Digitization of Scalar Fields for Quantum Computing

TL;DR

This paper evaluates methods for digitizing scalar quantum fields on NISQ devices, comparing basis choices and improvements to minimize digitization errors, and determines qubit requirements for accurate low-energy state simulations.

Contribution

It introduces a comprehensive analysis of digitization strategies for scalar fields on quantum computers, emphasizing the effectiveness of Hamiltonian digitization via Quantum Fourier Transform.

Findings

Digitization errors scale logarithmically with qubits per site.

Complete Hamiltonian digitization outperforms improved actions.

Approximately 4-7 qubits per site suffice for low-error simulations.

Abstract

Qubit, operator and gate resources required for the digitization of lattice $\lambda\phi^4$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan, Lee and Preskill, with a focus towards noisy intermediate-scale quantum (NISQ) devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spentzouris, Amundson and Harnik building on the work of Somma, provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan, Lee and Preskill, and eigenstates of a harmonic oscillator, to describe $0+1$- and $d+1$-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in Lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in $\lambda\phi^4$, but are found to be inferior to a complete digitization-improvement of the Hamiltonian using a Quantum Fourier Transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as $|\log|\log |\epsilon_{\rm dig}|||\sim n_Q$ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as $|\log |\epsilon_{\rm latt}||\sim N_Q$ (total number of qubits in the simulation). For localized(delocalized) field-space wavefunctions, it is found that $n_Q\sim4(7)$ qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices.