Exploiting Symmetry Reduces the Cost of Training QAOA
A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy...
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| Vydané v: | IEEE transactions on quantum engineering Ročník 2; s. 1 - 9 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
IEEE
2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2689-1808, 2689-1808 |
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| Abstract | A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement for <inline-formula><tex-math notation="LaTeX">p=1</tex-math></inline-formula> on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers. |
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| AbstractList | A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement for <tex-math notation="LaTeX">$p=1$</tex-math> on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers. A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement for <inline-formula><tex-math notation="LaTeX">p=1</tex-math></inline-formula> on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers. A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement for [Formula Omitted] on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers. A promising approach to the practical application of the quantum approximate optimization algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. In this article, we propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization recursive QAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10 000 nodes. Our approach provides an improvement for p=1 on 71.7% of the graphs considered, with a median speedup of 4.06, on a benchmark, where 62.5% of the graphs are known to be hard for automorphism solvers. |
| Author | Shaydulin, Ruslan Wild, Stefan M. |
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| BackLink | https://www.osti.gov/servlets/purl/1817859$$D View this record in Osti.gov |
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| References_xml | – ident: ref45 doi: 10.1109/MC.2019.2908942 – volume: 325 start-page: 2338 year: 0 ident: ref41 article-title: Planning for compilation of a quantum algorithm for graph coloring publication-title: Proc 24th Eur Conf Artif Intell Ser Front Artif Intell Appl – year: 2019 ident: ref14 article-title: What do QAOA energies reveal about graphs – ident: ref36 doi: 10.1103/PhysRevX.10.021067 – year: 2015 ident: ref32 article-title: Graph isomorphism in quasipolynomial time – year: 2020 ident: ref18 – year: 2019 ident: ref9 article-title: Learning to learn with quantum neural networks via classical neural networks – ident: ref8 doi: 10.1145/3425607 – ident: ref30 doi: 10.1137/1.9781611972870.13 – ident: ref4 doi: 10.1038/s41567-020-01105-y – ident: ref1 doi: 10.1038/s41586-019-1666-5 – year: 2008 ident: ref28 article-title: Search space contraction in canonical labeling of graphs – year: 2020 ident: ref47 article-title: Nauty and Traces user's guide (version 2.7) – year: 2020 ident: ref11 article-title: Tensor network quantum simulator with step-dependent parallelization – year: 2014 ident: ref3 article-title: A quantum approximate optimization algorithm – ident: ref46 doi: 10.1103/PRXQuantum.1.020304 – ident: ref19 doi: 10.1016/j.jsc.2013.09.003 – year: 2020 ident: ref17 article-title: Hybrid quantum-classical algorithms for approximate graph coloring – ident: ref42 doi: 10.1103/PhysRevA.101.012320 – ident: ref40 doi: 10.1002/qute.201900029 – ident: ref2 doi: 10.1088/2058-9565/abe519 – ident: ref13 doi: 10.1088/2058-9565/ab8c2b – ident: ref31 doi: 10.1155/2014/934637 – volume: 30 start-page: 45 year: 1981 ident: ref27 article-title: Practical graph isomorphism publication-title: Congressus Numerantium – year: 0 ident: ref48 article-title: [Online]. Available – year: 2018 ident: ref44 article-title: Quantum algorithms for scientific computing and approximate optimization – ident: ref24 doi: 10.1002/rsa.10054 – ident: ref49 doi: 10.1007/s10107-009-0273-x – year: 2019 ident: ref12 article-title: Alibaba cloud quantum development platform: Applications to quantum algorithm design – ident: ref10 doi: 10.1609/aaai.v34i03.5616 – ident: ref25 doi: 10.1007/BF01895716 – ident: ref26 doi: 10.1007/BFb0062536 – start-page: 647 year: 1958 ident: ref50 article-title: Symmetry in integer linear programming publication-title: 50 Years of Integer Programming 19582008 – ident: ref5 doi: 10.1109/HPEC.2019.8916288 – year: 2020 ident: ref37 article-title: The quantum approximate optimization algorithm needs to see the whole graph: A typical case – ident: ref6 doi: 10.1109/IGSC48788.2019.8957201 – year: 2019 ident: ref16 article-title: Obstacles to state preparation and variational optimization from symmetry protection – ident: ref7 doi: 10.1088/2058-9565/ab7559 – ident: ref33 doi: 10.1145/2897518.2897542 – ident: ref34 doi: 10.1038/ncomms5213 – year: 2020 ident: ref20 article-title: Classical symmetries and QAOA – year: 0 ident: ref39 article-title: Community detection across emerging quantum architectures publication-title: Proc 3rd Int Workshop Post Moore's Era Supercomputing – year: 2020 ident: ref38 article-title: The quantum approximate optimization algorithm needs to see the whole graph: Worst case examples – ident: ref29 doi: 10.1145/996566.996712 – year: 2018 ident: ref22 article-title: For fixed control parameters the quantum approximate optimization algorithm's objective function value concentrates for typical instances – year: 2018 ident: ref21 article-title: On the representation of Boolean and real functions as Hamiltonians for quantum computing – ident: ref23 doi: 10.1016/0022-0000(91)90023-X – ident: ref15 doi: 10.1103/PhysRevLett.125.260505 – ident: ref43 doi: 10.3390/a12020034 – year: 2018 ident: ref35 article-title: Performance of the quantum approximate optimization algorithm on the maximum cut problem |
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| SubjectTerms | Algorithms Approximation algorithms Automorphisms Benchmarks Graph theory Graphs Hardware Linear programming Optimization Parameters quantum approximate optimization algorithm quantum approximate optimization algorithm (QAOA) Quantum computers quantum computing quantum optimization Qubit Simulation Solvers Subgroups Symmetry Tensors Training |
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| Title | Exploiting Symmetry Reduces the Cost of Training QAOA |
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