Digitization of Scalar Fields for Quantum Computing

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.