbox": [50, 152, 431, 165], "score": 1.0, "content": "tion can be the energy state of an electron in an atom, the polarization of a photon, or", "type": "text"}], "index": 4}, {"bbox": [50, 165, 431, 177], "spans": [{"bbox": [50, 165, 431, 177], "score": 1.0, "content": "any other bi-state quantum system. When a qubit is measured (or observed), its state", "type": "text"}], "index": 5}, {"bbox": [49, 176, 432, 191], "spans": [{"bbox": [49, 176, 395, 191], "score": 1.0, "content": "is always found in one of two clearly distinct states, usually transcribed as ", "type": "text"}, {"bbox": [395, 176, 411, 190], "score": 0.88, "content": "|0>", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [411, 176, 432, 191], "score": 1.0, "content": " and", "type": "text"}], "index": 6}, {"bbox": [49, 189, 431, 203], "spans": [{"bbox": [49, 189, 65, 202], "score": 0.59, "content": "|1>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [65, 189, 431, 203], "score": 1.0, "content": ". These are direct analogs of the 0 and 1 states of a classical bit but they are also", "type": "text"}], "index": 7}, {"bbox": [50, 203, 431, 215], "spans": [{"bbox": [50, 203, 431, 215], "score": 1.0, "content": "orthogonal states of a 2-dimensional Hilbert space and they are called basis states for", "type": "text"}], "index": 8}, {"bbox": [50, 216, 431, 228], "spans": [{"bbox": [50, 216, 431, 228], "score": 1.0, "content": "the qubit. Before the qubit is measured, its state can be in a composition of its basis", "type": "text"}], "index": 9}, {"bbox": [50, 229, 128, 239], "spans": [{"bbox": [50, 229, 128, 239], "score": 1.0, "content": "states denoted as:", "type": "text"}], "index": 10}], "index": 6.5}, {"type": "interline_equation", "bbox": [166, 241, 321, 275], "lines": [{"bbox": [166, 241, 321, 275], "spans": [{"bbox": [166, 241, 321, 275], "score": 0.94, "content": "{\\big|}{\\mathbf{q}}{\\big\\rangle}={\\mathbf{a}}{\\big|}0{\\big\\rangle}+{\\mathbf{b}}{\\big|}1{\\big\\rangle}={\\mathbf{a}}{\\binom{\\!{\\mathsf{T}}{1}}{0}}+{\\mathbf{b}}{\\binom{\\!{\\mathsf{0}}}{1}}={\\left[\\!{\\mathbf{a}}\\!\\right]}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [49, 282, 432, 335], "lines": [{"bbox": [50, 282, 431, 297], "spans": [{"bbox": [50, 284, 383, 297], "score": 1.0, "content": "In Eq.1 a and b are complex numbers called probability amplitudes; ", "type": "text"}, {"bbox": [383, 282, 399, 296], "score": 0.89, "content": "|\\mathbf{a}|^{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [399, 284, 431, 297], "score": 1.0, "content": " is the", "type": "text"}], "index": 12}, {"bbox": [50, 295, 430, 310], "spans": [{"bbox": [50, 296, 231, 310], "score": 1.0, "content": "probability of the qubit to appear in state", "type": "text"}, {"bbox": [232, 296, 247, 309], "score": 0.87, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [247, 296, 338, 310], "score": 1.0, "content": " when observed, and", "type": "text"}, {"bbox": [338, 295, 354, 309], "score": 0.9, "content": "|\\mathfrak{b}|^{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [354, 296, 430, 310], "score": 1.0, "content": " is the probability", "type": "text"}], "index": 13}, {"bbox": [49, 309, 431, 323], "spans": [{"bbox": [49, 309, 128, 323], "score": 1.0, "content": "to appear in state", "type": "text"}, {"bbox": [128, 309, 144, 322], "score": 0.58, "content": "|1>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [144, 309, 431, 323], "score": 1.0, "content": ". Equation 1 also presents the matrix notation of the qubit states.", "type": "text"}], "index": 14}, {"bbox": [49, 322, 256, 336], "spans": [{"bbox": [49, 322, 256, 336], "score": 1.0, "content": "A series of qubits is called a quantum register.", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [50, 339, 238, 353], "lines": [{"bbox": [51, 340, 239, 354], "spans": [{"bbox": [51, 340, 239, 354], "score": 1.0, "content": "An n-qubit quantum register is denoted as:", "type": "text"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [89, 359, 390, 394], "lines": [{"bbox": [89, 359, 390, 394], "spans": [{"bbox": [89, 359, 390, 394], "score": 0.91, "content": "\\left|Q_{n}\\right\\rangle=c_{0}{\\big|}0\\cdots000{\\big\\rangle}+c_{1}{\\big|}0\\cdots001{\\big\\rangle}+\\cdots+c_{2^{n}-1}{\\big|}1\\cdots111{\\big\\rangle}=\\sum_{i=0}^{2^{n}-1}c_{i}{\\big|}i{\\big\\rangle}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [50, 400, 431, 459], "lines": [{"bbox": [49, 401, 430, 414], "spans": [{"bbox": [49, 401, 77, 414], "score": 1.0, "content": "It has", "type": "text"}, {"bbox": [77, 401, 88, 411], "score": 0.84, "content": "2^{\\mathtt{n}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [89, 401, 430, 414], "score": 1.0, "content": " observable states, corresponding to the basis states of Eq.2, each one having", "type": "text"}], "index": 18}, {"bbox": [50, 412, 430, 427], "spans": [{"bbox": [50, 413, 120, 427], "score": 1.0, "content": "a probability of", "type": "text"}, {"bbox": [120, 412, 137, 426], "score": 0.91, "content": "\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [137, 413, 430, 427], "score": 1.0, "content": " when measured. Again, this can be considered as a vector of an n-", "type": "text"}], "index": 19}, {"bbox": [51, 426, 246, 461], "spans": [{"bbox": [51, 438, 190, 451], "score": 1.0, "content": "dimensional Hilbert space with ", "type": "text"}, {"bbox": [191, 426, 246, 461], "score": 0.74, "content": "\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.", "type": "inline_equation", "height": 35, "width": 55}], "index": 20}], "index": 19}, {"type": "text", "bbox": [49, 467, 431, 493], "lines": [{"bbox": [50, 468, 430, 482], "spans": [{"bbox": [50, 469, 343, 482], "score": 1.0, "content": "A single qubit can be considered as a trivial quantum register with", "type": "text"}, {"bbox": [344, 468, 362, 479], "score": 0.84, "content": "\\mathtt{n}{=}1", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [363, 469, 395, 482], "score": 1.0, "content": ". When", "type": "text"}, {"bbox": [395, 468, 414, 479], "score": 0.75, "content": "\\mathfrak{n}^{>1}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [414, 469, 430, 482], "score": 1.0, "content": " the", "type": "text"}], "index": 21}, {"bbox": [50, 482, 298, 493], "spans": [{"bbox": [50, 482, 298, 493], "score": 1.0, "content": "quantum register can be considered as a series of qubits:", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "interline_equation", "bbox": [84, 497, 398, 518], "lines": [{"bbox": [84, 497, 398, 518], "spans": [{"bbox": [84, 497, 398, 518], "score": 0.9, "content": "\\left|\\!\\left|Q_{n}\\right\\rangle\\!\\right=\\!\\left|q_{n-1}\\right\\rangle\\otimes\\!\\left|q_{n-2}\\right\\rangle\\!\\cdots\\!\\left|q_{i}\\right\\rangle\\!\\cdots\\!\\left|q_{1}\\right\\rangle\\otimes\\!\\left|q_{0}\\right\\rangle\\!=\\!\\left|q_{n-1}q_{n-2}\\cdots q_{i}\\cdots q_{1}q_{0}\\right\\rangle", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [50, 526, 212, 538], "lines": [{"bbox": [50, 526, 211, 539], "spans": [{"bbox": [50, 526, 79, 539], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [79, 526, 89, 536], "score": 0.83, "content": "\\otimes", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [89, 526, 211, 539], "score": 1.0, "content": " denotes the tensor product.", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [49, 544, 432, 608], "lines": [{"bbox": [50, 545, 431, 558], "spans": [{"bbox": [50, 545, 431, 558], "score": 1.0, "content": "Quantum systems are able to simultaneously occupy different quantum states. This is", "type": "text"}], "index": 25}, {"bbox": [50, 558, 431, 572], "spans": [{"bbox": [50, 558, 431, 572], "score": 1.0, "content": "known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state", "type": "text"}], "index": 26}, {"bbox": [50, 570, 431, 584], "spans": [{"bbox": [50, 570, 431, 584], "score": 1.0, "content": "of Eq.2 for the quantum register represent superpositions of the basis states over the", "type": "text"}], "index": 27}, {"bbox": [50, 583, 431, 596], "spans": [{"bbox": [50, 583, 431, 596], "score": 1.0, "content": "same set of qubits. A quantum register can be in a superposition of two or more basis", "type": "text"}], "index": 28}, {"bbox": [50, 595, 431, 608], "spans": [{"bbox": [50, 596, 171, 608], "score": 1.0, "content": "states (with a maximum of", "type": "text"}, {"bbox": [172, 595, 182, 606], "score": 0.82, "content": "2^{\\ n}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [183, 596, 431, 608], "score": 1.0, "content": ", where n is the number of its qubits). The qubits of the", "type": "text"}], "index": 29}], "index": 27}], "layout_bboxes": [], "page_idx": 1, "page_size": [481.0, 680.