Digitization of Scalar Fields for NISQ-Era Quantum Computing

Abstract

Qubit, operator and gate resources required for the digitization of lattice $\lambda\phi^4$ scalar field theories onto quantum computers in the NISQ era are considered, building upon the foundational work by Jordan, Lee and Preskill. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spetzouris, 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 $1+1$-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). We find that fewer than $n_Q =10$ qubits per spatial lattice site are sufficient to reduce digitization errors below noise levels expected in NISQ-era quantum devices for both localized and delocalized field-space wavefunctions. For localized wavefunctions, $n_Q = 4$ qubits are likely to be sufficient for calculations requiring modest precision.