Efficient quantum interpolation of natural data

Ramos-Calderer, Sergi

doi:10.48550/arxiv.2203.06196

Cited by 2 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[25,26], and was generalized for loading images on a quantum computer in Ref. [52]. This circuit is similar to the circuit that we have proposed in Fig.…”

Section: A Comparison With Previous Workmentioning

confidence: 97%

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

Moosa¹,

W.²,

Chen³

et al. 2023

Preprint

View full text Add to dashboard Cite

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%.

show abstract

“…[25,26], and was generalized for loading images on a quantum computer in Ref. [52]. This circuit is similar to the circuit that we have proposed in Fig.…”

Section: A Comparison With Previous Workmentioning

confidence: 97%

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

Moosa¹,

W.²,

Chen³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…One such method is based on Fourier interpolation or trigonometric interpolation, which uses a Fourier transform to approximate the values of a function everywhere given the values of the function on a subset of points in its domain. A quantum circuit implementing this interpolation technique was proposed in [25,26], and was generalized for loading images on a quantum computer in [52]. This circuit is similar to the circuit that we have proposed in figure 1(a) but with U c replaced with QFT m ′ U f , where U f is any unitary that loads the values of the function f sampled at 2 m ′ equidistant grid points to an m ′ -qubit state, and QFT m ′ is a QFT operator acting on m ′ qubits.…”

Section: Comparison With Previous Workmentioning

confidence: 99%

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

Moosa,

Watts,

Chen

et al. 2023

Quantum Sci. Technol.

View full text Add to dashboard Cite

The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms 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 2Dn-point uniform discretization of a D-dimensional function specified by a D-dimensional Fourier series. A free parameter m < n determines the number of Fourier coefficients, 2D(m+1), used to represent the function. The FSL method uses a quantum circuit of depth at most 2(n-2)+ ⌈log2(n-m)⌉ + 2D(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 (2Dn) points. The FSL circuit consists of at most Dn+2D(m+1)+1-1 single-qubit and Dn(n+1)/2 + 2D(m+1)+1 - 3D(m+1) - 2 two-qubit gates. 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%.

show abstract

Efficient quantum interpolation of natural data

Cited by 2 publications

References 25 publications

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

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

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

Contact Info

Product

Resources

About