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [166, 241, 321, 275], "lines": [{"bbox": [166, 241, 321, 275], "spans": [{"bbox": [166, 241, 321, 275], "score": 0.94, "content": "{\\big|}{\\mathbf{q}}{\\big\\rangle}={\\mathbf{a}}{\\big|}0{\\big\\rangle}+{\\mathbf{b}}{\\big|}1{\\big\\rangle}={\\mathbf{a}}{\\binom{\\!{\\mathsf{T}}{1}}{0}}+{\\mathbf{b}}{\\binom{\\!{\\mathsf{0}}}{1}}={\\left[\\!{\\mathbf{a}}\\!\\right]}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [89, 359, 390, 394], "lines": [{"bbox": [89, 359, 390, 394], "spans": [{"bbox": [89, 359, 390, 394], "score": 0.91, "content": "\\left|Q_{n}\\right\\rangle=c_{0}{\\big|}0\\cdots000{\\big\\rangle}+c_{1}{\\big|}0\\cdots001{\\big\\rangle}+\\cdots+c_{2^{n}-1}{\\big|}1\\cdots111{\\big\\rangle}=\\sum_{i=0}^{2^{n}-1}c_{i}{\\big|}i{\\big\\rangle}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [84, 497, 398, 518], "lines": [{"bbox": [84, 497, 398, 518], "spans": [{"bbox": [84, 497, 398, 518], "score": 0.9, "content": "\\left|\\!\\left|Q_{n}\\right\\rangle\\!\\right=\\!\\left|q_{n-1}\\right\\rangle\\otimes\\!\\left|q_{n-2}\\right\\rangle\\!\\cdots\\!\\left|q_{i}\\right\\rangle\\!\\cdots\\!\\left|q_{1}\\right\\rangle\\otimes\\!\\left|q_{0}\\right\\rangle\\!=\\!\\left|q_{n-1}q_{n-2}\\cdots q_{i}\\cdots q_{1}q_{0}\\right\\rangle", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [50, 52, 70, 63], "lines": [{"bbox": [49, 51, 71, 65], "spans": [{"bbox": [49, 51, 71, 65], "score": 1.0, "content": "470", "type": "text"}]}]}, {"type": "discarded", "bbox": [223, 51, 432, 64], "lines": [{"bbox": [224, 51, 430, 66], "spans": [{"bbox": [224, 51, 430, 66], "score": 1.0, "content": "11th Panhellenic Conference in Informatics", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [49, 78, 432, 105], "lines": [], "index": 0.5, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [50, 79, 431, 107], "lines_deleted": true}, {"type": "title", "bbox": [49, 117, 240, 132], "lines": [{"bbox": [50, 118, 239, 133], "spans": [{"bbox": [50, 118, 239, 133], "score": 1.0, "content": "2. Quantum Computation Basics", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_1", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [49, 137, 432, 239], "lines": [{"bbox": [50, 139, 430, 151], "spans": [{"bbox": [50, 139, 430, 151], "score": 1.0, "content": "The quantum analog of a bit is called a quantum bit or qubit. Its physical implementa-", "type": "text"}], "index": 3}, {"bbox": [50, 152, 431, 165], "spans": [{"bbox": [50, 152, 431, 165], "score": 1.0, "content": "tion can be the energy state of an electron in an atom, the polarization of a photon, or", "type": "text"}], "index": 4}, {"bbox": [50, 165, 431, 177], "spans": [{"bbox": [50, 165, 431, 177], "score": 1.0, "content": "any other bi-state quantum system. When a qubit is measured (or observed), its state", "type": "text"}], "index": 5}, {"bbox": [49, 176, 432, 191], "spans": [{"bbox": [49, 176, 395, 191], "score": 1.0, "content": "is always found in one of two clearly distinct states, usually transcribed as ", "type": "text"}, {"bbox": [395, 176, 411, 190], "score": 0.88, "content": "|0>", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [411, 176, 432, 191], "score": 1.0, "content": " and", "type": "text"}], "index": 6}, {"bbox": [49, 189, 431, 203], "spans": [{"bbox": [49, 189, 65, 202], "score": 0.59, "content": "|1>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [65, 189, 431, 203], "score": 1.0, "content": ". These are direct analogs of the 0 and 1 states of a classical bit but they are also", "type": "text"}], "index": 7}, {"bbox": [50, 203, 431, 215], "spans": [{"bbox": [50, 203, 431, 215], "score": 1.0, "content": "orthogonal states of a 2-dimensional Hilbert space and they are called basis states for", "type": "text"}], "index": 8}, {"bbox": [50, 216, 431, 228], "spans": [{"bbox": [50, 216, 431, 228], "score": 1.0, "content": "the qubit. Before the qubit is measured, its state can be in a composition of its basis", "type": "text"}], "index": 9}, {"bbox": [50, 229, 128, 239], "spans": [{"bbox": [50, 229, 128, 239], "score": 1.0, "content": "states denoted as:", "type": "text"}], "index": 10}], "index": 6.5, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [49, 139, 432, 239]}, {"type": "interline_equation", "bbox": [166, 241, 321, 275], "lines": [{"bbox": [166, 241, 321, 275], "spans": [{"bbox": [166, 241, 321, 275], "score": 0.94, "content": "{\\big|}{\\mathbf{q}}{\\big\\rangle}={\\mathbf{a}}{\\big|}0{\\big\\rangle}+{\\mathbf{b}}{\\big|}1{\\big\\rangle}={\\mathbf{a}}{\\binom{\\!{\\mathsf{T}}{1}}{0}}+{\\mathbf{b}}{\\binom{\\!{\\mathsf{0}}}{1}}={\\left[\\!{\\mathbf{a}}\\!\\right]}", "type": "interline_equation"}], "index": 11}], "index": 11, "page_num": "page_1", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [49, 282, 432, 335], "lines": [{"bbox": [50, 282, 431, 297], "spans": [{"bbox": [50, 284, 383, 297], "score": 1.0, "content": "In Eq.1 a and b are complex numbers called probability amplitudes; ", "type": "text"}, {"bbox": [383, 282, 399, 296], "score": 0.89, "content": "|\\mathbf{a}|^{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [399, 284, 431, 297], "score": 1.0, "content": " is the", "type": "text"}], "index": 12}, {"bbox": [50, 295, 430, 310], "spans": [{"bbox": [50, 296, 231, 310], "score": 1.0, "content": "probability of the qubit to appear in state", "type": "text"}, {"bbox": [232, 296, 247, 309], "score": 0.87, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [247, 296, 338, 310], "score": 1.0, "content": " when observed, and", "type": "text"}, {"bbox": [338, 295, 354, 309], "score": 0.9, "content": "|\\mathfrak{b}|^{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [354, 296, 430, 310], "score": 1.0, "content": " is the probability", "type": "text"}], "index": 13}, {"bbox": [49, 309, 431, 323], "spans": [{"bbox": [49, 309, 128, 323], "score": 1.0, "content": "to appear in state", "type": "text"}, {"bbox": [128, 309, 144, 322], "score": 0.58, "content": "|1>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [144, 309, 431, 323], "score": 1.0, "content": ". Equation 1 also presents the matrix notation of the qubit states.", "type": "text"}], "index": 14}, {"bbox": [49, 322, 256, 336], "spans": [{"bbox": [49, 322, 256, 336], "score": 1.0, "content": "A series of qubits is called a quantum register.", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [49, 282, 431, 336]}, {"type": "text", "bbox": [50, 339, 238, 353], "lines": [{"bbox": [51, 340, 239, 354], "spans": [{"bbox": [51, 340, 239, 354], "score": 1.0, "content": "An n-qubit quantum register is denoted as:", "type": "text"}], "index": 16}], "index": 16, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [51, 340, 239, 354]}, {"type": "interline_equation", "bbox": [89, 359, 390, 394], "lines": [{"bbox": [89, 359, 390, 394], "spans": [{"bbox": [89, 359, 390, 394], "score": 0.91, "content": "\\left|Q_{n}\\right\\rangle=c_{0}{\\big|}0\\cdots000{\\big\\rangle}+c_{1}{\\big|}0\\cdots001{\\big\\rangle}+\\cdots+c_{2^{n}-1}{\\big|}1\\cdots111{\\big\\rangle}=\\sum_{i=0}^{2^{n}-1}c_{i}{\\big|}i{\\big\\rangle}", "type": "interline_equation"}], "index": 