diff --git "a/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt" "b/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/69A0T4oBgHgl3EQfOP-G/content/tmp_files/load_file.txt" @@ -0,0 +1,911 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf,len=910 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='02158v1 [quant-ph] 5 Jan 2023 Limits of Fault-Tolerance on Resource-Constrained Quantum Circuits for Classical Problems Uthirakalyani G†, Anuj K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Nayak†, Avhishek Chatterjee, and Lav R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Varshney, Senior Member IEEE Abstract—Existing lower bounds on redundancy in fault- tolerant quantum circuits are applicable when both the input and the intended output are quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' These bounds may not necessarily hold, however, when the input and the intended output are classical bits, as in the Deutsch-Jozsa, Grover, or Shor algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here we show that indeed, noise thresholds obtained from existing bounds do not apply to a simple fault-tolerant implementation of the Deutsch-Jozsa algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then we obtain the first lower bound on the minimum required redundancy for fault-tolerant quantum circuits with classical inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Recent results show that due to physical resource constraints in quantum circuits, increasing redundancy can increase noise, which in turn may render many fault-tolerance schemes useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' So it is of both practical and theoretical interest to characterize the effect of resource constraints on the fundamental limits of fault-tolerant quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus as an application of our lower bound, we characterize the fundamental limit of fault- tolerant quantum circuits with classical inputs and outputs under resource constraint-induced noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Keywords—fault-tolerant computing, redundancy, resource constraints I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' INTRODUCTION Initial ideas [1], [2], and especially mathematical demon- strations of advantages of quantum computing over classical computing [3], [4], have spurred considerable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' How- ever, noise in quantum circuits heavily restricts the class of problems that can be solved using quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Indeed, the formal term NISQ (Noisy Intermediate Scale Quantum) has been introduced to describe the current era where quantum processors are noise-limited [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' To limit the corruption of quantum states due to noise, the pursuit of fault-tolerant quantum circuits has led to a large literature in quantum error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Early papers demonstrated that one can achieve arbitrary computational accuracy when physical noise is below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Achievability of any desired fault tolerance in these initial works required a poly-logarithmic redundancy with respect to the size of the quantum circuit [6]–[9], but more recent works extend such threshold theorems to require only a constant overhead [10], [11], reminiscent of work in classical fault- tolerant computing [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' † The student authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Uthirakalyani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' G and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Chatterjee are with the Department of Electrical Engineering, Indian Institute of Technology, Madras, Chennai 600036, India (emails:{ee19d404@smail,avhishek@ee}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='iitm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Nayak and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Varshney are with Coordinated Science Labora- tory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (emails:{anujk4, varshney}@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This work was supported in part by National Science Foundation grant PHY-2112890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In this direction, some works provide fundamental limits (lower bounds) on redundancy for arbitrarily accurate compu- tation [14]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, all of these lower bounds are for quantum input/output, rather than classical input/output which is common for a large class of algorithms, such as those due to Deutsch-Jozsa [4], Shor [19], and Grover [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, we demonstrate by example that lower bounds obtained so far in quantum fault tolerance are not applicable for quantum circuits with classical input/output, and provide a general alternate bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As far as we know, this is the first lower bound on fault tolerance for quantum circuits with classical input/output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The effects of noise on computational accuracy of quantum circuits are typically studied assuming the noise per physical qubit is constant with respect to the the size of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Un- fortunately, this is not true in many quantum devices today— often due to limited physical resources such as energy [21], volume [22], or available bandwidth [23]—that have physical noise levels that grow as the quantum computer grows [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Fellous-Asiani, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' introduce physical models of such scale- dependent noise and also aim to extend threshold theorems to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, the characterization of computational error (per logical qubit error) is empirical in nature, lacking precise mathematical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Moreover, the characterization depends on specific implementation and is restricted to con- catenated codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, using our new redundancy lower bound and tools from optimization theory, we characterize the limits of scale-dependence on fault-tolerant quantum circuits with classical input/output, agnostic to specific implementation and error correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The two motivations for the present work are therefore to obtain lower bounds on the required redundancy of a quantum circuit for computation with classical input/output, and to investigate the effect of resource constraints (like energy or volume) on this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The distance between the distributions of the output clas- sical bit corresponding to two different quantum input states vanishes exponentially with the depth of the circuit when noise is above a threshold [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Similarly for the trace distance be- tween the output quantum states [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' These results, however, do not apply when the depth of the circuit is small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' they also do not provide lower bounds on required redundancy for sub- threshold noise when fault-tolerant computation is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This work focuses on shallow quantum circuits whose input and output are classical, aiming for converse results for classical computation using quantum circuits that yield lower bounds on required redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In [14], [18], lower bounds on required redundancy that also led to improved noise thresholds were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, the fault-tolerance criteria in [14], [18] are not appropriate for our setting, as Section II argues using the example of the well-known Deutsch-Jozsa algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The experimental finding that noise increases with more redundancy under resource constraints implies that simple per (logical) qubit redundancy cannot achieve arbitrary computa- tional accuracy even if noise per physical qubit is below the fault-tolerance threshold, in contrast to conventional threshold theorems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This limitation is due to two opposing forces: improvement in accuracy due to increased redundancy and worse overall noise with redundancy due to scale-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In this regard, we aim to find the sweet spot on redundancy for a desired computational accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Sec- tion II motivates our work through a counterexample that illus- trates the need for a new redundancy lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Section III then gives the mathematical models of computation, noise, and resource constraints that form the basis of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then, the primary contributions follow: Section IV proves a converse bound on redundancy required for classical computation on quantum circuits, drawing on one-shot capacity of classical-quantum chan- nels (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Section V through VII analyze the converse results on the limits of scale-dependence for fault-tolerant computation, including closed-form and numerical solutions for some canonical quantum device models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Finally, Section VIII concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' NEED FOR A NEW REDUNDANCY LOWER BOUND: A COUNTEREXAMPLE Consider a quantum circuit that suffers from erasure noise (with erasure probability p) right before the final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' From one of the best known noise threshold bounds [14] and the capacity results for erasure channels, it follows that for an erasure noise per physical qubit p > 1 2, fault-tolerant computation is not possible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the required redundancy is not finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, we demonstrate that a simple adaptation of the Deutsch-Jozsa algorithm on this quantum circuit (with erasure noise before the final measurement) can have a prob- ability of error less than any ǫ > 0 even if p > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A schema of quantum circuit that implements Deutsch-Jozsa algorithm with erasure noise, to demonstrate the need for a new redundancy bound for fault-tolerant quantum computation with classical I/Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The Deutsch-Jozsa algorithm is used to determine if the given function oracle, f : {0, 1}n → {0, 1} is constant (0 or 1 for all input strings) or balanced (0 for half the input strings and 1 for the rest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' From [25, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='51], the quantum state before measurement under the absence of noise is ψ1 = � z,x∈{0,1}n (−1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='z+f(x) 2n |z⟩ |y⟩ , where |y⟩ = |0⟩−|1⟩ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Measuring the first n qubits yields either |0⟩⊗n if f(·) is constant, or an n-qubit state from {|0⟩ , |1⟩}⊗n \\ {|0⟩⊗n} if f(·) is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose the quantum states are corrupted by erasure right before measurement as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' then the state of the circuit becomes ψ2 = � z,x∈{0,1}n (−1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='z+f(x) 2n |z⟩(e) |y⟩(e) , where |z⟩(e) and |y⟩(e) are the corrupted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' erased) versions of |z⟩ and |y⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For example, if f(·) is constant, |z⟩(e) = |e00e00 · · ·e0⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', each qubit state |0⟩ is replaced i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' with probability p by qubit state |e⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Now, consider our modified algorithm: 1) Run Deutsch-Jozsa algorithm T times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) If no |1⟩ state was measured in any run, declare function oracle f(·) a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' When f(·) is balanced, the measurement in the no-erasure case must have one or more |1⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Such an oracle can be incorrectly declared as constant when all of these |1⟩ states are erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' So the probability of error is: Pe = P{f(·) is declared constant |f(·) is balanced}, = P \uf8f1 \uf8f2 \uf8f3 � j,t |zj⟩(e) t ̸= |1⟩ ���f(·) is balanced \uf8fc \uf8fd \uf8fe , ≤ P �� t |zj⟩(e) t = |e⟩ ��� |zj⟩ = |1⟩ � , for some j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since erasures are independent, Pe ≤ T � t=1 P � |zj⟩(e) t = |e⟩ ��� |zj⟩ = |1⟩ � = pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Choosing T ≥ ��� ln ǫ ln p ���, one can achieve Pe ≤ ǫ for any p ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This counterexample proves that the bound for redundancy N ≥ n Q(N) proposed in [14] does not hold for quantum computation with classical input/output, since for an erasure channel, the quantum capacity Q(N) = max{0, 1 − 2p} = 0 as p > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This motivates the need for a different bound which holds for classical I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that this does not imply the prior bounds are incorrect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' the apparent contradiction is due differences in the definition of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Prior work [14] uses a notion of distance (or similarity) between the output quantum states of noiseless and noisy circuits to quantify accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This requirement is too stringent when the output bits are classical and error probability is a more suitable performance criterion [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As such, we obtain a lower bound on the redundancy under the error probability criterion and then study the effect of resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Model of Computation Consider the computational model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2, which is a quantum circuit with classical inputs and classical outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is denoted by CQC : {0, 1}n → {0, 1}n or equivalently CQC(x) for x ∈ {0, 1}n, where n is the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The goal of the circuit is to realize a function f : {0, 1}n → {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The circuit consists of l layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The first layer takes n clas- sical inputs (x) as orthogonal quantum states |0⟩ and |1⟩ along with N − n ancillas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' It maps the input to a density operator of dimension 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Any subsequent layer i, for 2 ≤ i ≤ l − 1, takes the output of the previous layer, layer i − 1 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The output of any layer i, 1 ≤ i ≤ l − 1, is a density operator of dimension 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The final layer, layer l, performs a POVM measurement and obtains classical output CQC(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Each layer i, with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', l − 1} is a noisy quantum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is modeled as a noiseless quantum operation Li on density operators of dimensions 2N followed by N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' quantum channels N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Finally, the last layer, layer l, performs a measurement (POVM), which yields a classical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus the quantum circuit can be represented as a composition of quantum operations as CQC(x) = Ll ◦ N ⊗N ◦ Ll−1 ◦ · · · ◦ L2 ◦ N ⊗N ◦ L1(x), where ◦ has the usual meaning of function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We use the notation QCl−1 to denote the combined opera- tions of layers 2 to l, given by Ll◦N ⊗N ◦Ll−1◦· · ·◦N ⊗N ◦L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noise models Here, we consider only Holevo-additive channels charac- terized by a single parameter p ∈ [0, 1] and whose Holevo capacity is monotonically decreasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We use the generic notation Np for such a channel with parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Examples include erasure, depolarizing, and symmetric GAD channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' a) Erasure Channel: