Quantum Approximate Optimization for Hard Problems in Linear Algebra

Borle, Ajinkya; Elfving, Vincent E.; Lomonaco, Samuel J.

doi:10.48550/arxiv.2006.15438

Cited by 4 publications

(4 citation statements)

References 48 publications

(101 reference statements)

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“…QAOA is a hybrid quantum-classical algorithm for solving a special class of problems such as combinatorial optimization. An approximate solution for such problems would be acceptable as they have been proven to be NP-hard [5,6].…”

Section: Introductionmentioning

confidence: 99%

Automatic depth optimization for a quantum approximate optimization algorithm

2022

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Noise on near-term quantum devices will inevitably limit the performance of Quantum Approximate Optimization Algorithm (QAOA). One significant consequence is that the performance of QAOA may fail to monotonically improve with depth. In particular, optimal depth can be found at a certain point where the noise effects just outweigh the benefits brought by increasing the depth. In this work, we propose to use the model selection algorithm to identify the optimal depth with a few iterations of regularization parameters. Numerical experiments show that the algorithm can efficiently locate the optimal depth under relaxation and dephasing noises.

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Section: Introductionmentioning

confidence: 99%

Automatic depth optimization for a quantum approximate optimization algorithm

2022

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“…Among several quantum algorithms implemented on noisy intermediate-scale quantum (NISQ) devices [1][2][3][4][5][6][7][8][9][10][11][12], the quantum approximate optimization algorithm (QAOA) offers an opportunity to approximately solve combinatorial optimization problems such as MaxCut, Max Independent Set, and Max k-cover [13][14][15][16][17][18][19][20][21][22]. QAOA tunes a set of classical parameters to optimize the cost function expectation value for a quantum state prepared by well-defined sequence of operators acting on a known initial state.…”

Section: Introductionmentioning

confidence: 99%

Multi-angle Quantum Approximate Optimization Algorithm

Herrman¹,

Lotshaw²,

Ostrowski³

et al. 2021

Preprint

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The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we introduce a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.

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“…Various strategies of error mitigation were proposed that can further improve the performance of algorithms when run on physical devices [125][126][127][128][129][130][131][132]. Finally, considering linear algebra problems, several VQAs were also proposed to solve LSEs [133][134][135][136][137][138], and ongoing efforts are directed towards improving their workflow. To date, several small scale linear equation solvers have been implemented using a nuclear magnetic resonance-based setup [139,140].…”

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confidence: 99%

Solving nonlinear differential equations with differentiable quantum circuits

Kyriienko,

Paine,

Elfving

2020

Preprint

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We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

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Quantum Approximate Optimization for Hard Problems in Linear Algebra

Cited by 4 publications

References 48 publications

Automatic depth optimization for a quantum approximate optimization algorithm

Automatic depth optimization for a quantum approximate optimization algorithm

Multi-angle Quantum Approximate Optimization Algorithm

Solving nonlinear differential equations with differentiable quantum circuits

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