Progress toward favorable landscapes in quantum combinatorial optimization

Lee, Juneseo; Magann, Alicia B.; Rabitz, Herschel; Arenz, Christian

doi:10.1103/physreva.104.032401

Cited by 37 publications

(32 citation statements)

References 48 publications

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“…(displayed version) In this thesis we expand upon the results that led to the paper [1] of Lee et al, arXiv:2105.01114 (2021. In particular, we give more details on the oracular formulation of variational quantum algorithms, and the relationship between properties of Ansätze and the strength of their corresponding oracles.…”

mentioning

confidence: 87%

“…Nevertheless, it has also been shown that the choice of the two-qubit gate CNOT was not unique in the above Lemma (1), and in fact:…”

Section: Quantum Computing Basicsmentioning

confidence: 99%

“…This has a similar form as a generic noncommutative Ansatz, where H X is indeed an element of the simple X-Ansatz, and Hj = H p is a noncommutative element. In QAOA the initial state is given by |φ = |+ where |+ = |+ ⊗ • • • ⊗ |+ with |+ being the eigenstate of X corresponding to an eigenvalue of +1 introduced as the Bell state in Observation (1), so that for sufficiently many alternations between H p and H X the ground state(s) of H p is reachable. In contrast to the X-and XZ-Ansätze defined in the previous section, for which the ground state is reachable at a k-body depth of one, a sufficiently large circuit depth is needed in QAOA before the ground state is reachable.…”

Section: Comparison With Qaoa and Initial State Dependencementioning

confidence: 99%

See 2 more Smart Citations

On Assessing the Quantum Advantage for MaxCut Provided by Quantum Neural Network Ansätze

Lee¹

2021

Preprint

Self Cite

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In this work we design a class of Ansätze to solve MaxCut on a parameterized quantum circuit (PQC). Gaining inspiration from properties of quantum optimal control landscapes, we consider the presence of optimization traps as a measure of complexity for hybrid variational quantum algorithms. In particular, we analytically show that no simple Ansatz, satisfying certain criteria, Declaration I declare that I have not violated the Honor Code during the composition of this work. This paper represents my own work in accordance with University regulations.

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mentioning

confidence: 87%

“…Nevertheless, it has also been shown that the choice of the two-qubit gate CNOT was not unique in the above Lemma (1), and in fact:…”

Section: Quantum Computing Basicsmentioning

confidence: 99%

Section: Comparison With Qaoa and Initial State Dependencementioning

confidence: 99%

See 1 more Smart Citation

On Assessing the Quantum Advantage for MaxCut Provided by Quantum Neural Network Ansätze

Lee¹

2021

Preprint

Self Cite

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“…However, as quantum typicality makes the energy landscape of VQA Hamiltonians flattened [11,23], the circuit parameter optimization via local gradient search becomes more difficult with highly entangled circuits [4][5][6][7][8]. A known remedy for the flattened energy landscape is overparametrization of variational ansatz [9,10,12,24], developing multiple steep directions that lead to the robust success of the gradient descent method [11]. This comes with a classical computational cost for storing and manipulating variables.…”

Section: B Optimization and Expressibilitymentioning

confidence: 99%

“…Entanglement is a valuable resource for achieving quantum advantage, but it also becomes a hurdle for successful optimization of circuit control parameters at the same time, specifically when random circuit states are much more highly entangled than the ground state of the Hamiltonian encoding the task [4][5][6][7][8]. Quantum information in such highly entangled states is scrambled, and a successful adjustment of circuit parameters via local gradient search typically requires over-parametrization [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Quantum Chaos and Circuit Parameter Optimization

Kim¹,

Oz²,

Rosa³

2022

Preprint

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We explore quantum chaos diagnostics of variational circuit states at random parameters and study their correlation with the circuit expressibility and the optimization of control parameters. By measuring the operator spreading coefficient and the eigenvalue spectrum of the modular Hamiltonian of the reduced density matrix, we identify the universal structure of random matrix models in high-depth circuit states. We construct different layer unitaries corresponding to the GOE and GUE distributions and quantify their VQA performance. Our study also highlights a potential tension between the OTOC and BGS-type diagnostics of quantum chaos.

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Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis‐Hastings Warm‐Starting Algorithm

Liang

Wang

et al. 2023

Adv Quantum Tech

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Quantum approximate optimization algorithm (QAOA) is a promising variational quantum algorithm for combinatorial optimization problems. However, the implementation of QAOA is limited due to the requirement that the problems be mapped to Ising Hamiltonians and the nonconvex optimization landscapes. Although the Ising Hamiltonians for many NP hard problems have been obtained, a general method to obtain the Ising Hamiltonians for constrained combinatorial optimization problems (CCOPs) has not yet been investigated. In this paper, a general method is introduced to obtain the Ising Hamiltonians for CCOPs and the Metropolis‐Hastings warm‐starting algorithm for QAOA is presented which can provably converge to the global optimal solutions. The effectiveness of this method is demonstrated by tackling the minimum weight vertex cover (MWVC) problem, the minimum vertex cover (MVC) problem, and the maximal independent set problem as examples. The Ising Hamiltonian for the MWVC problem is obtained first time by using this method. The advantages of the Metropolis‐Hastings warm‐starting algorithm presented here is numerically analyzed through solving 30 randomly generated MVC cases with 1‐depth QAOA.

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Progress toward favorable landscapes in quantum combinatorial optimization

Cited by 37 publications

References 48 publications

On Assessing the Quantum Advantage for MaxCut Provided by Quantum Neural Network Ansätze

On Assessing the Quantum Advantage for MaxCut Provided by Quantum Neural Network Ansätze

Quantum Chaos and Circuit Parameter Optimization

Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis‐Hastings Warm‐Starting Algorithm

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