17}], "index": 17, "page_num": "page_1", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [50, 400, 431, 459], "lines": [{"bbox": [49, 401, 430, 414], "spans": [{"bbox": [49, 401, 77, 414], "score": 1.0, "content": "It has", "type": "text"}, {"bbox": [77, 401, 88, 411], "score": 0.84, "content": "2^{\\mathtt{n}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [89, 401, 430, 414], "score": 1.0, "content": " observable states, corresponding to the basis states of Eq.2, each one having", "type": "text"}], "index": 18}, {"bbox": [50, 412, 430, 427], "spans": [{"bbox": [50, 413, 120, 427], "score": 1.0, "content": "a probability of", "type": "text"}, {"bbox": [120, 412, 137, 426], "score": 0.91, "content": "\\left|\\mathbf{c}_{\\mathrm{i}}\\right|^{2}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [137, 413, 430, 427], "score": 1.0, "content": " when measured. Again, this can be considered as a vector of an n-", "type": "text"}], "index": 19}, {"bbox": [51, 426, 246, 461], "spans": [{"bbox": [51, 438, 190, 451], "score": 1.0, "content": "dimensional Hilbert space with ", "type": "text"}, {"bbox": [191, 426, 246, 461], "score": 0.74, "content": "\\sum_{i=0}^{2^{n}-1}\\bigl|c_{i}\\bigr|^{2}=1\\,.", "type": "inline_equation", "height": 35, "width": 55}], "index": 20}], "index": 19, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [49, 401, 430, 461]}, {"type": "text", "bbox": [49, 467, 431, 493], "lines": [{"bbox": [50, 468, 430, 482], "spans": [{"bbox": [50, 469, 343, 482], "score": 1.0, "content": "A single qubit can be considered as a trivial quantum register with", "type": "text"}, {"bbox": [344, 468, 362, 479], "score": 0.84, "content": "\\mathtt{n}{=}1", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [363, 469, 395, 482], "score": 1.0, "content": ". When", "type": "text"}, {"bbox": [395, 468, 414, 479], "score": 0.75, "content": "\\mathfrak{n}^{>1}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [414, 469, 430, 482], "score": 1.0, "content": " the", "type": "text"}], "index": 21}, {"bbox": [50, 482, 298, 493], "spans": [{"bbox": [50, 482, 298, 493], "score": 1.0, "content": "quantum register can be considered as a series of qubits:", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [50, 468, 430, 493]}, {"type": "interline_equation", "bbox": [84, 497, 398, 518], "lines": [{"bbox": [84, 497, 398, 518], "spans": [{"bbox": [84, 497, 398, 518], "score": 0.9, "content": "\\left|\\!\\left|Q_{n}\\right\\rangle\\!\\right=\\!\\left|q_{n-1}\\right\\rangle\\otimes\\!\\left|q_{n-2}\\right\\rangle\\!\\cdots\\!\\left|q_{i}\\right\\rangle\\!\\cdots\\!\\left|q_{1}\\right\\rangle\\otimes\\!\\left|q_{0}\\right\\rangle\\!=\\!\\left|q_{n-1}q_{n-2}\\cdots q_{i}\\cdots q_{1}q_{0}\\right\\rangle", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_1", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [50, 526, 212, 538], "lines": [{"bbox": [50, 526, 211, 539], "spans": [{"bbox": [50, 526, 79, 539], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [79, 526, 89, 536], "score": 0.83, "content": "\\otimes", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [89, 526, 211, 539], "score": 1.0, "content": " denotes the tensor product.", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [50, 526, 211, 539]}, {"type": "text", "bbox": [49, 544, 432, 608], "lines": [{"bbox": [50, 545, 431, 558], "spans": [{"bbox": [50, 545, 431, 558], "score": 1.0, "content": "Quantum systems are able to simultaneously occupy different quantum states. This is", "type": "text"}], "index": 25}, {"bbox": [50, 558, 431, 572], "spans": [{"bbox": [50, 558, 431, 572], "score": 1.0, "content": "known as a superposition of states. In fact, the state of Eq.1 for the qubit and the state", "type": "text"}], "index": 26}, {"bbox": [50, 570, 431, 584], "spans": [{"bbox": [50, 570, 431, 584], "score": 1.0, "content": "of Eq.2 for the quantum register represent superpositions of the basis states over the", "type": "text"}], "index": 27}, {"bbox": [50, 583, 431, 596], "spans": [{"bbox": [50, 583, 431, 596], "score": 1.0, "content": "same set of qubits. A quantum register can be in a superposition of two or more basis", "type": "text"}], "index": 28}, {"bbox": [50, 595, 431, 608], "spans": [{"bbox": [50, 596, 171, 608], "score": 1.0, "content": "states (with a maximum of", "type": "text"}, {"bbox": [172, 595, 182, 606], "score": 0.82, "content": "2^{\\ n}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [183, 596, 431, 608], "score": 1.0, "content": ", where n is the number of its qubits). The qubits of the", "type": "text"}], "index": 29}, {"bbox": [50, 80, 431, 93], "spans": [{"bbox": [50, 80, 431, 93], "score": 1.0, "content": "quantum register remain in superposition until they are measured (intentionally or", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [50, 92, 430, 105], "spans": [{"bbox": [50, 92, 430, 105], "score": 1.0, "content": "not). At the time of measurement the state of the register collapses (or is resolved) to", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [50, 104, 413, 118], "spans": [{"bbox": [50, 104, 413, 118], "score": 1.0, "content": "one of its basis states randomly, according to the probability assigned to that state.", "type": "text", "cross_page": true}], "index": 2}], "index": 27, "page_num": "page_1", "page_size": [481.0, 680.0], "bbox_fs": [50, 545, 431, 608]}]} | ./arxiv_full_mineru_outputs_20/images/0705.3360_1.png | images/0705.3360_1.png |
|
0705.3360.pdf | 2 | quantum register remain in superposition until they are measured (intentionally or
not). At the time of measurement the state of the register collapses (or is resolved) to
one of its basis states randomly, according to the probability assigned to that state.
It is not necessary to measure every single qubit of a quantum register in order to
trigger its collapse to a basis state. For example, consider this case:
Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)
basis states, $$|00000>$$ , $$|10000>$$ and $$\left|1111\right>$$ with equal probability amplitudes; each of
the three states has a $$33\%$$ chance to be observed. Now, suppose we measure the
qubits one by one starting from the leftmost. The leftmost qubit has a $$67\%$$ chance to
be $$|1>$$ and $$33\%$$ to be $$|0>$$ . Let’s say we measure it and find a $$\left|0>\right.$$ . We say that the
leftmost qubit has collapsed to $$|0>$$ . But it is not the only qubit that has collapsed; the
rest four qubits must be all $$\left|0>\right.$$ too, since these are the only states consistent with the
leftmost $$|0>$$ . We say that the four rightmost qubits are entangled with the leftmost
one. In other words, they are linked together in a way that each of the qubits loses its
individuality. Measurement of one affects the others as long as they remain entangled
together. Note that if instead of measuring the leftmost qubit we had decided to
measure the rightmost one and found it $$\left|0>\right.$$ , three other qubits would collapse to $$|0>$$
as well, but the leftmost qubit would still remain in superposition. But this does not
mean that it was not affected by the measurement; it now has a $$50\%{-}50\%$$ chance of
being observed in $$\lvert0>$$ or $$|1>$$ instead of the initial $$33\%{-}67\%$$ .
Superposition does not always imply entanglement. For example, consider the state of
Eq.2: we have to measure each and every one of the n qubits in order to determine the
exact state of the register. In this case there is no entanglement.
Quantum systems in superposition or entangled states are said to be coherent. This is
a very fragile condition and can be easily disturbed by interaction with the
environment (which is considered an act of measurement). Such an accidental
disturbance is called decoherence and results to losing information to the
environment. Keeping a quantum register coherent is very difficult, especially if its
size is large.
