diff --git "a/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/load_file.txt" "b/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4dAyT4oBgHgl3EQfpPgc/content/tmp_files/load_file.txt" @@ -0,0 +1,1299 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf,len=1298 +page_content='Quantum Annealing vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' QAOA: 127 Qubit Higher-Order Ising Problems on NISQ Computers Elijah Pelofske∗1, Andreas B¨artschi†1, and Stephan Eidenbenz1 1CCS-3 Information Sciences, Los Alamos National Laboratory Abstract Quantum annealing (QA) and Quantum Alternating Operator Ansatz (QAOA) are both heuristic quantum algorithms intended for sampling optimal solutions of combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In this article we implement a rigorous direct comparison between QA on D-Wave hardware and QAOA on IBMQ hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The studied problems are instances of a class of Ising problems, with variable assignments of +1 or −1, that contain cubic ZZZ interactions (higher order terms) and match both the native connectivity of the Pegasus topology D- Wave chips and the heavy hexagonal lattice of the IBMQ chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The novel QAOA implementation on the heavy hexagonal lattice has a CNOT depth of 6 per round and allows for usage of an entire heavy hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Experimentally, QAOA is executed on an ensemble of randomly generated Ising instances with a grid search over 1 and 2 round angles using all 127 programmable superconducting transmon qubits of ibm washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The error suppression technique digital dynamical decoupling (DDD) is also tested on all QAOA circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' QA is executed on the same Ising instances with the programmable superconducting flux qubit devices D-Wave Advantage system4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 and Advantage system6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 using modified annealing schedules with pauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We find that QA outperforms QAOA on all problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We also find that DDD enables 2-round QAOA to outperform 1-round QAOA, which is not the case without DDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 1 Introduction Quantum annealing (QA) in the transverse field Ising model (TFIM) is an analog computation technology which utilizes quantum fluctuations in order to search for ground state solutions of a problem Hamiltonian [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' D-Wave quantum annealers are programmable hardware implementations of quantum annealing which use superconducting flux qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Quantum alternating operator ansatz (QAOA) is a hybrid quantum classical algorithm for sampling combina- torial optimization problems [5, 6], the quantum component of which can be instantiated with a programmable gate-based universal quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The quantum approximate optimization algorithm [7] was the first vari- ational algorithm of this type, which was then generalized to the quantum alternating operator ansatz algorithm [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' QAOA is effectively a Trotterization of the Quantum Adiabatic Algorithm, and is overall similar to Quantum Annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In particular both algorithms address combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The exact characteristics of how both QA and QAOA will scale to large system sizes is currently not fully understood, in particular because quantum hardware is still in the NISQ era [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For example, there is evidence that QAOA may be more difficult for classical computers to simulate than quantum annealing, which could make it a viable candidate for quantum advantage [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Therefore it is of interest to investigate differences between QAOA and QA, and determine how these algorithms will scale [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' There have been experimental QAOA implementations which used up to 27 qubits [18] and 23 qubits [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' There have also been QAOA experiments which had circuit depth up to 159 [20] and 148 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The contributions of this article are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We provide a direct comparison between QAOA and Quantum Annealing in terms of experiments on D-Wave and IBMQ hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' This comparison uses a comparable parameter search space for QA and QAOA, uses no minor embedding for quantum annealing, and uses short depth QAOA circuits, thus providing a fair comparison of the two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We show that QAOA is better than random sampling, and quantum annealing clearly outperforms QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' ∗Email: epelofske@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='gov †Email: baertschi@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='gov 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='00520v1 [quant-ph] 2 Jan 2023 Device name Topology/chip name Available qubits Available couplers/ CNOTs Computation type Advantage system4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 Pegasus P16 5627 40279 QA Advantage system6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 Pegasus P16 5616 40135 QA ibm washington Eagle r1 heavy-hexagonal 127 142 Universal gate-model Table 1: NISQ hardware summary at the time the experiments were executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The hardware yield (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=', the number of available qubits or two qubit interactions) for all of these devices can be less than the logical lattice because of hardware defects, and can also change over time if device calibration changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The QAOA algorithm we present is tailored for short depth circuit construction on the heavy hexagonal lattice, therefore allowing full usage of any heavy hexagonal topology quantum processor in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We use all 127 qubits of the ibm washington chip in order to execute the largest QAOA circuit, in terms of qubits, to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The problem instances that are used to compare quantum annealing and QAOA are specifically constructed to include higher order terms, specifically three variable (cubic) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' QAOA can directly implement higher order terms, and quantum annealing requires order reduction using auxiliary variables to implement these higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' This is the largest experimental demonstration of QAOA with higher order terms to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In order to mitigate errors when executing the QAOA circuits, we utilize digital dynamical decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' This is the largest usage of dynamical decoupling in terms of qubit system size to date, and the results show that digital dynamical decoupling improves performance for two round QAOA, suggesting that it will be useful for computations with large numbers of qubits in the noisy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In Section 2 the QAOA and QA hardware implementations are detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Section 3 details the experimental results and how the two algorithms compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Section 4 concludes with what the results indicate and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The figures in this article are generated using matplotlib [22, 23], and Qiskit [24] in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 2 Methods The problem instances are defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='2 the QAOA circuit algorithm and hardware parameters are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3 the quantum annealing implementation is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 Problem instances The NISQ computers which are used in this comparison are detailed in Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' the clear difference between the D-Wave quantum annealers and ibm washington is the number of qubits that are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The additional qubits available on the quantum annealers will allow us to embed multiple problem instances onto the chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The current IBMQ devices have a graph topology referred to as the heavy-hexagonal lattice [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Therefore, for a direct QAOA and QA comparison we would want to be able to create QAOA circuits which match the logical heavy-hexagonal lattice and the quantum annealer graph topology of Pegasus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For this direct comparison we target D-Wave quantum annealers with Pegasus graph hardware [26, 27] connectivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The two current D-Wave quantum annealers with Pegasus hardware graphs have chip id names Advantage system6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 and Advantage system4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The goal for this direct comparison is that ideally we want problems which can be instantiated on all three of the devices in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In particular, we want these implementations to not be unfairly costly in terms of implementation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For example we do not want to introduce unnecessary qubit swapping in the QAOA circuit because that would introduce larger circuit depths which would introduce more decoherence in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We also do not want to introduce unnecessary minor-embedding in the problems for quantum annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The other property of these problem instances that is of interest is an introduction of higher order terms, specifically cubic ZZZ interactions [28] also referred to as multi-body interactions [29], in addition to random linear and quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' These higher order terms require both QAOA and QA to be handle these higher order variable interactions, which is an additional test on the capability of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' QAOA can naturally handle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='Figure 1: Left: ibm washington graph connectivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' where qubits are connected by CNOT (also referred to as cx) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The ideal lattice is called the heavy-hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that there are two missing graph edges from the lattice between qubits 8-9 and 109-114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The total number of qubits (nodes) is 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The edges of the graph are three colored (red, blue, and green) such that no node shares two or more edges with the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The node colorings of light and dark gray show that the heavy hexagonal lattice is bipartite (meaning it can be partitioned into two disjoint sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The three edge coloring is consistent with the QAOA circuit construction in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Right: Example of a single random problem instance with cubic terms (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (1)) on the ibm washington graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The linear and quadratic terms are shown using two distinct colors (red and green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The nodes and edges colored red denote a weight of −1 and the nodes and edges colored green denote a weight of +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The cubic terms are represented by ovals around the three qubits which define the cubic variable interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Like the linear and quadratic terms, the color of the oval representing the cubic terms represents the sign of the weight on the terms, where green is +1 and red is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' higher order terms [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Implementing high order terms