Circuit depth scaling for quantum approximate optimization

Akshay, V.; Philathong, H.; Campos, E.; Rabinovich, D. S.; Zacharov, I.; Zhang, Xiaoming; Biamonte, Jacob

doi:10.1103/physreva.106.042438

Cited by 10 publications

(4 citation statements)

References 28 publications

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“…Assessing scaling of the ground-state probability with size

will be an essential aspect of extending this approach to larger sizes

. This includes numerical simulations to quantify how the ground-state probabilities depend on the number of spins

and number of QAOA layers

; previous works have shown

maintains a large ground-state probability

0.7

for simple models in different contexts [ 37 , 38 ], but future work is needed to test scaling in the current model. Benchmarking on quantum computers is also essential to understand how real noise processes effect scalability.…”

Section: Discussionmentioning

confidence: 99%

“…We choose N = 9 spins as this is the logical minimum number of spins required to construct a unit cell of the Shastry-Sutherland lattice and also it is a feasible size for the Quantinuum quantum computer (with 12 qubits available at the time this work was completed). We focus on p = 1 layers of the QAOA algorithm; for larger N instances more layers p of QAOA will be needed to maintain a significant success probability [36][37][38]. Implementations on quantum computers will also have to overcome predicted limitations due to noise [39][40][41][42] including an exponential scaling in the number of measurements with circuit size, depth, and other factors [43].…”

Section: Introductionmentioning

confidence: 99%

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Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

Lotshaw

Khalid

et al. 2022

Phil. Trans. R. Soc. A.

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Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ( ≲ 100 ) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.

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“…Assessing scaling of the ground-state probability with size

will be an essential aspect of extending this approach to larger sizes

. This includes numerical simulations to quantify how the ground-state probabilities depend on the number of spins

and number of QAOA layers

; previous works have shown

maintains a large ground-state probability

0.7

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

Lotshaw

Khalid

et al. 2022

Phil. Trans. R. Soc. A.

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“…The Quantum Approximate Optimization Algorithm (QAOA) was first introduced by Farhi et al [1] as a quantum-classical hybrid algorithm, which consists of a quantum circuit with an outer classical optimization loop, to approximate the solution of combinatorial optimization problems. Since then, many studies have been conducted to discuss its quantum advantage and its implementability on near-term Noisy Intermediate Scale Quantum (NISQ) devices [2][3][4][5][6][7][8]. QAOA is shown to guarantee the approximation ratio α > 0.6924 for circuit depth p = 1 in the Max-cut problem on 3-regular graphs [1].…”

Section: Introductionmentioning

confidence: 99%

A Depth-Progressive Initialization Strategy for Quantum Approximate Optimization Algorithm

et al. 2023

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The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms.

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“…Thus, it is crucial to design parameterized circuits that are able to adequately prepare or approximate the quantum state of interest, while not being overly expressive so as not to compromise its trainability. To this end, a characterization of the expressibility of these algorithms has been performed in several contexts, and strategies to systematically quantify it have been proposed [1,2,23,68,84]. A way to constrain the expressibility of a parameterized circuit is to employ problem-tailored ansätze, such as the Hamiltonian Variational Ansatz [103], among other strategies such as exploiting symmetries in the problem [26,33,59,95,98] or removing redundant parameters [31,32].…”

Section: Introductionmentioning

confidence: 99%

Characterization of variational quantum algorithms using free fermions

Matos

Self

Papić

et al. 2023

Quantum

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We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras, we show that the Quantum Approximate Optimization Algorithm (QAOA) on a one-dimensional lattice – with and without decoupled angles – is able to prepare all fermionic Gaussian states respecting the symmetries of the circuit. Leveraging these results, we numerically study the interplay between these symmetries and the locality of the target state, and find that an absence of symmetries makes nonlocal states easier to prepare. An efficient classical simulation of Gaussian states, with system sizes up to 80 and deep circuits, is employed to study the behavior of the circuit when it is overparameterized. In this regime of optimization, we find that the number of iterations to converge to the solution scales linearly with system size. Moreover, we observe that the number of iterations to converge to the solution decreases exponentially with the depth of the circuit, until it saturates at a depth which is quadratic in system size. Finally, we conclude that the improvement in the optimization can be explained in terms of better local linear approximations provided by the gradients.

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Circuit depth scaling for quantum approximate optimization

Cited by 10 publications

References 28 publications

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

A Depth-Progressive Initialization Strategy for Quantum Approximate Optimization Algorithm

Characterization of variational quantum algorithms using free fermions

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