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arXiv:1803.10731 · Physical Review A

Quantum approximate optimization with Gaussian boson sampling

Juan Miguel Arrazola, Thomas R. Bromley, Patrick Rebentrost

Published Jul 19, 2018• 57 citations
arXivHTMLPDF

Abstract

Hard optimization problems are often approached by finding approximate solutions. Here, we highlight the concept of proportional sampling and discuss how it can be used to improve the performance of stochastic algorithms for optimization. We introduce an NP-Hard problem called Max-Haf and show that Gaussian boson sampling (GBS) can be used to enhance any stochastic algorithm for this problem. These results are applied by enhancing the random search, simulated annealing, and greedy algorithms. With numerical simulations, we confirm that all algorithms are improved when employing GBS, and that a GBS-enhanced random search performs the best despite being the one with the simplest underlying classical routine.

DOI
10.1103/physreva.98.012322

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Hard optimization problems are often approached by finding approximate solutions. Here, we highlight the concept of proportional sampling and discuss how it can be used to improve the performance of stochastic algorithms for optimization. We introduce an NP-Hard problem called Max-Haf and show that Gaussian boson sampling (GBS) can be used to enhance any stochastic algorithm for this problem. These results are applied by enhancing the random search, simulated annealing, and greedy algorithms. With numerical simulations, we confirm that all algorithms are improved when employing GBS, and that a GBS-enhanced random search performs the best despite being the one with the simplest underlying classical routine.
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Boson sampling devices are a prime candidate for exhibiting quantum supremacy, yet their application for solving problems of practical interest is less well understood. Here we show that Gaussian boson sampling (GBS) can be used for dense subgraph identification. Focusing on the NP-hard densest k-subgraph problem, we find that stochastic algorithms are enhanced through GBS, which selects dense subgraphs with high probability. These findings rely on a link between graph density and the number of perfect matchings-enumerated by the Hafnian-which is the relevant quantity determining sampling probabilities in GBS. We test our findings by constructing GBS-enhanced versions of the random search and simulated annealing algorithms and apply them through numerical simulations of GBS to identify the densest subgraph of a 30 vertex graph.
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Gaussian Boson Sampling (GBS), which can be realized with a photonic quantum computing model, perform some special kind of sampling tasks. In [4], we introduced algorithms that use GBS samples to approximate Gaussian expectation problems. We found a non-empty open subset of the problem space where these algorithms achieve exponential speedup over the standard Monte Carlo (MC) method. This speedup is defined in terms of the guaranteed sample size to reach the same accuracy εεε and success probability δδδ under the (ε,δ)(ε, δ)(ε,δ) multiplicative error approximation scheme. In this paper, we enhance our original approach by optimizing the average photon number in the GBS distribution to match the specific Gaussian expectation problem. We provide updated estimates of the guaranteed sample size for these improved algorithms and quantify the proportion of problem space where they outperform MC. Numerical results indicate that the proportion of the problem space where our improved algorithms have an advantage is substantial, and the advantage gained is significant. Notably, for certain special cases, our methods consistently outperform MC across nearly 100\% of the problem space.
We study the application of Gaussian Boson Sampling (GBS) to the densest k-subgraph problem (DkSP). GBS with hard post-selection suffers from poor sampling efficiency due to strict cardinality constraints. To address this limitation, we introduce effective classical post-processing strategies that transform, otherwise discarded, near-k samples into feasible solutions. A comprehensive set of simulations is carried out, demonstrating that these approaches achieve near-optimal solution quality while improving sampling efficiency by approximately 4X compared to post-selection on community-structured graphs, and also post-selection often fails to reach the optimal solution on sparse random graphs even with large number of samples. Furthermore, the proposed methods perform on par with, and in some cases outperform, established classical approaches for graphs up to moderate size. Overall, the results indicate that while GBS with post-selection alone is insufficient, its combination with lightweight classical refinement can be highly effective. This underscores the potential of hybrid quantum-classical frameworks and positions GBS as a promising sampling primitive for combinatorial graph optimization.
Gaussian boson sampling (GBS) is not only a feasible protocol for demonstrating quantum computational advantage, but also mathematically associated with certain graph-related and quantum chemistry problems. In particular, it is proposed that the generated samples from the GBS could be harnessed to enhance the classical stochastic algorithms in searching some graph features. Here, we use Jiǔzhāng, a noisy intermediate-scale quantum computer, to solve graph problems. The samples are generated from a 144-mode fully connected photonic processor, with photon click up to 80 in the quantum computational advantage regime. We investigate the open question of whether the GBS enhancement over the classical stochastic algorithms persists-and how it scales-with an increasing system size on noisy quantum devices in the computationally interesting regime. We experimentally observe the presence of GBS enhancement with a large photon-click number and a robustness of the enhancement under certain noise. Our work is a step toward testing real-world problems using the existing noisy intermediate-scale quantum computers and hopes to stimulate the development of more efficient classical and quantum-inspired algorithms.