with QA requires introducing auxiliary variables in order to perform order reduction to get a problem structure that is comprised of only linear and quadratic terms, so that it can be implemented on the hardware, but whose optimal solutions match the optimal solutions of the original high order polynomial [3, 31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Taking each of these characteristics into account, we create a class of random problems which follow the native device connectivities in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The problem instances we will be considering are Ising problems defined on the hardware connectivity graph of the heavy hexagonal lattice of the device, which for these experiments will be ibm washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' C(x) = � v∈N cv · xv + � (i,j)∈E ci,j · xi · xj + � l∈D cl · xl · xn1(l) · xn2(l) (1) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (1) defines the class of problem Isings as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' N is the set of qubits, or variables, that exist on the heavy hexagonal layout topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' E is the edge set of all two qubit (CNOT) gates that can allow two qubits, indexed as i and j, to interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Any heavy hexagonal lattice is a bipartite graph with vertices V = V2 ∪ V3 where V2 consists of vertices with a maximum degree of 2, and V3 consists of vertices with a maximum degree of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' D is the set of vertices in V2 which all have degree exactly equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' n1 is a function which gives the qubit (variable) index of the first of the two neighbors of a degree-2 node, and n2 provides the qubit (variable) index of the second of the two neighbors of any degree-2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' cv, ci,j, and ct are all functions representing the random selection of the linear, quadratic, and cubic coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' These coefficients could be drawn from any distribution - in this case we draw the coefficients from {+1, −1} with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The decision variables are xi, where the possible variable states are the spins −1 or +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Combined, any vector of variable states x can be evaluated given this objective function formulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The heavy hexagonal topology of ibm washington, along with an overlay showing one of the random problem instances with cubic terms defined on ibm washington, is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Each term coefficient was chosen to 3 be either +1 or −1 in order to mitigate the potential problem of limited precision for the programming control on all of the NISQ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 10 random instances of this class of problems are generated and sampled using QAOA and QA, the implementations of each will be discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='2 QAOA Given a combinatorial optimization problem over inputs x ∈ {0, 1}n, let f(x): {0, 1}n → R be the objective function which evaluates the cost of solution x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For a maximization (or minimization) problem, the goal is to find a variable assignment vector x for which f(x) is maximized (or minimized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The QAOA algorithm consists of the following components: An initial state |ψ⟩ A phase separating Hamiltonian: HP |x⟩ = f(x) |x⟩ A mixing Hamiltonian: HM An integer p ≥ 1, the number of rounds to run the algorithm Two real vectors ⃗γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=', γp) and ⃗β = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=', βp), each with length p The algorithm consists of preparing the initial state |ψ⟩, then applying p rounds of the alternating simulation of the phase separating Hamiltonian and the mixing Hamiltonian: |⃗γ, ⃗β⟩ = e−iβpHM e−iγpHP � �� � round p · · e−iβ1HM e−iγ1HP � �� � round 1 |ψ⟩ (2) Within reach round, HP is applied first, which separates the basis states of the state vector by phases e−iγf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' HM then provides parameterized interference between solutions of different cost values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' After p rounds, the state |⃗γ, ⃗β⟩ is measured in the computational basis and returns a sample solution y of cost value f(y) with probability | ⟨y|⃗γ, ⃗β⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The aim of QAOA is to prepare the state |⃗γ, ⃗β⟩ from which we can sample a solution y with high cost value f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Therefore, in order to use QAOA the task is to find angles ⃗γ and ⃗β such that the expectation value ⟨⃗γ, ⃗β|HP |⃗γ, ⃗β⟩ is large (−HP for minimization problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In the limit p → ∞, QAOA is effectively a Trotterization of of the Quantum Adiabatic Algorithm, and in general as we increase p we expect to see a corresponding increase in the probability of sampling the optimal solution [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The challenge is the classical outer loop component of finding the good angles ⃗γ and ⃗β for all rounds p, which has a high computational cost as p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Variational quantum algorithms, such as QAOA, have been a subject of large amount of attention, in large part because of the problem domains that variational algorithms can address (such as combinatorial optimization) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' One of the challenges however with variational quantum algorithms is that the classical component of parameter selection, in the case of QAOA this is the angle finding problem, is not solved and is even more difficult when noise is present in the computation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Typically the optimal angles for QAOA are computed exactly for small problem instances [15, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' However, in this case the angle finding approach we will use is a reasonably high resolution gridsearch over the possible angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note however that a fine gridsearch scales exponentially with the