Improving the Performance of Deep Quantum Optimization Algorithms with Continuous Gate Sets

Lacroix, Nathan; Hellings, Christoph; Andersen, Christian Kraglund; Paolo, Agustín Di; Remm, Ants; Lazar, Stefania; Krinner, Sebastian; Norris, Graham J.; Gabureac, Mihai; Heinsoo, Johannes; Blais, Alexandre; Eichler, Christopher; Wallraff, Andreas

doi:10.1103/prxquantum.1.020304

Cited by 83 publications

(69 citation statements)

References 29 publications

(42 reference statements)

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“…Therefore, in the near term, quantum computers will most likely only run low-depth QAOA. Low-depth QAOA results are improved by robust control [41] and by mapping β and γ to parameters of the control pulses [42,43], a method available to cloud-based quantum computers [44] with pulse-level control [45,46].…”

Section: Introductionmentioning

confidence: 99%

Warm-starting quantum optimization

Egger

Mareček

Woerner

2021

Quantum

146

111

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There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

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Section: Introductionmentioning

confidence: 99%

Warm-starting quantum optimization

Egger

Mareček

Woerner

2021

Quantum

146

111

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“…Problems considered include MaxCut [15], [35], [36], Maximum-k-Cut [17], Maximum Independent Set [37], [38], Community Detection [5], [39], [40], Graph Vertex k-Coloring [41], Maximum k-Colorable Subgraph [42], Graph Partitioning [8], and many more [43]- [45]. The combination of hardness and sparsity make graph problems especially appealing as an early application of QAOA, as evidenced by the fact that a number of recent experimental demonstrations apply QAOA to graph problems [4], [46]. For problems defined explicitly on unweighted graphs, the group of automorphisms of the graph is a subgroup of the group of symmetries of the problem.…”

Section: Accelerating Qaoa Training By Using Fast Graph Automorphism Solversmentioning

confidence: 99%

Exploiting Symmetry Reduces the Cost of Training QAOA

Shaydulin

Wild

2021

IEEE Trans. Quantum Eng.

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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. 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 RQAOA. 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.

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“…10, we show the comparison in the performance of our minimal encoding scheme and the QAOA protocol for multiple randomly generated A matrices of size n c = 8. This is in contrast to problems artificially curated to match the topology of the quantum device commonly used in experimental implementations [21,26,36,37,49]. Similar to the simulations shown in Appendix C, we use a noise model that, in addition to the finite gate fidelity, also includes thermal relaxations and readout errors.…”

Section: Encoding)mentioning

confidence: 99%

“…We emphasize that the search protocol in both the QAOA and minimal encoding scheme have been performed in presence of the simulated noise. This is also in contrast to some recent experimental QAOA demonstrations where the optimization is first performed with an ideal simulation and only the optimized circuit is executed on the quantum hardware [21,26,49]. For the minimal encoding, we used 15 starting points of randomly chosen parameters.…”

Section: Encoding)mentioning

confidence: 99%

Qubit-efficient encoding schemes for binary optimisation problems

Tan

Lemonde

Thanasilp

et al. 2021

Quantum

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We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of nc classical variables can be implemented on O(log⁡nc) number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42-variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.

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Improving the Performance of Deep Quantum Optimization Algorithms with Continuous Gate Sets

Cited by 83 publications

References 29 publications

Warm-starting quantum optimization

Warm-starting quantum optimization

Exploiting Symmetry Reduces the Cost of Training QAOA

Qubit-efficient encoding schemes for binary optimisation problems

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