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

Abstract

The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms. We introduce the Fourier Series Loader (FSL) method for preparing quantum states that exactly encode multi-dimensional Fourier series using linear-depth quantum circuits. 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 $m<n$ 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 $2(n-2)+\lceil {\log }_2(n-m)\rceil +2^{D(m+1)+2}-2D(m+1)$, 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. 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 H$1$-$1$ and H$1$-$2$ 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 $\approx 99\%$, and a 2D function on 10 qubits with a classical fidelity $\approx 94\%$.

arXiv commentV2: published version

References in corpus (49)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• The density-matrix renormalization group in the age of matrix product states
• Quantum algorithm for solving linear systems of equations
• On the Measurement of Qubits
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Efficient quantum algorithms for simulating sparse Hamiltonians
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Quantum Data Fitting
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Semiclassical Fourier Transform for Quantum Computation
• Efficient Simulation of Quantum Systems by Quantum Computers
• The randomized measurement toolbox
• Quantum-state preparation with universal gate decompositions
• Variational quantum algorithms for nonlinear problems
• Preconditioned quantum linear system algorithm
• Quantum Risk Analysis
• Creating superpositions that correspond to efficiently integrable probability distributions
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Option Pricing using Quantum Computers
• Quantum algorithm for linear differential equations with exponentially improved dependence on precision
• Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment
• A divide-and-conquer algorithm for quantum state preparation
• Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications
• High-precision quantum algorithms for partial differential equations
• Efficient phase-factor evaluation in quantum signal processing
• Encoding of Matrix Product States into Quantum Circuits of One- and Two-Qubit Gates
• Quantum singular value decomposition of non-sparse low-rank matrices
• Quantum tomography of an entangled three-spin state in silicon
• Quantum simulation of the single-particle Schrodinger equation
• Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions
• Neural network quantum state tomography in a two-qubit experiment
• Scalable randomized benchmarking of quantum computers using mirror circuits
• Product Decomposition of Periodic Functions in Quantum Signal Processing
• Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
• Attention-based Quantum Tomography
• Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm
• Optimizing Quantum Circuits for Arithmetic
• Finding Angles for Quantum Signal Processing with Machine Precision
• The Efficient Preparation of Normal Distributions in Quantum Registers
• Wavefunction preparation and resampling using a quantum computer
• Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations
• Preparing Arbitrary Continuous Functions in Quantum Registers With Logarithmic Complexity
• Sample-Optimal Average-Case Sparse Fourier Transform in Two Dimensions
• Efficient quantum interpolation of natural data
• Quantum self-learning Monte Carlo with quantum Fourier transform sampler
• A Generalized Quantum Inner Product and Applications to Financial Engineering
• Practical considerations for the preparation of multivariate Gaussian states on quantum computers
• Quantum Amplitude Interpolation
• All-optical neural network quantum state tomography

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