number of QAOA rounds p, and therefore is not advisable for practical high round QAOA [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Exactly computing what the optimal angles are for problems of this size would be quite computationally intensive, especially with the introduction of higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' We leave the problem of exactly computing the optimal QAOA angles up to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Figure 2 describes the short depth QAOA circuit construction for sampling the higher order Ising test instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' This algorithm can be applied to any heavy hexagonal lattice topology, which allows for executing the QAOA circuits on the 127 variable instances on the IBMQ ibm washington backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For the class of Isings with higher order terms defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1, the QAOA angle ranges which are used are γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' , γp ∈ [0, π) and β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' , βp−1 ∈ [0, π), βp ∈ [0, π 2 ) where p is the number of QAOA rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that the halving of the angle search space for β applies when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For optimizing the angles using the naive grid search for p = 1, β0 is varied over 60 linearly spaced angles ∈ [0, π 2 ] and γ0 is varied over 120 linearly spaced angles ∈ [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For the high resolution gridsearch for p = 2, β1 is varied over 5 linearly spaced angles ∈ [0, π 2 ] and γ0, γ1, and β0 are varied over 11 linearly spaced angles ∈ [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Therefore, for p = 2 the angle gridsearch uses 6655 separate circuit executions (for each of the 10 problem instances), and for p = 1 the angle gridsearch uses 7200 separate circuit executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Each circuit execution used 10, 000 samples in order to compute a robust distribution for each angle combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 4 |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ H H H H H H H H H H H dB dC dD dE dF dG dH dI dJ dBA dDE dF C dJI dBC dF I dHG dJK dDC dHI dA dK dBAC dDCE dHGI dJIK dF CI γdBC γdDE γdF I γdJK γdHI γdDC γdBA γdF C γdHG γdA γdB γdC γdD γdE γdF γdG γdH γdI γdJ γdK Z Z γdJI γdBAC γdDCE γdF CI γdHGI γdJIK Z Z Z Z Row Column Time d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' = ±1 γd Z β X = Rz(2γd) = Rx(2β) Init Phase Separator Mixer β X β β β β β β β β β β Eval � �� � ⟨β,γ|HC|β,γ⟩ Figure 2: A 1-round QAOA circuit: (left) The problem instance is a hardware-native bipartite graph with an arbitrary 3-edge-coloring given by K˝onig’s line coloring theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (right) Any quadratic term (colored edge) gives rise to a combination of two CNOTs and a Rz-rotation in the phase separator, giving a CNOT depth of 6 due to the degree-3 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' When targeting the degree-2 nodes with the CNOT gates, these constructions can be nested, leading to no overhead when implementing the three-qubit terms: these always have a degree-2 node in the middle (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In order to mitigate decoherence on idle qubits, digital dynamical decoupling (DDD) is also tested for all QAOA circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Dynamical Decoupling is an open loop quantum control technique error suppression technique for mitigating decoherence on idle qubits [38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Dynamical decoupling can be implemented with pulse level quantum control, and digital dynamical decoupling can be implemented simply with circuit level instructions of sequences of gates which are identities [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that digital dynamical decoupling is an approximation of pulse level dynamical decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Dynamical decoupling has been experimentally demonstrated for superconducting qubit quantum processors including IBMQ devices [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Dynamical decoupling in particular is applicable for QAOA circuits because they can be relatively sparse and therefore have idle qubits [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' DDD does not always effective at consistently reducing errors during computation (for example because of other control errors present on the device [40, 42]), and therefore the raw QAOA circuits are compared against the QAOA circuits with DDD in the experiments section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' In order to apply the DDD sequences to the OpenQASM [45] QAOA circuits, the PadDynamicalDecoupling 1 method from Qiskit [24] is used, with the pulse alignment parameter set based on the ibm washington backend properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The native gateset of all current IBMQ backends is x, rz, cx, sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The circuit scheduling algorithm that is used for inserting the digital dynamical decoupling sequences is ALAP, which schedules the stop time of instructions as late as possible 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' There are other scheduling algorithms that could be applied which may increase the efficacy of dynamical decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that the rz gate is a virtual gate which is not executed on the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' There are different DDD gate sequences that can be applied, including Y-Y or X-X sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Because the X Pauli gate is already a native gate of the IBMQ device, the X-X DDD sequence is used for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that the variable states for the optimization problems are either −1 or +1, but the circuit measurement states are either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Therefore once the measurements are made on the QAOA circuits, for each variable in each sample the variable state mapping of 0 → 1, 1 → −1 is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' For circuit execution on the superconducting transom qubit ibm washington, circuits are batched into jobs where each job is composed of a group