Fast digital methods for adiabatic state preparation

Wan, Kianna; Kim, Isaac

doi:10.48550/arxiv.2004.04164

Cited by 11 publications

(22 citation statements)

References 29 publications

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“…One reason we might try doing this is because the qubitized quantum walks that we discuss in Section II are sometimes cheaper to implement than Trotter steps for some problems. One approach to using LCU methods for adiabatic state preparation might be to directly attempt to simulate the time-dependent Hamiltonian evolution using a Dyson series approach, as was recently suggested for the purpose of adiabatic state preparation in [56]. However, this would require fairly complicated circuits due to the many time registers that one must keep to index the time-ordered exponential operator.…”

Section: Heuristic Adiabatic Optimization Using Quantum Walksmentioning

confidence: 99%

Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

Sanders,

Berry,

Costa

et al. 2020

Preprint

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Here we explore which heuristic quantum algorithms for combinatorial optimization might be most practical to try out on a small fault-tolerant quantum computer. We compile circuits for several variants of quantum accelerated simulated annealing including those using qubitization or Szegedy walks to quantize classical Markov chains and those simulating spectral gap amplified Hamiltonians encoding a Gibbs state. We also optimize fault-tolerant realizations of the adiabatic algorithm, quantum enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. Many of these methods are bottlenecked by calls to the same subroutines; thus, optimized circuits for those primitives should be of interest regardless of which heuristic is most effective in practice. We compile these bottlenecks for several families of optimization problems and report for how long and for what size systems one can perform these heuristics in the surface code given a range of resource budgets. Our results discourage the notion that any quantum optimization heuristic realizing only a quadratic speedup will achieve an advantage over classical algorithms on modest superconducting qubit surface code processors without significant improvements in the implementation of the surface code. For instance, under quantum-favorable assumptions (e.g., that the quantum algorithm requires exactly quadratically fewer steps), our analysis suggests that quantum accelerated simulated annealing would require roughly a day and a million physical qubits to optimize spin glasses that could be solved by classical simulated annealing in about four CPU-minutes.

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Section: Heuristic Adiabatic Optimization Using Quantum Walksmentioning

confidence: 99%

Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

Sanders,

Berry,

Costa

et al. 2020

Preprint

View full text Add to dashboard Cite

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“…DAS can be used to implement adiabatic quantum computation [12][13][14] and adiabatic state preparation [15,16] on a digital quantum computer. The scaling of Trotter error in DAS has previously been analyzed in terms of the operator norm [17,18].…”

Section: Introductionmentioning

confidence: 99%

Spectral Analysis of Product Formulas for Quantum Simulation

Yi¹,

Crosson²

2021

Preprint

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We consider Hamiltonian simulation using the first order Lie-Trotter product formula under the assumption that the initial state has a high overlap with an energy eigenstate, or a collection of eigenstates in a narrow energy band. This assumption is motivated by quantum phase estimation (QPE) and digital adiabatic simulation (DAS). Treating the effective Hamiltonian that generates the Trotterized time evolution using rigorous perturbative methods, we show that the Trotter step size needed to estimate an energy eigenvalue within precision using QPE can be improved in scaling from to 1/2 for a large class of systems (including any Hamiltonian which can be decomposed as a sum of local terms or commuting layers that each have real-valued matrix elements). For DAS we improve the asymptotic scaling of the Trotter error with the total number of gates M from O(M −1 ) to O(M −2 ), and for any fixed circuit depth we calculate an approximately optimal step size that balances the error contributions from Trotterization and the adiabatic approximation. These results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap, thereby contributing to the explanation of the observed similarities between the quantum approximate optimization algorithm and diabatic quantum annealing at small system sizes. Our analysis depends on the perturbation of eigenvectors as well as eigenvalues, and on quantifying the error using state fidelity (instead of the matrix norm of the difference of unitaries which is sensitive to an overall global phase).

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“…In 1996, Lloyd proposed the first explicit quantum algorithm for simulating local Hamiltonians [45]. Since then, various quantum simulation algorithms have been developed [9-12, 17, 23, 30, 47-50, 56], with potential applications in studying condensed matter physics [22], quantum chemistry [19,51], quantum field theories [36,37], superstring/M-theory [28], as well as in designing other classical [70] and quantum algorithms [2,8,14,21,29,33,42,74].…”

Section: Introductionmentioning

confidence: 99%

Nearly tight Trotterization of interacting electrons

Su,

Huang,

Campbell

2020

Preprint

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We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use O n 5/3 η 2/3 + n 4/3 η 2/3 gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when n = O η 2 . We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

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Fast digital methods for adiabatic state preparation

Cited by 11 publications

References 29 publications

Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

Spectral Analysis of Product Formulas for Quantum Simulation

Nearly tight Trotterization of interacting electrons

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