In a quantum erasure channel (QEC), each qubit flips to |e⟩⟨e|, which is orthogonal to every ρ ∈ L(Cd), with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, whenever a qubit gets corrupted, the location of corruption is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Np(ρ) = (1 − p)ρ + ρTr[ρ] |e⟩⟨e| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The classical capacity is [28]: χ(Np) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (1) b) Depolarizing Channel: When a qubit undergoes de- polarizing noise, it is replaced by a maximally mixed state I/2 with probability p [28]: Np(ρ) = (1 − p)ρ + p 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In contrast to the erasure channel, the receiver (or the decoder) is not aware of the location of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The Holevo informa- tion of the depolarizing channel is: χ(Np) = 1 − h2 � p 2 � , (2) where h2(·) is the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that the Holevo information is similar to the capacity of a binary symmetric channel with crossover probability p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' c) Generalized Amplitude Damping Channel (GADC): Amplitude damping channels model the transformation of an excited atom to ground state by spontaneous emission of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The changes are expressed using |0⟩ for the ground (no photon) state and |1⟩ for the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' If the initial state of the environment |0⟩⟨0|, is replaced by the state θµ ≜ (1 − µ) |0⟩⟨0| + µ |1⟩⟨1| , µ ∈ [0, 1] where, µ is thermal noise, we get the generalized ADC described using the following four Kraus operators [29]: A1 = � 1 − µ �1 0 0 √1 − p � , A2 = � 1 − µ �0 √p 0 0 � , A3 = √µ �√1 − p 0 0 1 � , A4 = √µ � 0 0 √p 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' GADC is not additive in general (for arbitrary µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, in the special case of symmetric generalized amplitude damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', generalized amplitude damping with µ = 1/2, it is a Holevo additive channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The classical capacity of symmetric GADC (µ = 1/2) is [29]: χ(Np) = 1 − h2 � 1−√1−p 2 � , (3) where p is the probability an atom decays from excited to ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that we have used p to describe different impairments in different channels, so p must be interpreted appropriately based on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Resource Constraints and Scale-Dependent Noise In [24], it was shown that resource constraints can lead to an increase in noise with increase in redundancy, scale- dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' A few models of scale-dependent noise have been studied in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Let k ≜ N/n ≥ 1 be the redundancy and p(k) be the noise strength when the redundancy is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (Recall that we consider Holevo-additive noise models that can be characterized by a single parameter 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=') In the polynomial model, p(k) = min(p0(1 + α(k − 1))γ, 1) and in the exponential model, p(k) = min(p0 exp(α(k − 1)γ), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, p0 ∈ [0, 1] is the noise strength in the absence of any redundancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', k = 1, and α and γ are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Motivated by practically useful noise models like erasure, depolarization, and models for scale dependence in [24], we consider the following generic scale-dependent noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noise Np is parameterized by a single parameter p ∈ [0, 1] and the Holevo information χ(Np) is non-increasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The parameter p is a function of redundancy k, given by min(p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), 1), where θ is a tuple of non-negative parameters from the set K, and 1) p0 = p(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) for all θ, 2) for any k ≥ 1, p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non-decreasing in any component of θ and in p0, given the other parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, p0 represents the noise without redundancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the initial noise without any resource constraint arising due to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' CQC model of computation: classical input, quantum computation, and classical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Clearly, the polynomial and exponential models are special cases with θ = (α, γ) ∈ K = R2 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The threshold for p0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', the minimum p0 beyond which reliable quantum computation is not possible, was studied in [24] assuming concatenated codes for error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here, we obtain a universal threshold for all fault tolerance schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' LOWER BOUND ON REQUIRED REDUNDANCY We first define the accuracy criterion for computation using quantum circuits with classical input/output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then we show how to convert the noisy computation problem to a communi- cation problem over i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This finally leads to the redundancy bound in Theorem 1, which we use to obtain thresholds for p0 under resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose f(·) is a classical function realized by a quantum circuit CQC(·) as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the ǫ-accuracy is: P{CQC(x) ̸= f(x)} < ǫ, for all x ∈ Zn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (4) holds for all x ∈ Zn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, the ǫ-accuracy condition holds for any subset of Zn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following lemma states a necessary condition for ǫ-accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Consider x(1), x(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' , x(R) ∈ Zn 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' |{f(x(i)) : 1 ≤ i ≤ R}| = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then a necessary condition for ǫ-accuracy condition (4) to hold is P{CQC(x(i)) ̸= f(x(i))} < ǫ, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that the domain is restricted to R inputs, such that the restricted mapping is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Using this bijectivity, we obtain the following simpler lemma, which connects ǫ-accurate computation with finite blocklength communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose there exists a CQC(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' P{CQC(x(i)) ̸= f(x(i))} < ǫ for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then there exists a classical circuit C(·) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' P{C(CQC(x(i))) ̸= x(i)} < ǫ, for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (5) Proof: Suppose ˆf(·) is a restriction of f(·) such that the mapping ˆf : {x(i), 1 ≤ i ≤ R} → {f(x(i)), 1 ≤ i ≤ R}, is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the inverse map ˆf −1(·) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Choosing a C(·) that implements ˆf −1(·), the probability of error can be equivalently expressed as P{C(CQC(x(i))) ̸= x(i)} < ǫ, for all 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We define ˆx(i) to be the output of C(CQC(x(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then the condition in (5) is equivalent to max x(i) P{x(i) ̸= ˆx(i)} < ǫ, 1 ≤ i ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This implies that a necessary condition to satisfy accuracy condition (4) is inf L1,C,QCl−1 max x(i) P{x(i) ̸= ˆx(i)} ≤ max x(i) P{x(i) ̸= ˆx(i)} < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that L1 is an encoding of classical bits into a quantum state, and QCl−1 followed by C(·) can be interpreted as the decoding of the noisy version of the same quantum state (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Hence, infL1,C,QCl−1 maxx(i) P{x(i) ̸= ˆx(i)} is equivalent to the maximum probability of error for transmitting message x(i), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' , R} over the channel N ⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Using this reduction, we lower-bound the redundancy for any classical computation using a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Let f : {0, 1}n → {0, 1}n be a classical function and Rf = |{f(x) : x ∈ {0, 1}n}| be the cardinality of the range of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then, for computing a classical function f with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Reduction of noisy computation model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2 to noisy communi- cation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' ǫ-accuracy using a quantum circuit corrupted by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Holevo- additive noise, the required number of physical qubits N is bounded as N > (1 − ǫ) log2(Rf) − h2(ǫ) χ(N) for all ǫ ∈ [0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: For any additive quantum channel N, an upper bound for classical communication over a quantum channel using an (M, N, ǫ) code is [28]: log2(|M|) ≤ χ(N ⊗N) + h2(ǫ) 1 − ǫ , where M is the message alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Assigning |M| = Rf yields ǫ > Pe ≥ 1 − χ(N ⊗N) + h2(Pe) log2 Rf , (6) ≥ 1 − χ(N ⊗N) + h2(ǫ) log2 Rf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The last inequality holds, since h2(·) is increasing in [0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Rearranging, we obtain χ(N ⊗N) > (1 − ǫ) log2 Rf − h2(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Noting that Holevo information is sub-additive, Nχ(N) > (1 − ǫ) log2 Rf − h2(ǫ), N > (1 − ǫ) log2 Rf − h2(ǫ) χ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (7) The bound in Theorem 1 states that the number of quantum buffers, N, needed for accurate computation of an n-bit function f is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' III, k ≜ N n is the redundancy of the quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, Theorem 1 can be seen as a lower bound on the required redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The quantity log2 Rf is the number of bits needed to encode the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We define η ≜ log2 Rf n as the compression factor of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' To understand the impact of scale-dependent noise, we will use the following corollary of Theorem 1 that gives a lower bound on the redundancy k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The condition for ǫ-accuracy in Theorem 1 is alternatively k > c(ǫ, η, n) χ(N) , (8) where c(ǫ, η, n) ≜ (1 − ǫ)η − h2(ǫ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Substituting log2 Rf = ηn and k = N/n in (7), and rearranging we obtain N > (1 − ǫ)ηn − h2(ǫ) χ(N) , k > c(ǫ, η, n) χ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' SCALE-DEPENDENT NOISE: CONVERSE REGIONS Let Np denote the channel