# 3. Quantum Computation Components and Algorithms
Higher order quantum computation machines can be devised based on quantum
registers: for instance quantum finite state automata can be produced by extending
probabilistic finite-state automata in the quantum domain. Analogous extensions can
be performed for other similar state machines (e.g. quantum cellular automata,
quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the
| <p>quantum register remain in superposition until they are measured (intentionally or
not). At the time of measurement the state of the register collapses (or is resolved) to
one of its basis states randomly, according to the probability assigned to that state.</p>
<p>It is not necessary to measure every single qubit of a quantum register in order to
trigger its collapse to a basis state. For example, consider this case:</p>
<p>Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)
basis states, $$|00000>$$ , $$|10000>$$ and $$\left|1111\right>$$ with equal probability amplitudes; each of
the three states has a $$33\%$$ chance to be observed. Now, suppose we measure the
qubits one by one starting from the leftmost. The leftmost qubit has a $$67\%$$ chance to
be $$|1>$$ and $$33\%$$ to be $$|0>$$ . Let’s say we measure it and find a $$\left|0>\right.$$ . We say that the
leftmost qubit has collapsed to $$|0>$$ . But it is not the only qubit that has collapsed; the
rest four qubits must be all $$\left|0>\right.$$ too, since these are the only states consistent with the
leftmost $$|0>$$ . We say that the four rightmost qubits are entangled with the leftmost
one. In other words, they are linked together in a way that each of the qubits loses its
individuality. Measurement of one affects the others as long as they remain entangled
together. Note that if instead of measuring the leftmost qubit we had decided to
measure the rightmost one and found it $$\left|0>\right.$$ , three other qubits would collapse to $$|0>$$
as well, but the leftmost qubit would still remain in superposition. But this does not
mean that it was not affected by the measurement; it now has a $$50\%{-}50\%$$ chance of
being observed in $$\lvert0>$$ or $$|1>$$ instead of the initial $$33\%{-}67\%$$ .</p>
<p>Superposition does not always imply entanglement. For example, consider the state of
Eq.2: we have to measure each and every one of the n qubits in order to determine the
exact state of the register. In this case there is no entanglement.</p>
<p>Quantum systems in superposition or entangled states are said to be coherent. This is
a very fragile condition and can be easily disturbed by interaction with the
environment (which is considered an act of measurement). Such an accidental
disturbance is called decoherence and results to losing information to the
environment. Keeping a quantum register coherent is very difficult, especially if its
size is large.</p>
<h1>3. Quantum Computation Components and Algorithms</h1>
<p>Higher order quantum computation machines can be devised based on quantum
registers: for instance quantum finite state automata can be produced by extending
probabilistic finite-state automata in the quantum domain. Analogous extensions can
be performed for other similar state machines (e.g. quantum cellular automata,
quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the</p>
| [{"type": "text", "coordinates": [49, 78, 432, 117], "content": "quantum register remain in superposition until they are measured (intentionally or\nnot). At the time of measurement the state of the register collapses (or is resolved) to\none of its basis states randomly, according to the probability assigned to that state.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [49, 122, 432, 149], "content": "It is not necessary to measure every single qubit of a quantum register in order to\ntrigger its collapse to a basis state. For example, consider this case:", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [130, 154, 356, 187], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [48, 196, 432, 390], "content": "Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)\nbasis states, $$|00000>$$ , $$|10000>$$ and $$\\left|1111\\right>$$ with equal probability amplitudes; each of\nthe three states has a $$33\\%$$ chance to be observed. Now, suppose we measure the\nqubits one by one starting from the leftmost. The leftmost qubit has a $$67\\%$$ chance to\nbe $$|1>$$ and $$33\\%$$ to be $$|0>$$ . Let\u2019s say we measure it and find a $$\\left|0>\\right.$$ . We say that the\nleftmost qubit has collapsed to $$|0>$$ . But it is not the only qubit that has collapsed; the\nrest four qubits must be all $$\\left|0>\\right.$$ too, since these are the only states consistent with the\nleftmost $$|0>$$ . We say that the four rightmost qubits are entangled with the leftmost\none. In other words, they are linked together in a way that each of the qubits loses its\nindividuality. Measurement of one affects the others as long as they remain entangled\ntogether. Note that if instead of measuring the leftmost qubit we had decided to\nmeasure the rightmost one and found it $$\\left|0>\\right.$$ , three other qubits would collapse to $$|0>$$\nas well, but the leftmost qubit would still remain in superposition. But this does not\nmean that it was not affected by the measurement; it now has a $$50\\%{-}50\\%$$ chance of\nbeing observed in $$\\lvert0>$$ or $$|1>$$ instead of the initial $$33\\%{-}67\\%$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [49, 393, 432, 432], "content": "Superposition does not always imply entanglement. For example, consider the state of\nEq.2: we have to measure each and every one of the n qubits in order to determine the\nexact state of the register. In this case there is no entanglement.", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [49, 437, 433, 514], "content": "Quantum systems in superposition or entangled states are said to be coherent. This is\na very fragile condition and can be easily disturbed by interaction with the\nenvironment (which is considered an act of measurement). Such an accidental\ndisturbance is called decoherence and results to losing information to the\nenvironment. Keeping a quantum register coherent is very difficult, especially if its\nsize is large.", "block_type": "text", "index": 6}, {"type": "title", "coordinates": [50, 526, 370, 542], "content": "3. Quantum Computation Components and Algorithms", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [49, 548, 432, 612], "content": "Higher order quantum computation machines can be devised based on quantum\nregisters: for instance quantum finite state automata can be produced by extending\nprobabilistic finite-state automata in the quantum domain. Analogous extensions can\nbe performed for other similar state machines (e.g. quantum cellular automata,\nquantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the", "block_type": "text", "index": 8}] | [{"type": "text", "coordinates": [50, 80, 431, 93], "content": "quantum register remain in superposition until they are measured (intentionally or", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [50, 92, 430, 105], "content": "not). At the time of measurement the state of the register collapses (or is resolved) to", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [50, 104, 413, 118], "content": "one of its basis states randomly, according to the probability assigned to that state.", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [49, 123, 431, 137], "content": "It is not necessary to measure every single qubit of a quantum register in order to", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [50, 136, 347, 150], "content": "trigger its collapse to a basis state. For example, consider this case:", "score": 1.0, "index": 5}, {"type": "interline_equation", "coordinates": [130, 154, 356, 187], "content": "\\left|\\mathbf{Q}_{s}\\right\\rangle\\!=\\!\\frac{1}{\\sqrt{3}}\\big|00000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|10000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|11111\\big\\rangle", "score": 0.94, "index": 6}, {"type": "text", "coordinates": [50, 199, 431, 212], "content": "Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [50, 212, 104, 224], "content": "basis states,", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [105, 210, 142, 224], "content": "|00000>", "score": 0.45, "index": 9}, {"type": "text", "coordinates": [142, 212, 148, 224], "content": ", ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [148, 211, 183, 223], "content": "|10000>", "score": 0.31, "index": 11}, {"type": "text", "coordinates": [184, 212, 205, 224], "content": " and ", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [205, 210, 241, 223], "content": "\\left|1111\\right>", "score": 0.43, "index": 13}, {"type": "text", "coordinates": [242, 212, 432, 224], "content": " with equal probability amplitudes; each of", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [50, 224, 152, 237], "content": "the three states has a ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [152, 223, 174, 235], "content": "33\\%", "score": 0.85, "index": 16}, {"type": "text", "coordinates": [174, 224, 431, 237], "content": " chance to be observed. Now, suppose we measure the", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [50, 237, 363, 250], "content": "qubits one by one starting from the leftmost. The leftmost qubit has a", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [363, 236, 385, 248], "content": "67\\%", "score": 0.87, "index": 19}, {"type": "text", "coordinates": [386, 237, 432, 250], "content": " chance to", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [49, 250, 63, 263], "content": "be ", "score": 1.0, "index": 21}, {"type": "inline_equation", "coordinates": [64, 249, 79, 262], "content": "|1>", "score": 0.81, "index": 22}, {"type": "text", "coordinates": [80, 250, 101, 263], "content": " and ", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [101, 249, 123, 260], "content": "33\\%", "score": 0.87, "index": 24}, {"type": "text", "coordinates": [124, 250, 152, 263], "content": " to be ", "score": 1.0, "index": 25}, {"type": "inline_equation", "coordinates": [152, 249, 168, 262], "content": "|0>", "score": 0.84, "index": 26}, {"type": "text", "coordinates": [168, 250, 336, 263], "content": ". Let\u2019s say we measure it and find a ", "score": 1.0, "index": 27}, {"type": "inline_equation", "coordinates": [336, 249, 352, 262], "content": "\\left|0>\\right.", "score": 