Implementing smooth functions of a Hermitian matrix on a quantum computer

Subramanian, Sathyawageeswar; Brierley, Stephen; Jozsa, Richard

doi:10.1088/2399-6528/ab25a2

Cited by 20 publications

(18 citation statements)

References 32 publications

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“…. , M − 1} at regular intervals on the circle and apply the trapezoidal rule to integral (14). Then, as we can see in the following, we have approximation f M (A).…”

Section: Approximation By Cauchy's Integral Theorem and The Trapezoidal Rulementioning

confidence: 94%

“…Quantum algorithms to obtain state |f with poly(log(1/ǫ)) runtime have already been shown in [13,14,15]. The difference between the proposed quantum algorithm and those quantum algorithms is whether the matrix decomposition of matrix function f (A) is used.…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, we can apply the proposed quantum algorithm to the case in which matrix A is not Hermitian. The method based on eigenvalue decomposition in [13,14] uses the property that matrix A is Hermitian. The method based on singular value decomposition in [15] is applicable when matrix function f (A) is defined as f (A) = U f (Σ)V † for singular value decomposition A = U ΣV † of matrix A.…”

Section: Introductionmentioning

confidence: 99%

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Quantum algorithm for matrix functions by Cauchy's integral formula

Takahira¹,

Ohashi²,

Sogabe³

et al. 2020

QIC

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For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

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“…. , M − 1} at regular intervals on the circle and apply the trapezoidal rule to integral (14). Then, as we can see in the following, we have approximation f M (A).…”

Section: Approximation By Cauchy's Integral Theorem and The Trapezoidal Rulementioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum algorithm for matrix functions by Cauchy's integral formula

Takahira¹,

Ohashi²,

Sogabe³

et al. 2020

QIC

View full text Add to dashboard Cite

show abstract

“…and T k x ð Þ is the k th degree Chebyshev polynomial of the first kind [67]. This expansion is useful because a "quantum walk" can exactly produce the effect of Chebyshev polynomials in Ĥ=d, where d is the sparsity of the matrix.…”

Section: Expansion By Chebyshev Polynomials Via Quantum Walkmentioning

confidence: 99%

State preparation and evolution in quantum computing: A perspective from Hamiltonian moments

Aulicino

Keen

2021

Int J of Quantum Chemistry

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Quantum algorithms on noisy intermediate-scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non-negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum-classical algorithms. One such hybrid quantum-classical algorithm, is based upon quantum computed Hamiltonian moments ϕj Ĥn jϕ D E n ¼ 1,2,ÁÁÁ ð Þ , with respect to quantum state j ϕi. In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four-site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM-Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.

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“…Usually, those kinds of algorithms include some black box Hamiltonian input models, which we call oracles, and ask how many queries we need to access the oracles. This type of algorithms includes algorithms based on quantum walks [156,157], multiproduct formula [158,159], Taylor expansion [141,160], fractional-query models [144], Chebyshev polynomial approximations [161], qubitization [12,146], and quantum signal processing [13,162]. Many elements in the web of such algorithms are conceptually or technically related.…”

Section: Jhep07(2021)140 F a Short Introduction On Hamiltonian Simulation Algorithmsmentioning

confidence: 99%

Toward simulating superstring/M-theory on a quantum computer

Gharibyan

Hanada

Honda

et al. 2021

J. High Energ. Phys.

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We present a novel framework for simulating matrix models on a quantum computer. Supersymmetric matrix models have natural applications to superstring/M-theory and gravitational physics, in an appropriate limit of parameters. Furthermore, for certain states in the Berenstein-Maldacena-Nastase (BMN) matrix model, several supersymmetric quantum field theories dual to superstring/M-theory can be realized on a quantum device. Our prescription consists of four steps: regularization of the Hilbert space, adiabatic state preparation, simulation of real-time dynamics, and measurements. Regularization is performed for the BMN matrix model with the introduction of energy cut-off via the truncation in the Fock space. We use the Wan-Kim algorithm for fast digital adiabatic state preparation to prepare the low-energy eigenstates of this model as well as thermofield double state. Then, we provide an explicit construction for simulating real-time dynamics utilizing techniques of block-encoding, qubitization, and quantum signal processing. Lastly, we present a set of measurements and experiments that can be carried out on a quantum computer to further our understanding of superstring/M-theory beyond analytic results.

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Implementing smooth functions of a Hermitian matrix on a quantum computer

Cited by 20 publications

References 32 publications

Quantum algorithm for matrix functions by Cauchy's integral formula

Quantum algorithm for matrix functions by Cauchy's integral formula

State preparation and evolution in quantum computing: A perspective from Hamiltonian moments

Toward simulating superstring/M-theory on a quantum computer

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