Quantum eigenvalue estimation via time series analysis

Somma, Rolando D.

doi:10.1088/1367-2630/ab5c60

Cited by 83 publications

(79 citation statements)

References 35 publications

(65 reference statements)

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“…We primarily foresee VFF being used to study the long-time evolution of the observables of a system. But one may also use VFF to reduce the gate complexity of eigenvalue estimation algorithms, such as quantum phase estimation 46 or time series analyses 47,48 . Such algorithms require simulating a Hamiltonian up to time T ¼ O 1 σ À Á to obtain eigenvalue estimates of accuracy σ.…”

Section: Implementationsmentioning

confidence: 99%

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Variational fast forwarding for quantum simulation beyond the coherence time

et al. 2020

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Trotterization-based, iterative approaches to quantum simulation (QS) are restricted to simulation times less than the coherence time of the quantum computer (QC), which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called variational fast forwarding (VFF), for decreasing the quantum circuit depth of QSs. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function’s operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. In addition, we implement VFF on Rigetti’s QC to demonstrate simulation beyond the coherence time. Finally, we show how to estimate energy eigenvalues using VFF.

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Section: Implementationsmentioning

confidence: 99%

“…However, these can be extracted using the time series analysis in ref. 48 . This method does not require large ancillary systems nor large numbers of controlled-unitary operations, and thus is a promising avenue for eigenvalue estimation in the NISQ era.…”

Section: Implementationsmentioning

confidence: 99%

Variational fast forwarding for quantum simulation beyond the coherence time

et al. 2020

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“…(2) and χ(t) = Tr[ρe −iHt ] is the expected value of the time evolution operator. The function P (ω) can be approximated by measuring the correlators χ(t) at different times t and calculating the discrete Fourier transform of these values [55,56]. Such correlators can be measured via an many-body-interferometric experiment akin to, for example, Ref.…”

Section: Regimes Of Resolution and Error Achievable By Quantum Devicesmentioning

confidence: 99%

Quantum advantage from energy measurements of many-body quantum systems

Novo

Bermejo-Vega

García−Patrón

2021

Quantum

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The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state – a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be exponentially fast-forwarded, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.

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“…53 Parrish and McMahon, 45 investigated a quantum filter diagonalization (QFD) formalism in which a basis of states is generated via an approximate real-time dynamics. QFD is inspired by classical filter diagonalization [54][55][56][57] as well as quantum time grid methods, [58][59][60][61] and may be viewed as a variant of the QLanczos algorithm in which imaginary time propagation is replaced with a real-time version.…”

Section: Introductionmentioning

confidence: 99%

A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons

Stair

Huang

Evangelista

2020

J. Chem. Theory Comput.

131

123

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We introduce a multireference selected quantum Krylov (MRSQK) algorithm suitable for quantum simulation of many-body problems. MRSQK is a low-cost alternative to the quantum phase estimation algorithm that generates a target state as a linear combination of non-orthogonal Krylov basis states. This basis is constructed from a set of reference states via real-time evolution avoiding the numerical optimization of parameters. An efficient algorithm for the evaluation of the off-diagonal matrix elements of the overlap and Hamiltonian matrices is discussed and a selection procedure is introduced to identify a basis of orthogonal references that ameliorates the linear dependency problem. Preliminary benchmarks on linear H 6 , H 8 , and BeH 2 indicate that MRSQK can predict the energy of these systems accurately using very compact Krylov bases.

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Quantum eigenvalue estimation via time series analysis

Cited by 83 publications

References 35 publications

Variational fast forwarding for quantum simulation beyond the coherence time

Variational fast forwarding for quantum simulation beyond the coherence time

Quantum advantage from energy measurements of many-body quantum systems

A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons

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