0.88, "index": 28}, {"type": "text", "coordinates": [352, 250, 432, 263], "content": ". We say that the", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [49, 262, 188, 276], "content": "leftmost qubit has collapsed to", "score": 1.0, "index": 30}, {"type": "inline_equation", "coordinates": [189, 261, 204, 274], "content": "|0>", "score": 0.84, "index": 31}, {"type": "text", "coordinates": [205, 262, 432, 276], "content": ". But it is not the only qubit that has collapsed; the", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [50, 276, 173, 288], "content": "rest four qubits must be all ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [173, 274, 188, 287], "content": "\\left|0>\\right.", "score": 0.83, "index": 34}, {"type": "text", "coordinates": [189, 276, 431, 288], "content": " too, since these are the only states consistent with the", "score": 1.0, "index": 35}, {"type": "text", "coordinates": [50, 287, 89, 300], "content": "leftmost ", "score": 1.0, "index": 36}, {"type": "inline_equation", "coordinates": [89, 287, 105, 300], "content": "|0>", "score": 0.86, "index": 37}, {"type": "text", "coordinates": [105, 287, 432, 300], "content": ". We say that the four rightmost qubits are entangled with the leftmost", "score": 1.0, "index": 38}, {"type": "text", "coordinates": [50, 300, 431, 313], "content": "one. In other words, they are linked together in a way that each of the qubits loses its", "score": 1.0, "index": 39}, {"type": "text", "coordinates": [50, 313, 431, 326], "content": "individuality. Measurement of one affects the others as long as they remain entangled", "score": 1.0, "index": 40}, {"type": "text", "coordinates": [49, 325, 432, 339], "content": "together. Note that if instead of measuring the leftmost qubit we had decided to", "score": 1.0, "index": 41}, {"type": "text", "coordinates": [50, 339, 230, 351], "content": "measure the rightmost one and found it ", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [230, 337, 245, 350], "content": "\\left|0>\\right.", "score": 0.85, "index": 43}, {"type": "text", "coordinates": [246, 339, 415, 351], "content": ", three other qubits would collapse to ", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [415, 337, 430, 350], "content": "|0>", "score": 0.85, "index": 45}, {"type": "text", "coordinates": [50, 351, 432, 363], "content": "as well, but the leftmost qubit would still remain in superposition. But this does not", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [49, 363, 338, 376], "content": "mean that it was not affected by the measurement; it now has a", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [338, 362, 384, 374], "content": "50\\%{-}50\\%", "score": 0.66, "index": 48}, {"type": "text", "coordinates": [384, 363, 432, 376], "content": " chance of", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [50, 376, 130, 389], "content": "being observed in", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [130, 375, 146, 388], "content": "\\lvert0>", "score": 0.87, "index": 51}, {"type": "text", "coordinates": [146, 376, 159, 389], "content": " or", "score": 1.0, "index": 52}, {"type": "inline_equation", "coordinates": [159, 375, 174, 388], "content": "|1>", "score": 0.63, "index": 53}, {"type": "text", "coordinates": [175, 376, 265, 389], "content": " instead of the initial", "score": 1.0, "index": 54}, {"type": "inline_equation", "coordinates": [266, 375, 311, 387], "content": "33\\%{-}67\\%", "score": 0.77, "index": 55}, {"type": "text", "coordinates": [311, 376, 314, 389], "content": ".", "score": 1.0, "index": 56}, {"type": "text", "coordinates": [50, 395, 432, 408], "content": "Superposition does not always imply entanglement. For example, consider the state of", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [50, 408, 431, 421], "content": "Eq.2: we have to measure each and every one of the n qubits in order to determine the", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [50, 420, 329, 433], "content": "exact state of the register. In this case there is no entanglement.", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [51, 439, 432, 451], "content": "Quantum systems in superposition or entangled states are said to be coherent. This is", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [50, 451, 431, 464], "content": "a very fragile condition and can be easily disturbed by interaction with the", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [50, 464, 432, 476], "content": "environment (which is considered an act of measurement). Such an accidental", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [51, 477, 430, 488], "content": "disturbance is called decoherence and results to losing information to the", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [49, 489, 432, 503], "content": "environment. Keeping a quantum register coherent is very difficult, especially if its", "score": 1.0, "index": 64}, {"type": "text", "coordinates": [50, 502, 106, 515], "content": "size is large.", "score": 1.0, "index": 65}, {"type": "text", "coordinates": [50, 527, 367, 543], "content": "3. Quantum Computation Components and Algorithms", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [51, 549, 431, 562], "content": "Higher order quantum computation machines can be devised based on quantum", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [50, 561, 431, 575], "content": "registers: for instance quantum finite state automata can be produced by extending", "score": 1.0, "index": 68}, {"type": "text", "coordinates": [51, 575, 431, 586], "content": "probabilistic finite-state automata in the quantum domain. Analogous extensions can", "score": 1.0, "index": 69}, {"type": "text", "coordinates": [50, 587, 431, 600], "content": "be performed for other similar state machines (e.g. quantum cellular automata,", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [50, 600, 431, 612], "content": "quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the", "score": 1.0, "index": 71}] | [] | [{"type": "block", "coordinates": [130, 154, 356, 187], "content": "", "caption": ""}, {"type": "inline", "coordinates": [105, 210, 142, 224], "content": "|00000>", "caption": ""}, {"type": "inline", "coordinates": [148, 211, 183, 223], "content": "|10000>", "caption": ""}, {"type": "inline", "coordinates": [205, 210, 241, 223], "content": "\\left|1111\\right>", "caption": ""}, {"type": "inline", "coordinates": [152, 223, 174, 235], "content": "33\\%", "caption": ""}, {"type": "inline", "coordinates": [363, 236, 385, 248], "content": "67\\%", "caption": ""}, {"type": "inline", "coordinates": [64, 249, 79, 262], "content": "|1>", "caption": ""}, {"type": "inline", "coordinates": [101, 249, 123, 260], "content": "33\\%", "caption": ""}, {"type": "inline", "coordinates": [152, 249, 168, 262], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [336, 249, 352, 262], "content": "\\left|0>\\right.", "caption": ""}, {"type": "inline", "coordinates": [189, 261, 204, 274], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [173, 274, 188, 287], "content": "\\left|0>\\right.", "caption": ""}, {"type": "inline", "coordinates": [89, 287, 105, 300], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [230, 337, 245, 350], "content": "\\left|0>\\right.", "caption": ""}, {"type": "inline", "coordinates": [415, 337, 430, 350], "content": "|0>", "caption": ""}, {"type": "inline", "coordinates": [338, 362, 384, 374], "content": "50\\%{-}50\\%", "caption": ""}, {"type": "inline", "coordinates": [130, 375, 146, 388], "content": "\\lvert0>", "caption": ""}, {"type": "inline", "coordinates": [159, 375, 174, 388], "content": "|1>", "caption": ""}, {"type": "inline", "coordinates": [266, 375, 311, 387], "content": "33\\%{-}67\\%", "caption": ""}] | [] | [481.0, 680.0] | [{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "It is not necessary to measure every single qubit of a quantum register in order to trigger its collapse to a basis state. For example, consider this case: ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\left|\\mathbf{Q}_{s}\\right\\rangle\\!=\\!\\frac{1}{\\sqrt{3}}\\big|00000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|10000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|11111\\big\\rangle\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible) basis states, $|00000>$ , $|10000>$ and $\\left|1111\\right>$ with equal probability amplitudes; each of the three states has a $33\\%$ chance to be observed. Now, suppose we measure the qubits one by one starting from the leftmost. The leftmost qubit has a $67\\%$ chance to be $|1>$ and $33\\%$ to be $|0>$ . Let\u2019s say we measure it and find a $\\left|0>\\right.$ . We say that the leftmost qubit has collapsed to $|0>$ . But it is not the only qubit that has collapsed; the rest four qubits must be all $\\left|0>\\right.$ too, since these are the only states consistent with the leftmost $|0>$ . We say that the four rightmost qubits are entangled with the leftmost one. In other words, they are linked together in a way that each of the qubits loses its individuality. Measurement of one affects the others as long as they remain entangled together. Note that if instead of measuring the leftmost qubit we had decided to measure the rightmost one and found it $\\left|0>\\right.$ , three other qubits would collapse to $|0>$ as well, but the leftmost qubit would still remain in superposition. But this does not mean that it was not affected by the measurement; it now has a $50\\%{-}50\\%$ chance of being observed in $\\lvert0>$ or $|1>$ instead of the initial $33\\%{-}67\\%$ . ", "page_idx": 2}, {"type": "text", "text": "Superposition does not always imply entanglement. For example, consider the state of Eq.2: we have to measure each and every one of the n qubits in order to determine the exact state of the register. In this case there is no entanglement. ", "page_idx": 2}, {"type": "text", "text": "Quantum systems in superposition or entangled states are said to be coherent. This is a very fragile condition and can be easily disturbed by interaction with the environment (which is considered an act of measurement). Such an accidental disturbance is called decoherence and results to losing information to the environment. Keeping a quantum register coherent is very difficult, especially if its size is large. ", "page_idx": 2}, {"type": "text", "text": "3. Quantum Computation Components and Algorithms ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Higher order quantum computation machines can be devised based on quantum registers: for instance quantum finite state automata can be produced by extending probabilistic finite-state automata in the quantum domain. Analogous extensions can be performed for other similar state machines (e.g. quantum cellular automata, quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the computation is eventually reduced to a series of basic operations to some qubits of a quantum register; this is what quantum gates do. 