of at most 250 circuits - the maximum number of circuits for a job on ibm washington is currently 300, but we use 250 in order to reduce job errors related to the size of jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Grouping circuits into jobs is helpful for reducing the total amount of compute time required to prepare and measure each circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' When submitting the circuits to the backend, they are all first locally transpiled via Qiskit [24] with optimization level=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' This transpilation converts the gateset to the ibm washington native gateset, and the transpiler optimization attempts to simplify the circuit where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The QAOA circuit execution on ibm washington spanned a large amount of time, and therefore the backend versions were not consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The exact backend software versions were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='13, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 1https://qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='org/documentation/locale/bn_BN/stubs/qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='transpiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='PadDynamicalDecoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='html 2https://qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='org/documentation/apidoc/transpiler_passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='html 5 dA dB dC dBA dBC dBAC = +1 dA dB dC dBA dBC +1 −1 −1 −1 −2 1 1 1 2 2 2 dA dB dC dBA dBC +1 −3 −3 −1 +1 6 0 −1 −1 2 −4 −4 dA dB dC dBA dBC dBAC = −1 dA dB dC dBA dBC +1 −1 −1 −1 −2 −1 1 1 2 2 2 dA dB dC dBA dBC +3 −1 −1 +1 −1 2 0 1 −1 −2 −4 −4 Figure 3: (left) Two different embeddings for cubic +1/−1 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Each embedding needs two slack variable qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Our overall embedding alternates between these two cubic term embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Any embedding with only one slack variable needs a 4-clique between the slack and the three original variables, which is not possible to embed for consecutive cubic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (right) Embedding structures of the problem instances with higher order terms embedded in parallel (independently) 6 times onto the logical Pegasus P16 graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The view of this graph has been slightly partitioned so that not all of the outer parts of the Pegasus chip are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The light grey qubits and couplers indicate unused hardware regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The cyan coloring on nodes and edges denote the vertical qubits and CNOTs on the ibm washington hardware graph (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The red coloring on nodes and edges denote the horizontal lines of qubits and CNOTs on ibm washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The green nodes and edges denote the order reduction auxiliary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Note that the top right hand and lower left hand qubits are not present on the ibm washington lattice - but for the purposes of generating the embeddings, these extra qubits are filled in to complete the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='3 Quantum Annealing Quantum annealing is a proposed type of quantum computation which uses quantum fluctuations, such as quantum tunneling, in order to search for the ground state of a user programmed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Quantum annealing, in the case of the transverse field Ising model implemented on D-Wave hardware, is explicitly described by the system given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The state begins at time zero purely in the transverse Hamiltonian state � i σx i , and then over the course of the anneal (parameterized by the annealing time) the user programmed Ising is applied according the function B(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Together, A(s) and B(s) define the anneal schedules of the annealing process, and s is referred to as the anneal fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The standard anneal schedule that is used is a linear interpolation between s = 0 and s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' H = −A(s) 2 � � i σx i � + B(s) 2 � Hising � (3) The adiabatic theorem states that if changes to the Hamiltonian of the system are sufficiently slow, the system will remain in the ground state of problem Hamiltonian, thereby providing a computational mechanism for comput- ing the ground state of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The user programmed Ising Hising, acting on n qubits, is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The quadratic terms and the linear terms combined define the optimization problem instance that the annealing procedure will ideally find the ground state of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' As with QAOA, the objective of quantum annealing is 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='0 Pause duration fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='0 Anneal fraction [s] Figure 4: All modified (forward) quantum annealing schedules which are tested in order to find the best anneal schedule with a pause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The symmetric pause inserted into the normal linearly interpolated schedule defining the A(s) and B(s) functions can provide better ground state sampling probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The anneal fraction at which this pause occurs is varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='9 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' The pause duration, as a fraction of the total annealing time, is also varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='9 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Although not shown in this figure, the annealing times are also varied between 10, 100, 1000, and 2000 microseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' to find the variable assignment vector x that minimizes the cost function which has the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfpPgc/content/2301.00520v1.pdf'} +page_content=' Hising = n � i hiσz i + n � i