Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations

García-Molina, Paula; Rodríguez-Mediavilla, Javier; García-Ripoll, Juan José

doi:10.1103/physreva.105.012433

Cited by 21 publications

(16 citation statements)

References 47 publications

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“…Finally, an iteration of this method which we call quantum natural CSPSA (QN-CSPSA) is given by Eqs. ( 43), ( 42), (33), (32), and (23).…”

Section: Qn-cspsamentioning

confidence: 99%

“…Thus, hybrid optimization algorithms are used whenever the objective function can be evaluated more efficiently on a quantum computer than on a classical one. This is the case for applications to quantum chemistry [6][7][8], quantum control [9][10][11], quantum simulation [12,13], entanglement detection [14][15][16], state estimation [17][18][19][20][21], quantum machine learning [22][23][24][25][26], error correction [27], graph theory [28][29][30], differential equations [31][32][33], and finances [34].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic optimization algorithms for quantum applications

Gidi¹,

Candia²,

Muñoz-Moller³

et al. 2022

Preprint

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Hybrid classical quantum optimization methods have become an important tool for efficiently solving problems in the current generation of NISQ computers. These methods use an optimization algorithm that is executed in a classical computer, which is fed with values of the objective function that are obtained in a quantum computer. Therefore, a proper choice of optimization algorithm is essential to achieve good performance. Here, we review the use of first-order, second-order, and quantum natural gradient stochastic optimization methods, which are defined in the field of real numbers, and propose new stochastic algorithms defined in the field of complex numbers. The performance of the methods is evaluated by means of their application to variational quantum eigensolver, quantum control of quantum states, and quantum state estimation. In general, complex number optimization algorithms perform better and second-order algorithms provide a large improvement over first-order methods in the case of variational quantum eigensolver.

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“…Finally, an iteration of this method which we call quantum natural CSPSA (QN-CSPSA) is given by Eqs. ( 43), ( 42), (33), (32), and (23).…”

Section: Qn-cspsamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic optimization algorithms for quantum applications

Gidi¹,

Candia²,

Muñoz-Moller³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…It is worth mentioning that the FSL method is analogous to a state preparation method based on Fourier interpolation [25,26]. The interpolation-based method loads the values of the target function at 2 m+1 discretized points and then interpolates to 2 n points using the quantum Fourier transform.…”

Section: Introductionmentioning

confidence: 99%

Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions

Moosa¹,

W.²,

Chen³

et al. 2023

Preprint

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The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms, including those for solving partial differential equations and Monte Carlo estimation. In this work, we introduce the Fourier Series Loader (FSL) method for preparing quantum states that exactly encode multi-dimensional Fourier series using linear-depth quantum circuits. Specifically, the FSL method prepares a (Dn)-qubit state encoding the 2 Dn -point uniform discretization of a D-dimensional function specified by a D-dimensional Fourier series. A free parameter, m, which must be less than n, determines the number of Fourier coefficients, 2 D(m+1) , used to represent the function. The FSL method uses a quantum circuit of depth at most 2(n − 2) + log 2 (n − m) + 2 D(m+1)+2 − 2D(m + 1), which is linear in the number of Fourier coefficients, and linear in the number of qubits (Dn) despite the fact that the loaded function's discretization is over exponentially many (2 Dn ) points. The FSL circuit consists of at most Dn + 2 D(m+1)+1 − 1 single-qubit and Dn(n + 1)/2 + 2 D(m+1)+1 − 3D(m + 1) − 2 two-qubit gates; we present a classical compilation algorithm with runtime O(2 3D(m+1) ) to determine the FSL circuit for a given Fourier series. The FSL method allows for the highly accurate loading of complex-valued functions that are well-approximated by a Fourier series with finitely many terms. We report results from noiseless quantum circuit simulations, illustrating the capability of the FSL method to load various continuous 1D functions, and a discontinuous 1D function, on 20 qubits with infidelities of less than 10 −6 and 10 −3 , respectively. We also demonstrate the practicality of the FSL method for near-term quantum computers by presenting experiments performed on the Quantinuum H1-1 and H1-2 trapped-ion quantum computers: we loaded a complex-valued function on 3 qubits with a fidelity of over 95%, as well as various 1D real-valued functions on up to 6 qubits with classical fidelities ≈ 99%, and a 2D function on 10 qubits with a classical fidelity ≈ 94%.

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“…[1], and introduced to enhance the solutions of partial differential equations on a quantum computer in Ref. [2].…”

Section: Introductionmentioning

confidence: 99%

Efficient quantum interpolation of natural data

Ramos-Calderer

2022

Preprint

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We present an efficient method to interpolate smooth distributions with a quantum computer that aims to complement data uploading techniques. The quantum algorithm operates in four steps: i) a small sample of the distribution is uploaded onto a reduced set of qubits; ii) a Quantum Fourier Transform (QFT) is applied; iii) qubits encoding vanishing high frequencies are added to the register; and iv) an inverse QFT is operated on the total register. This leverages on the efficiency of the QFT and opens the door of quantum advantage to a relevant family of distributions. We demonstrate the power of the method with an efficient interpolation of a toy gaussian model, discussing uploading methods and precision. We then apply the QFT interpolation technique to quantum encoded images and present further advantages when processing data in superposition.

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Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations

Cited by 21 publications

References 47 publications

Stochastic optimization algorithms for quantum applications

Stochastic optimization algorithms for quantum applications

Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions

Efficient quantum interpolation of natural data

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