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[{"bbox": [50, 80, 431, 93], "score": 1.0, "content": "quantum register remain in superposition until they are measured (intentionally or", "type": "text"}], "index": 0}, {"bbox": [50, 92, 430, 105], "spans": [{"bbox": [50, 92, 430, 105], "score": 1.0, "content": "not). At the time of measurement the state of the register collapses (or is resolved) to", "type": "text"}], "index": 1}, {"bbox": [50, 104, 413, 118], "spans": [{"bbox": [50, 104, 413, 118], "score": 1.0, "content": "one of its basis states randomly, according to the probability assigned to that state.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [49, 122, 432, 149], "lines": [{"bbox": [49, 123, 431, 137], "spans": [{"bbox": [49, 123, 431, 137], "score": 1.0, "content": "It is not necessary to measure every single qubit of a quantum register in order to", "type": "text"}], "index": 3}, {"bbox": [50, 136, 347, 150], "spans": [{"bbox": [50, 136, 347, 150], "score": 1.0, "content": "trigger its collapse to a basis state. For example, consider this case:", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "interline_equation", "bbox": [130, 154, 356, 187], "lines": [{"bbox": [130, 154, 356, 187], "spans": [{"bbox": [130, 154, 356, 187], "score": 0.94, "content": "\\left|\\mathbf{Q}_{s}\\right\\rangle\\!=\\!\\frac{1}{\\sqrt{3}}\\big|00000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|10000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|11111\\big\\rangle", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [48, 196, 432, 390], "lines": [{"bbox": [50, 199, 431, 212], "spans": [{"bbox": [50, 199, 431, 212], "score": 1.0, "content": "Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)", "type": "text"}], "index": 6}, {"bbox": [50, 210, 432, 224], "spans": [{"bbox": [50, 212, 104, 224], "score": 1.0, "content": "basis states,", "type": "text"}, {"bbox": [105, 210, 142, 224], "score": 0.45, "content": "|00000>", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [142, 212, 148, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [148, 211, 183, 223], "score": 0.31, "content": "|10000>", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [184, 212, 205, 224], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 210, 241, 223], "score": 0.43, "content": "\\left|1111\\right>", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [242, 212, 432, 224], "score": 1.0, "content": " with equal probability amplitudes; each of", "type": "text"}], "index": 7}, {"bbox": [50, 223, 431, 237], "spans": [{"bbox": [50, 224, 152, 237], "score": 1.0, "content": "the three states has a ", "type": "text"}, {"bbox": [152, 223, 174, 235], "score": 0.85, "content": "33\\%", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [174, 224, 431, 237], "score": 1.0, "content": " chance to be observed. Now, suppose we measure the", "type": "text"}], "index": 8}, {"bbox": [50, 236, 432, 250], "spans": [{"bbox": [50, 237, 363, 250], "score": 1.0, "content": "qubits one by one starting from the leftmost. The leftmost qubit has a", "type": "text"}, {"bbox": [363, 236, 385, 248], "score": 0.87, "content": "67\\%", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [386, 237, 432, 250], "score": 1.0, "content": " chance to", "type": "text"}], "index": 9}, {"bbox": [49, 249, 432, 263], "spans": [{"bbox": [49, 250, 63, 263], "score": 1.0, "content": "be ", "type": "text"}, {"bbox": [64, 249, 79, 262], "score": 0.81, "content": "|1>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [80, 250, 101, 263], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [101, 249, 123, 260], "score": 0.87, "content": "33\\%", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [124, 250, 152, 263], "score": 1.0, "content": " to be ", "type": "text"}, {"bbox": [152, 249, 168, 262], "score": 0.84, "content": "|0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [168, 250, 336, 263], "score": 1.0, "content": ". Let\u2019s say we measure it and find a ", "type": "text"}, {"bbox": [336, 249, 352, 262], "score": 0.88, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [352, 250, 432, 263], "score": 1.0, "content": ". We say that the", "type": "text"}], "index": 10}, {"bbox": [49, 261, 432, 276], "spans": [{"bbox": [49, 262, 188, 276], "score": 1.0, "content": "leftmost qubit has collapsed to", "type": "text"}, {"bbox": [189, 261, 204, 274], "score": 0.84, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [205, 262, 432, 276], "score": 1.0, "content": ". But it is not the only qubit that has collapsed; the", "type": "text"}], "index": 11}, {"bbox": [50, 274, 431, 288], "spans": [{"bbox": [50, 276, 173, 288], "score": 1.0, "content": "rest four qubits must be all ", "type": "text"}, {"bbox": [173, 274, 188, 287], "score": 0.83, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [189, 276, 431, 288], "score": 1.0, "content": " too, since these are the only states consistent with the", "type": "text"}], "index": 12}, {"bbox": [50, 287, 432, 300], "spans": [{"bbox": [50, 287, 89, 300], "score": 1.0, "content": "leftmost ", "type": "text"}, {"bbox": [89, 287, 105, 300], "score": 0.86, "content": "|0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [105, 287, 432, 300], "score": 1.0, "content": ". We say that the four rightmost qubits are entangled with the leftmost", "type": "text"}], "index": 13}, {"bbox": [50, 300, 431, 313], "spans": [{"bbox": [50, 300, 431, 313], "score": 1.0, "content": "one. In other words, they are linked together in a way that each of the qubits loses its", "type": "text"}], "index": 14}, {"bbox": [50, 313, 431, 326], "spans": [{"bbox": [50, 313, 431, 326], "score": 1.0, "content": "individuality. Measurement of one affects the others as long as they remain entangled", "type": "text"}], "index": 15}, {"bbox": [49, 325, 432, 339], "spans": [{"bbox": [49, 325, 432, 339], "score": 1.0, "content": "together. Note that if instead of measuring the leftmost qubit we had decided to", "type": "text"}], "index": 16}, {"bbox": [50, 337, 430, 351], "spans": [{"bbox": [50, 339, 230, 351], "score": 1.0, "content": "measure the rightmost one and found it ", "type": "text"}, {"bbox": [230, 337, 245, 350], "score": 0.85, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [246, 339, 415, 351], "score": 1.0, "content": ", three other qubits would collapse to ", "type": "text"}, {"bbox": [415, 337, 430, 350], "score": 0.85, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}], "index": 17}, {"bbox": [50, 351, 432, 363], "spans": [{"bbox": [50, 351, 432, 363], "score": 1.0, "content": "as well, but the leftmost qubit would still remain in superposition. But this does not", "type": "text"}], "index": 18}, {"bbox": [49, 362, 432, 376], "spans": [{"bbox": [49, 363, 338, 376], "score": 1.0, "content": "mean that it was not affected by the measurement; it now has a", "type": "text"}, {"bbox": [338, 362, 384, 374], "score": 0.66, "content": "50\\%{-}50\\%", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [384, 363, 432, 376], "score": 1.0, "content": " chance of", "type": "text"}], "index": 19}, {"bbox": [50, 375, 314, 389], "spans": [{"bbox": [50, 376, 130, 389], "score": 1.0, "content": "being observed in", "type": "text"}, {"bbox": [130, 375, 146, 388], "score": 0.87, "content": "\\lvert0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [146, 376, 159, 389], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [159, 375, 174, 388], "score": 0.63, "content": "|1>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [175, 376, 265, 389], "score": 1.0, "content": " instead of the initial", "type": "text"}, {"bbox": [266, 375, 311, 387], "score": 0.77, "content": "33\\%{-}67\\%", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [311, 376, 314, 389], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 13}, {"type": "text", "bbox": [49, 393, 432, 432], "lines": [{"bbox": [50, 395, 432, 408], "spans": [{"bbox": [50, 395, 432, 408], "score": 1.0, "content": "Superposition does not always imply entanglement. For example, consider the state of", "type": "text"}], "index": 21}, {"bbox": [50, 408, 431, 421], "spans": [{"bbox": [50, 408, 431, 421], "score": 1.0, "content": "Eq.2: we have to measure each and every one of the n qubits in order to determine the", "type": "text"}], "index": 22}, {"bbox": [50, 420, 329, 433], "spans": [{"bbox": [50, 420, 329, 433], "score": 1.0, "content": "exact state of the register. In this case there is no entanglement.", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [49, 437, 433, 514], "lines": [{"bbox": [51, 439, 432, 451], "spans": [{"bbox": [51, 439, 432, 451], "score": 1.0, "content": "Quantum systems in superposition or entangled states are said to be coherent. This is", "type": "text"}], "index": 24}, {"bbox": [50, 451, 431, 464], "spans": [{"bbox": [50, 451, 431, 464], "score": 1.0, "content": "a very fragile condition and can be easily disturbed by interaction with the", "type": "text"}], "index": 25}, {"bbox": [50, 464, 432, 476], "spans": [{"bbox": [50, 464, 432, 476], "score": 1.0, "content": "environment (which is considered an act of measurement). Such an accidental", "type": "text"}], "index": 26}, {"bbox": [51, 477, 430, 488], "spans": [{"bbox": [51, 477, 430, 488], "score": 1.0, "content": "disturbance is called decoherence and results to losing information to the", "type": "text"}], "index": 27}, {"bbox": [49, 489, 432, 503], "spans": [{"bbox": [49, 489, 432, 503], "score": 1.0, "content": "environment. Keeping a quantum register coherent is very difficult, especially if its", "type": "text"}], "index": 28}, {"bbox": [50, 502, 106, 515], "spans": [{"bbox": [50, 502, 106, 515], "score": 1.0, "content": "size is large.", "type": "text"}], "index": 29}], "index": 26.5}, {"type": "title", "bbox": [50, 526, 370, 542], "lines": [{"bbox": [50, 527, 367, 543], "spans": [{"bbox": [50, 527, 367, 543], "score": 1.0, "content": "3. Quantum Computation Components and Algorithms", "type": "text"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [49, 548, 432, 612], "lines": [{"bbox": [51, 549, 431, 562], "spans": [{"bbox": [51, 549, 431, 562], "score": 1.0, "content": "Higher order quantum computation machines can be devised based on quantum", "type": "text"}], "index": 31}, {"bbox": [50, 561, 431, 575], "spans": [{"bbox": [50, 561, 431, 575], "score": 1.0, "content": "registers: for instance quantum finite state automata can be produced by extending", "type": "text"}], "index": 32}, {"bbox": [51, 575, 431, 586], "spans": [{"bbox": [51, 575, 431, 586], "score": 1.0, "content": "probabilistic finite-state automata in the quantum domain. Analogous extensions can", "type": "text"}], "index": 33}, {"bbox": [50, 587, 431, 600], "spans": [{"bbox": [50, 587, 431, 600], "score": 1.0, "content": "be performed for other similar state machines (e.g. quantum cellular automata,", "type": "text"}], "index": 34}, {"bbox": [50, 600, 431, 612], "spans": [{"bbox": [50, 600, 431, 612], "score": 1.0, "content": "quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the", "type": "text"}], "index": 35}], "index": 33}], "layout_bboxes": [], "page_idx": 2, "page_size": [481.0, 680.