parameterized by a noise in- tensity term p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Examples include the probability of erasure, p, for erasure channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' the probability of a quantum state being replaced by a maximally mixed state, p, for depolarizing channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' and the amplitude decay parameter, p, for symmetric GAD channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' If the error per physical qubit p is a constant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' k, then χ(Np) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, one can ensure that the necessary condition for ǫ-accuracy in (8) is satisfied by sufficiently increasing redundancy (choosing large k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' On the other hand, if p scales (increases) with redundancy, then χ(N) decreases with k (we denote the dependency on p(k) as χ(Np(k))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, satisfying ǫ-accuracy condition k > c(ǫ, η, n)/χ(Np(k)) is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In fact, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 4 plots the error probability lower bound (6) with both scale- independent and scale-dependent erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Notice that when the physical noise is independent of k, the Pe lower bound rapidly decreases with an increase in redundancy, whereas when the noise is scale dependent, the probability of error initially decreases with increasing redundancy k, but then grows beyond a certain optimum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' With this motivation, we explore the limitations of ǫ-accurate computation under scale- dependent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We specifically aim to characterize the set of (p0, θ) for which ǫ-accurate computation is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' This is equiv- alent to the noise threshold in traditional models, with scale- independent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following corollary to Theorem 1 provides a converse in terms of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose we have, ¯Θ ≜ � (p0, θ) ∈ K ���min k≥1 g(k, p0, θ, ǫ) ≥ 0 � , where g(k, p0, θ, ǫ) ≜ c(ǫ, η, n) k − χ(Np(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then ǫ-accurate computation is not possible for θ ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Also, if (p0, θ) ∈ ¯Θ then (p′ 0, θ′) ∈ ¯Θ if (p′ 0, θ′) ≥ (p0, θ) in a component-wise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='75 Redundancy (k) 10−2 10−1 100 Pe (lower bound) p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='15 p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='20 p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Comparison between Pe lower bound with (solid lines) and without (dashed-lines) scale-dependent physical noise for erasure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: From Definition 2, we must prove that if Pe < ǫ, then (p0, θ) /∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Considering the scale-dependent noise in Corollary 1, we have if Pe < ǫ, then k > c(ǫ, η, n) χ(Np((k)), g(k, p0, θ, ǫ) = c(ǫ, η, n) k − χ(Np(k)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (9) For any θ ∈ K, (9) is satisfied only if min k≥1 g(k, p0, θ, ǫ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In other words, (p0, θ) /∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As χ(Np) is non-increasing in p and p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non-decreasing in each component, (p′ 0, θ′) ≥ (p0, θ) in a component-wise sense implies (p′ 0, θ′) ∈ ¯Θ whenever (p0, θ) ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We refer to ¯Θ as the converse region since ǫ-accurate classical computation on quantum circuits is not possible if the parameters of the scale-dependent noise are in ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' As any fault- tolerant implementation has to avoid this region, characterizing ¯Θ is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' By Corollary 2, for characterizing ¯Θ, it is enough to find the minimum p0 for each θ such that (p0, θ) ∈ ¯Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' More precisely, for a fixed θ, the threshold pth(θ) can be defined as: pth(θ) := inf{p0 | (p0, θ) ∈ ¯Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (10) The threshold pth(θ) (or pth for brevity) can be obtained by solving the following optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' minimize p0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' min k≥1, 0≤p(k)≤1 gθ(k, p0) ≥ 0, (11) where, gθ(k, p0) := g(k, p0, θ, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Consider the following optimization problem PL : min k≥1, 0≤p(k)≤1 gθ(k, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Clearly, (11) has the optimization problem PL, which we refer to as the lower-level optimization problem, as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, (11) is a bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For a given set of θ the solution to PL is a function of p0, which we denote as g∗ θ(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Thus, the bi-level optimization problem in (11) can also be written as min p0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' g∗ θ(p0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (12) In general, to compute the threshold pth one needs to solve (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' However, for erasure noise and some special classes of p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), closed-form expressions for pth