Quantum state preparation for multivariate functions

TL;DR

This paper introduces simplified quantum state preparation protocols for multivariate functions, enabling efficient encoding of complex distributions without complex quantum subroutines, demonstrated on real quantum hardware.

Contribution

The authors develop new protocols for preparing multivariate quantum states using linear combinations of basis functions, avoiding complex quantum subroutines and improving practicality.

Findings

Successfully prepared various distributions including Student's t, Ricker wavelets, and electron wavefunctions.

Demonstrated bivariate Gaussian state preparation on a 24-qubit quantum processor.

Protocols are effective both asymptotically and with near-term quantum resources.

Abstract

A fundamental step of any quantum algorithm is the preparation of qubit registers in a suitable initial state. Often qubit registers represent a discretization of continuous variables and the initial state is defined by a multivariate function. We develop protocols for preparing quantum states whose amplitudes encode multivariate functions by linearly combining block-encodings of Fourier and Chebyshev basis functions. Without relying on arithmetic circuits, quantum Fourier transforms, or multivariate quantum signal processing, our algorithms are simpler and more effective than previous proposals. We analyze requirements both asymptotically and pragmatically in terms of near/medium-term resources. Numerically, we prepare bivariate Student's t-distributions, 2D Ricker wavelets and electron wavefunctions in a 3D Coulomb potential, which are initial states with potential applications in finance, physics and chemistry simulations. Finally, we prepare bivariate Gaussian distributions on the Quantinuum H2-1 trapped-ion quantum processor using 24 qubits and up to 237 two-qubit gates.