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [130, 154, 356, 187], "lines": [{"bbox": [130, 154, 356, 187], "spans": [{"bbox": [130, 154, 356, 187], "score": 0.94, "content": "\\left|\\mathbf{Q}_{s}\\right\\rangle\\!=\\!\\frac{1}{\\sqrt{3}}\\big|00000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|10000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|11111\\big\\rangle", "type": "interline_equation"}], "index": 5}], "index": 5}], "discarded_blocks": [{"type": "discarded", "bbox": [412, 53, 430, 63], "lines": [{"bbox": [410, 51, 432, 65], "spans": [{"bbox": [410, 51, 432, 65], "score": 1.0, "content": "471", "type": "text"}]}]}, {"type": "discarded", "bbox": [50, 52, 143, 64], "lines": [{"bbox": [51, 51, 143, 65], "spans": [{"bbox": [51, 51, 143, 65], "score": 1.0, "content": "AI/Knowledge bases", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [49, 78, 432, 117], "lines": [], "index": 1, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [50, 80, 431, 118], "lines_deleted": true}, {"type": "text", "bbox": [49, 122, 432, 149], "lines": [{"bbox": [49, 123, 431, 137], "spans": [{"bbox": [49, 123, 431, 137], "score": 1.0, "content": "It is not necessary to measure every single qubit of a quantum register in order to", "type": "text"}], "index": 3}, {"bbox": [50, 136, 347, 150], "spans": [{"bbox": [50, 136, 347, 150], "score": 1.0, "content": "trigger its collapse to a basis state. For example, consider this case:", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [49, 123, 431, 150]}, {"type": "interline_equation", "bbox": [130, 154, 356, 187], "lines": [{"bbox": [130, 154, 356, 187], "spans": [{"bbox": [130, 154, 356, 187], "score": 0.94, "content": "\\left|\\mathbf{Q}_{s}\\right\\rangle\\!=\\!\\frac{1}{\\sqrt{3}}\\big|00000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|10000\\big\\rangle\\!+\\!\\frac{1}{\\sqrt{3}}\\big|11111\\big\\rangle", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_2", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [48, 196, 432, 390], "lines": [{"bbox": [50, 199, 431, 212], "spans": [{"bbox": [50, 199, 431, 212], "score": 1.0, "content": "Equation 4 specifies a 5-qubit register in superposition of three (of the 32 possible)", "type": "text"}], "index": 6}, {"bbox": [50, 210, 432, 224], "spans": [{"bbox": [50, 212, 104, 224], "score": 1.0, "content": "basis states,", "type": "text"}, {"bbox": [105, 210, 142, 224], "score": 0.45, "content": "|00000>", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [142, 212, 148, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [148, 211, 183, 223], "score": 0.31, "content": "|10000>", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [184, 212, 205, 224], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 210, 241, 223], "score": 0.43, "content": "\\left|1111\\right>", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [242, 212, 432, 224], "score": 1.0, "content": " with equal probability amplitudes; each of", "type": "text"}], "index": 7}, {"bbox": [50, 223, 431, 237], "spans": [{"bbox": [50, 224, 152, 237], "score": 1.0, "content": "the three states has a ", "type": "text"}, {"bbox": [152, 223, 174, 235], "score": 0.85, "content": "33\\%", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [174, 224, 431, 237], "score": 1.0, "content": " chance to be observed. Now, suppose we measure the", "type": "text"}], "index": 8}, {"bbox": [50, 236, 432, 250], "spans": [{"bbox": [50, 237, 363, 250], "score": 1.0, "content": "qubits one by one starting from the leftmost. The leftmost qubit has a", "type": "text"}, {"bbox": [363, 236, 385, 248], "score": 0.87, "content": "67\\%", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [386, 237, 432, 250], "score": 1.0, "content": " chance to", "type": "text"}], "index": 9}, {"bbox": [49, 249, 432, 263], "spans": [{"bbox": [49, 250, 63, 263], "score": 1.0, "content": "be ", "type": "text"}, {"bbox": [64, 249, 79, 262], "score": 0.81, "content": "|1>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [80, 250, 101, 263], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [101, 249, 123, 260], "score": 0.87, "content": "33\\%", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [124, 250, 152, 263], "score": 1.0, "content": " to be ", "type": "text"}, {"bbox": [152, 249, 168, 262], "score": 0.84, "content": "|0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [168, 250, 336, 263], "score": 1.0, "content": ". Let\u2019s say we measure it and find a ", "type": "text"}, {"bbox": [336, 249, 352, 262], "score": 0.88, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [352, 250, 432, 263], "score": 1.0, "content": ". We say that the", "type": "text"}], "index": 10}, {"bbox": [49, 261, 432, 276], "spans": [{"bbox": [49, 262, 188, 276], "score": 1.0, "content": "leftmost qubit has collapsed to", "type": "text"}, {"bbox": [189, 261, 204, 274], "score": 0.84, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [205, 262, 432, 276], "score": 1.0, "content": ". But it is not the only qubit that has collapsed; the", "type": "text"}], "index": 11}, {"bbox": [50, 274, 431, 288], "spans": [{"bbox": [50, 276, 173, 288], "score": 1.0, "content": "rest four qubits must be all ", "type": "text"}, {"bbox": [173, 274, 188, 287], "score": 0.83, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [189, 276, 431, 288], "score": 1.0, "content": " too, since these are the only states consistent with the", "type": "text"}], "index": 12}, {"bbox": [50, 287, 432, 300], "spans": [{"bbox": [50, 287, 89, 300], "score": 1.0, "content": "leftmost ", "type": "text"}, {"bbox": [89, 287, 105, 300], "score": 0.86, "content": "|0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [105, 287, 432, 300], "score": 1.0, "content": ". We say that the four rightmost qubits are entangled with the leftmost", "type": "text"}], "index": 13}, {"bbox": [50, 300, 431, 313], "spans": [{"bbox": [50, 300, 431, 313], "score": 1.0, "content": "one. In other words, they are linked together in a way that each of the qubits loses its", "type": "text"}], "index": 14}, {"bbox": [50, 313, 431, 326], "spans": [{"bbox": [50, 313, 431, 326], "score": 1.0, "content": "individuality. Measurement of one affects the others as long as they remain entangled", "type": "text"}], "index": 15}, {"bbox": [49, 325, 432, 339], "spans": [{"bbox": [49, 325, 432, 339], "score": 1.0, "content": "together. Note that if instead of measuring the leftmost qubit we had decided to", "type": "text"}], "index": 16}, {"bbox": [50, 337, 430, 351], "spans": [{"bbox": [50, 339, 230, 351], "score": 1.0, "content": "measure the rightmost one and found it ", "type": "text"}, {"bbox": [230, 337, 245, 350], "score": 0.85, "content": "\\left|0>\\right.", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [246, 339, 415, 351], "score": 1.0, "content": ", three other qubits would collapse to ", "type": "text"}, {"bbox": [415, 337, 430, 350], "score": 0.85, "content": "|0>", "type": "inline_equation", "height": 13, "width": 15}], "index": 17}, {"bbox": [50, 351, 432, 363], "spans": [{"bbox": [50, 351, 432, 363], "score": 1.0, "content": "as well, but the leftmost qubit would still remain in superposition. But this does not", "type": "text"}], "index": 18}, {"bbox": [49, 362, 432, 376], "spans": [{"bbox": [49, 363, 338, 376], "score": 1.0, "content": "mean that it was not affected by the measurement; it now has a", "type": "text"}, {"bbox": [338, 362, 384, 374], "score": 0.66, "content": "50\\%{-}50\\%", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [384, 363, 432, 376], "score": 1.0, "content": " chance of", "type": "text"}], "index": 19}, {"bbox": [50, 375, 314, 389], "spans": [{"bbox": [50, 376, 130, 389], "score": 1.0, "content": "being observed in", "type": "text"}, {"bbox": [130, 375, 146, 388], "score": 0.87, "content": "\\lvert0>", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [146, 376, 159, 389], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [159, 375, 174, 388], "score": 0.63, "content": "|1>", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [175, 376, 265, 389], "score": 1.0, "content": " instead of the initial", "type": "text"}, {"bbox": [266, 375, 311, 387], "score": 0.77, "content": "33\\%{-}67\\%", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [311, 376, 314, 389], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 13, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [49, 199, 432, 389]}, {"type": "text", "bbox": [49, 393, 432, 432], "lines": [{"bbox": [50, 395, 432, 408], "spans": [{"bbox": [50, 395, 432, 408], "score": 1.0, "content": "Superposition does not always imply entanglement. For example, consider the state of", "type": "text"}], "index": 21}, {"bbox": [50, 408, 431, 421], "spans": [{"bbox": [50, 408, 431, 421], "score": 1.0, "content": "Eq.2: we have to measure each and every one of the n qubits in order to determine the", "type": "text"}], "index": 22}, {"bbox": [50, 420, 329, 433], "spans": [{"bbox": [50, 420, 329, 433], "score": 1.0, "content": "exact state of the register. In this case there is no entanglement.", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [50, 395, 432, 433]}, {"type": "text", "bbox": [49, 437, 433, 514], "lines": [{"bbox": [51, 439, 432, 451], "spans": [{"bbox": [51, 439, 432, 451], "score": 1.0, "content": "Quantum systems in superposition or entangled