Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution

Motta, Mário; Sun, Chong; Tan, Adrian T. K.; O’Rourke, Matthew J.; Ye, Erika; Minnich, Austin J.; Brandão, Fernando G. S. L.; Chan, Garnet Kin‐Lic

doi:10.1038/s41567-019-0704-4

Cited by 556 publications

(632 citation statements)

References 46 publications

(48 reference statements)

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“…To implement Eq. (6) on the NISQ devices, Chan et al [26] introduces a unitary operator e −iAs(θ)/n to approximate it, where the operator A s (θ) (acting on k qubits) can be extended in the Pauli basis with relevant parameters a s…”

Section: Resultsmentioning

confidence: 99%

“…Taking parameters a s (θ) i1...i k as variables, finding out A s (θ) can be recognized as an optimization procedure, and parameters a s (θ) i1...i k can be efficiently determined by solving the linear equation (S + S † )a s (θ) = −b, where the matrix entries S i1. Therefore, the complexity of implementing the 2L-QRBM by using QITE algorithm is quasi-polynomial in n (the number of Trotter steps) [26]. Experimental results.…”

Section: Resultsmentioning

confidence: 99%

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Quantum restricted Boltzmann machine is universal for quantum computation

Wu¹,

Wei²,

Qin³

et al. 2020

Preprint

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The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the nontrivial correlations encoded in the many-body wave functions with high complexity. Quantum neural network provides a powerful tool to represent the large-scale wave function, which has aroused widespread concerns in the quantum superiority era. A significant open problem is what exactly the representational power boundary of the single-layer quantum neural network is. In this paper, we design a 2-local Hamiltonian and then give a kind of Quantum Restricted Boltzmann Machine (QRBM, i.e. single-layer quantum neural network) based on it. The proposed QRBM has the following two salient features. (1) It is proved universal for implementing quantum computation tasks. (2) It can be efficiently implemented on the Noisy Intermediate-Scale Quantum (NISQ) devices. We successfully utilize the proposed QRBM to compute the wave functions for the notable cases of physical interest including the ground state as well as the Gibbs state (thermal state) of molecules on the superconducting quantum chip. The experimental results illustrate the proposed QRBM can compute the above wave functions with an acceptable error.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Quantum restricted Boltzmann machine is universal for quantum computation

Wu¹,

Wei²,

Qin³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…One computational issue of the scheme is the resolution of the high-dimensional minimisation problem (4.1). Other imaginary-time evolution algorithms [13] bypassing this optimisation might provide an efficient alternative for large-scale applications. We leave these two issues for further investigations.…”

Section: Ansatz and Initial Condition Algorithm Input ← Payoff Functimentioning

confidence: 99%

A Quantum Algorithm for Linear PDEs Arising in Finance

2019

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We propose a hybrid quantum-classical algorithm, originated from quantum chemistry, to price European and Asian options in the Black-Scholes model. Our approach is based on the equivalence between the pricing partial differential equation and the Schrödinger equation in imaginary time. We devise a strategy to build a shallow quantum circuit approximation to this equation, only requiring few qubits. This constitutes a promising candidate for the application of Quantum Computing techniques (with large number of qubits affected by noise) in Quantitative Finance.Date: December 6, 2019. 2010 Mathematics Subject Classification. 35Q40, 91G20, 91G80. Key words and phrases. quantum algorithms, option pricing, PDE, Schrödinger equation. The views and opinions expressed here are the authors' and do not represent the opinions of their employers. They are not responsible for any use that may be made of these contents. No part of this presentation is intended to influence investment decisions or promote any product or service. The authors would like to thank Alexei Kondratyev for stimulating discussions.

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“…An approach was recently proposed in Ref. [14] by implementing an effective unitary evolution e −ih eff ∆τ to simulate the effect of imaginary time evolution e −h∆τ on a quantum state. We wish to develop an alternative approach that does not require the searching of the effective Hamiltonian h eff .…”

Section: Introductionmentioning

confidence: 99%

“…The iteration procedure with varying ∆t ∈ {10.7, 3.5, 1.3, 0.37, 0.091, 0.04, 0.01} but fixed r = 0 (red) & 2 (blue) for the Hamiltonian H 5 in Eq (14)…”

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confidence: 99%

Quantum algorithm for spectral projection by measuring an ancilla iteratively

Chen

Wei

2020

Phys. Rev. A

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We propose a quantum algorithm for projecting a quantum system to eigenstates of any Hermitian operator, provided one can access the associated control-unitary evolution for the ancilla and the system, as well as the measurement of the controlling ancillary qubit. Such a Hadamard-test like primitive is iterated so as to achieve the spectral projection, and the distribution of the projected eigenstates obeys the Born rule. This algorithm can be used as a subroutine in the quantum annealing procedure by measurement to drive the system to the ground state of a final Hamiltonian, and we simulate this for quantum many-body spin chains.

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Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution

Cited by 556 publications

References 46 publications

Quantum restricted Boltzmann machine is universal for quantum computation

Quantum restricted Boltzmann machine is universal for quantum computation

A Quantum Algorithm for Linear PDEs Arising in Finance

Quantum algorithm for spectral projection by measuring an ancilla iteratively

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