Classically-inspired Mixers for QAOA Beat Goemans-Williamson's Max-Cut at Low Circuit Depths

Tate, Reuben; Moondra, Jai; Gard, Bryan T.; Mohler, Greg; Gupta, Ruta

doi:10.48550/arxiv.2112.11354

“…[93] which also found that faster error correcting codes are needed to make heuristics for combinatorial optimization competitive. This also suggests that warm-starting methods [94][95][96] will be required to get a quantum advantage in combinatorial optimization with heuristic algorithms.…”

Section: Discussionmentioning

confidence: 99%

Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Weidenfeller¹,

Valor²,

Gacon³

et al. 2022

Preprint

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Quantum computers may provide good solutions to combinatorial optimization problems by leveraging the Quantum Approximate Optimization Algorithm (QAOA). The QAOA is often presented as an algorithm for noisy hardware. However, hardware constraints limit its applicability to problem instances that closely match the connectivity of the qubits. Furthermore, the QAOA must outpace classical solvers. Here, we investigate swap strategies to map dense problems into linear, grid and heavy-hex coupling maps. A line-based swap strategy works best for linear and two-dimensional grid coupling maps. Heavy-hex coupling maps require an adaptation of the line swap strategy. By contrast, three-dimensional grid coupling maps benefit from a different swap strategy. Using known entropic arguments we find that the required gate fidelity for dense problems lies deep below the fault-tolerant threshold. We also provide a methodology to reason about the execution-time of QAOA. Finally, we present a QAOA Qiskit Runtime program and execute the closed-loop optimization on cloud-based quantum computers with transpiler settings optimized for QAOA. This work highlights some obstacles to improve to make QAOA competitive, such as gate fidelity, gate speed, and the large number of shots needed. The Qiskit Runtime program gives us a tool to investigate such issues at scale on noisy superconducting qubit hardware.

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“…[97] which also found that faster error correcting codes are needed to make heuristics for combinatorial optimization competitive. This also suggests that algorithmic improvements to QAOA [95,98] such as warm-start methods [99][100][101] will be required to get a quantum advantage in combinatorial optimization with heuristic algorithms.…”

Section: Discussionmentioning

confidence: 99%

Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Weidenfeller

¹

,

Valor

²

,

Gacon

³

et al. 2022

Quantum

View full text Add to dashboard Cite

Quantum computers may provide good solutions to combinatorial optimization problems by leveraging the Quantum Approximate Optimization Algorithm (QAOA). The QAOA is often presented as an algorithm for noisy hardware. However, hardware constraints limit its applicability to problem instances that closely match the connectivity of the qubits. Furthermore, the QAOA must outpace classical solvers. Here, we investigate swap strategies to map dense problems into linear, grid and heavy-hex coupling maps. A line-based swap strategy works best for linear and two-dimensional grid coupling maps. Heavy-hex coupling maps require an adaptation of the line swap strategy. By contrast, three-dimensional grid coupling maps benefit from a different swap strategy. Using known entropic arguments we find that the required gate fidelity for dense problems lies deep below the fault-tolerant threshold. We also provide a methodology to reason about the execution-time of QAOA. Finally, we present a QAOA Qiskit Runtime program and execute the closed-loop optimization on cloud-based quantum computers with transpiler settings optimized for QAOA. This work highlights some obstacles to improve to make QAOA competitive, such as gate fidelity, gate speed, and the large number of shots needed. The Qiskit Runtime program gives us a tool to investigate such issues at scale on noisy superconducting qubit hardware.

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“…A handful of other approaches have been studied in Refs. [26,[28][29][30]. We hope that these works will stimulate more effort in combining the strengths of the two computing paradigms to advance the state of the art for combinatorial optimization.…”

Section: Discussionmentioning

confidence: 99%

“…Surprisingly, this has been rarely done in previous works (except that classical techniques for continuous optimization have been used to tune the parameters in the ansatz circuits). A few exceptions are the recent works on warm-starting quantum optimization [26,[28][29][30]. In particular, Egger et al [26] proposed the warmstarted QAOA (WS-QAOA) in which the initial state and mixing operator are constructed based on the continuous solution of the Quadratic Programming (QP) or Semidefinite Programming (SDP) relaxation of the original problem, and showed that this algorithm outperforms standard QAOA at low depths.…”

Section: Introductionmentioning

confidence: 99%

Classically-Boosted Quantum Optimization Algorithm

Wang¹

2022

Preprint

0

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Considerable effort has been made recently in the development of heuristic quantum algorithms for solving combinatorial optimization problems. Meanwhile, these problems have been studied extensively in classical computing for decades. In this paper, we explore a natural approach to leveraging existing classical techniques to enhance quantum optimization. Specifically, we run a classical algorithm to find an approximate solution and then use a quantum circuit to search its "neighborhood" for higher-quality solutions. We propose the Classically-Boosted Quantum Optimization Algorithm (CBQOA) that is based on this idea and can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on. A crucial component of this algorithm is an efficiently-implementable continuous-time quantum walk (CTQW) on a properly-constructed graph that connects the feasible solutions. CBQOA utilizes this CTQW and the output of an efficient classical procedure to create a suitable superposition of the feasible solutions which is then processed in certain way. This algorithm has the merits that it solves constrained problems without modifying their cost functions, confines the evolution of the quantum state to the feasible subspace, and does not rely on efficient indexing of the feasible solutions. We demonstrate the applications of CBQOA to Max 3SAT and Max Bisection, and provide empirical evidence that it outperforms previous approaches on these problems.

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Classically-inspired Mixers for QAOA Beat Goemans-Williamson's Max-Cut at Low Circuit Depths

Cited by 6 publications

References 20 publications

Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware

Classically-Boosted Quantum Optimization Algorithm

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