can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For erasure noise, thresholds are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0) = p0 (constant), then pth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, α) = p0(1 + α(k − 1)), then pth = \uf8f1 \uf8f2 \uf8f3 1 − c, if α ≥ c 1−c, and ( √cα−√cα−α+1) 2 (α−1)2 , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3) If p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, γ) = p0kγ, then pth = \uf8f1 \uf8f2 \uf8f3 1 − c, if γ ≥ c 1−c, and ( γ c ) γ (γ+1)γ+1 , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Here c = c(ǫ, η, n), defined in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Note that θ = ∅, α and γ in cases 1), 2) and 3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Consider a procedure to find a closed-form expres- sion for pth as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1) Minimize gθ(k, p0) over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is non- decreasing in k, it is enough to minimize gθ(k, p0) over [1, kmax], where kmax = max{k | p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) ≤ 1} (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The minimum occurs at either k = 1, k = kmax or a stationary point of gθ(k, p0) in [1, kmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2) Substitute the minimizer k into gθ(k, p0) ≥ 0, which yields an equation in p0, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3) Solving the equation for p0 yields a closed-form expres- sion for pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The derivation of pth for corresponding p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ) is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' For a general p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' p0, θ), however, a closed-form expression for pth in terms of θ cannot be obtained, and therefore, pth must be computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' We develop Algorithm 1 to obtain pth by solving bi-level optimization problem (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' In Algorithm 1, we solve the alternate formulation (12) using the bisection method, while assuming access to an oracle that computes g∗ θ(p0) for any p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Later, we also develop efficient algorithms that solve PL and obtain g∗ θ(p0) for any p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Algorithm 1 computes the threshold pth (up to an error of δp0), for a pre-determined set of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' First, a channel-specific Lipschitz constant L is computed using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' (15), (17), or (19) for a given (p0, θ), which determines how quickly PL is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Lines 7–17 describe the bisection method to compute pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Depending on whether PL is convex or non-convex, Algorithm 2 or Algorithm 3 is used to compute g∗ θ(p0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' The following theorem provides a proof of global conver- gence of Algorithm 1, with only a monotonicity assumption in θ (note that continuity in θ is not needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Suppose a quantum circuit is corrupted by a scale- dependent noise-per-physical qubit, p(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' θ) that is monotonic in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Then for any given θ ∈ K, the sequence {p0i} generated using Algorithm 1 converges to the threshold pth in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Proof: Algorithm 1 generates a non-increasing sequence {p+ 0i} and a non-decreasing sequence {p− 0i}, which at every iteration yields g∗(p+ 0i) ≥ 0 and g∗(p− 0i) < 0, with p0i = p+ 0i +p− 0i 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since the bisection method halves the difference between p+ 0i and p− 0i at every iteration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=', p+ 0i+1 − p− 0i+1 = p+ 0i −p− 0i 2 ), we have that for all ǫ > 0, there exists an i0 such that for all i ≥ i0, we get p+ 0i −p− 0i < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Also, since both {p+ 0i} and {p− 0i} are bounded, they converge, and since for all i ≥ i0, p+ 0i −p− 0i < ǫ, they converge to a common limit point (say p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Since, K is closed, (p∗ 0, α, γ) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Due to the monotonicity of g∗ θ(p0) (non-decreasing with p0), the following inequality holds: g∗ θ(p− 0i) ≤ g∗ θ(p∗ 0) ≤ g∗ θ(p+ 0i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Therefore, g∗ θ(p0) < 0, for all p0 < p∗, and g∗ θ(p0) ≥ 0, for all p0 > p∗, which is by definition p∗ = pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Obtaining g∗ θ(·) requires solving PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Next, we present efficient algorithms for solving PL for erasure, depolarizing, and symmetric GAD channels, and numerically obtain the converse surface for those noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' Algorithm 1 Algorithm to obtain pth/¯Θs numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 1: Initialize the set K′ ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 2: Initialize max iters, δp0, δ, ¯Θs = {}, k ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 3: for each θ ∈ K′ do 4: Initialize i ← 0, ∆p0 ← 1, p0 ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content='5, 5: p− 0 ← 0, p+ 0 ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69A0T4oBgHgl3EQfOP-G/content/2301.02158v1.pdf'} +page_content=' 6: % BISECTION METHOD 7: while ∆p0 > δp0 and i