states are said to be coherent. This is", "type": "text"}], "index": 24}, {"bbox": [50, 451, 431, 464], "spans": [{"bbox": [50, 451, 431, 464], "score": 1.0, "content": "a very fragile condition and can be easily disturbed by interaction with the", "type": "text"}], "index": 25}, {"bbox": [50, 464, 432, 476], "spans": [{"bbox": [50, 464, 432, 476], "score": 1.0, "content": "environment (which is considered an act of measurement). Such an accidental", "type": "text"}], "index": 26}, {"bbox": [51, 477, 430, 488], "spans": [{"bbox": [51, 477, 430, 488], "score": 1.0, "content": "disturbance is called decoherence and results to losing information to the", "type": "text"}], "index": 27}, {"bbox": [49, 489, 432, 503], "spans": [{"bbox": [49, 489, 432, 503], "score": 1.0, "content": "environment. Keeping a quantum register coherent is very difficult, especially if its", "type": "text"}], "index": 28}, {"bbox": [50, 502, 106, 515], "spans": [{"bbox": [50, 502, 106, 515], "score": 1.0, "content": "size is large.", "type": "text"}], "index": 29}], "index": 26.5, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [49, 439, 432, 515]}, {"type": "title", "bbox": [50, 526, 370, 542], "lines": [{"bbox": [50, 527, 367, 543], "spans": [{"bbox": [50, 527, 367, 543], "score": 1.0, "content": "3. Quantum Computation Components and Algorithms", "type": "text"}], "index": 30}], "index": 30, "page_num": "page_2", "page_size": [481.0, 680.0]}, {"type": "text", "bbox": [49, 548, 432, 612], "lines": [{"bbox": [51, 549, 431, 562], "spans": [{"bbox": [51, 549, 431, 562], "score": 1.0, "content": "Higher order quantum computation machines can be devised based on quantum", "type": "text"}], "index": 31}, {"bbox": [50, 561, 431, 575], "spans": [{"bbox": [50, 561, 431, 575], "score": 1.0, "content": "registers: for instance quantum finite state automata can be produced by extending", "type": "text"}], "index": 32}, {"bbox": [51, 575, 431, 586], "spans": [{"bbox": [51, 575, 431, 586], "score": 1.0, "content": "probabilistic finite-state automata in the quantum domain. Analogous extensions can", "type": "text"}], "index": 33}, {"bbox": [50, 587, 431, 600], "spans": [{"bbox": [50, 587, 431, 600], "score": 1.0, "content": "be performed for other similar state machines (e.g. quantum cellular automata,", "type": "text"}], "index": 34}, {"bbox": [50, 600, 431, 612], "spans": [{"bbox": [50, 600, 431, 612], "score": 1.0, "content": "quantum Turing machines, etc) [Gruska (1999)]. Regardless the machine, the", "type": "text"}], "index": 35}, {"bbox": [51, 81, 430, 93], "spans": [{"bbox": [51, 81, 430, 93], "score": 1.0, "content": "computation is eventually reduced to a series of basic operations to some qubits of a", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [51, 94, 264, 106], "spans": [{"bbox": [51, 94, 264, 106], "score": 1.0, "content": "quantum register; this is what quantum gates do.", "type": "text", "cross_page": true}], "index": 1}], "index": 33, "page_num": "page_2", "page_size": [481.0, 680.0], "bbox_fs": [50, 549, 431, 612]}]} | ./arxiv_full_mineru_outputs_20/images/0705.3360_2.png | images/0705.3360_2.png |
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0705.3360.pdf | 3 | "computation is eventually reduced to a series of basic operations to some qubits of a\nquantum regi(...TRUNCATED) | "<p>computation is eventually reduced to a series of basic operations to some qubits of a\nquantum r(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [50, 79, 431, 104], \"content\": \"computation is eventually(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [51, 81, 430, 93], \"content\": \"computation is eventually (...TRUNCATED) | [] | "[{\"type\": \"inline\", \"coordinates\": [412, 310, 428, 323], \"content\": \"|1>\", \"caption\": \(...TRUNCATED) | [] | [481.0, 680.0] | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 3}, {\"type\": \"text\", \"text\": \"Quantum ga(...TRUNCATED) | "[{\"category_id\": 2, \"poly\": [141.12033081054688, 148.28504943847656, 192.39727783203125, 148.28(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [50, 79, 431, 104], \"lines\": [{\"bbox\": [51(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_3.png | images/0705.3360_3.png |
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0705.3360.pdf | 4 | "the intended result. This is called probability amplitude amplification [Gruska\n(1999)].\nGrover(...TRUNCATED) | "<p>the intended result. This is called probability amplitude amplification [Gruska\n(1999)].\nGrove(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [69, 79, 432, 265], \"content\": \"the intended result. This(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [85, 79, 431, 92], \"content\": \"the intended result. This (...TRUNCATED) | [] | "[{\"type\": \"inline\", \"coordinates\": [239, 105, 265, 116], \"content\": \"\\\\scriptstyle\\\\ma(...TRUNCATED) | [] | [481.0, 680.0] | "[{\"type\": \"text\", \"text\": \"the intended result. This is called probability amplitude amplifi(...TRUNCATED) | "[{\"category_id\": 2, \"poly\": [1145.6829833984375, 147.68580627441406, 1197.993408203125, 147.685(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [69, 79, 432, 265], \"lines\": [{\"bbox\": [85(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_4.png | images/0705.3360_4.png |
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0705.3360.pdf | 5 | "Grover’s algorithm [Grover (1997)] and its variations are ideal for efficient content-\naddressab(...TRUNCATED) | "<p>Grover’s algorithm [Grover (1997)] and its variations are ideal for efficient content-\naddres(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [50, 79, 431, 205], \"content\": \"Grover\\u2019s algorithm (...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [51, 80, 430, 92], \"content\": \"Grover\\u2019s algorithm [(...TRUNCATED) | [] | "[{\"type\": \"inline\", \"coordinates\": [74, 193, 84, 204], \"content\": \"2^{\\\\mathrm{n}}\", \"(...TRUNCATED) | [] | [481.0, 680.0] | "[{\"type\": \"text\", \"text\": \"Grover\\u2019s algorithm [Grover (1997)] and its variations are i(...TRUNCATED) | "[{\"category_id\": 1, \"poly\": [141.4107666015625, 220.2403564453125, 1199.1956787109375, 220.2403(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [50, 79, 431, 205], \"lines\": [{\"bbox\": [51(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_5.png | images/0705.3360_5.png |
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0705.3360.pdf | 6 | "cooperation [Miakisz et al. (2006)] and coordination games [Huberman & Hogg\n(2003)], simulating ec(...TRUNCATED) | "<p>cooperation [Miakisz et al. (2006)] and coordination games [Huberman & Hogg\n(2003)], simulating(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [49, 78, 432, 244], \"content\": \"cooperation [Miakisz et a(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [49, 79, 431, 93], \"content\": \"cooperation [Miakisz et al(...TRUNCATED) | [] | [] | [] | [481.0, 680.0] | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 6}, {\"type\": \"text\", \"text\": \"The last c(...TRUNCATED) | "[{\"category_id\": 0, \"poly\": [140.0205535888672, 1552.326171875, 366.07598876953125, 1552.326171(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [49, 78, 432, 244], \"lines\": [{\"bbox\": [49(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_6.png | images/0705.3360_6.png |
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0705.3360.pdf | 7 | "a very narrow choice of recent papers and research directions. For a lengthier\noverview on QC appl(...TRUNCATED) | "<p>a very narrow choice of recent papers and research directions. For a lengthier\noverview on QC a(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [50, 79, 430, 104], \"content\": \"a very narrow choice of r(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [49, 79, 431, 94], \"content\": \"a very narrow choice of re(...TRUNCATED) | [] | [] | [] | [481.0, 680.0] | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 7}, {\"type\": \"text\", \"text\": \"The ideas (...TRUNCATED) | "[{\"category_id\": 0, \"poly\": [141.0516815185547, 798.3380737304688, 318.3129577636719, 798.33807(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [50, 79, 430, 104], \"lines\": [{\"bbox\": [49(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_7.png | images/0705.3360_7.png |
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0705.3360.pdf | 8 | "Karafyllidis, I.G. (2005a), Quantum Computers – Basic Principles, Klidarithmos,\nAthens (in Greek(...TRUNCATED) | "<p>Karafyllidis, I.G. (2005a), Quantum Computers – Basic Principles, Klidarithmos,\nAthens (in Gr(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [50, 79, 433, 446], \"content\": \"Karafyllidis, I.G. (2005a(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [49, 79, 431, 94], \"content\": \"Karafyllidis, I.G. (2005a)(...TRUNCATED) | [] | "[{\"type\": \"inline\", \"coordinates\": [103, 206, 111, 215], \"content\": \"\\\\risingdotseq\", \(...TRUNCATED) | [] | [481.0, 680.0] | [{"type": "text", "text": "", "page_idx": 8}] | "[{\"category_id\": 2, \"poly\": [1144.7769775390625, 146.99856567382812, 1198.39306640625, 146.9985(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [50, 79, 433, 446], \"lines\": [{\"bbox\": [49(...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.3360_8.png | images/0705.3360_8.png |
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0705.0734.pdf | 0 | "# Soft constraint abstraction based on semiring\nhomomorphism ∗\n\nSanjiang Li and Mingsheng Ying(...TRUNCATED) | "<h1>Soft constraint abstraction based on semiring\nhomomorphism ∗</h1>\n<p>Sanjiang Li and Mingsh(...TRUNCATED) | "[{\"type\": \"title\", \"coordinates\": [147, 163, 463, 202], \"content\": \"Soft constraint abstra(...TRUNCATED) | "[{\"type\": \"text\", \"coordinates\": [148, 160, 463, 182], \"content\": \"Soft constraint abstrac(...TRUNCATED) | [] | "[{\"type\": \"inline\", \"coordinates\": [282, 412, 289, 416], \"content\": \"\\\\alpha\", \"captio(...TRUNCATED) | [] | [612.0, 792.0] | "[{\"type\": \"text\", \"text\": \"Soft constraint abstraction based on semiring homomorphism \\u221(...TRUNCATED) | "[{\"category_id\": 1, \"poly\": [439.8004455566406, 859.0845336914062, 1257.0494384765625, 859.0845(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"title\", \"bbox\": [147, 163, 463, 202], \"lines\": [{\"bbox\": (...TRUNCATED) | ./arxiv_full_mineru_outputs_20/images/0705.0734_0.png | images/0705.0734_0.png |
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