Digitization of scalar fields for quantum computing

Abstract

Qubit, operator and gate resources required for the digitization of lattice λφ 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 λφ 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 | dig ||| ∼ 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 | latt || ∼ N Q (total number of qubits in the simulation). For localized(delocalized) field-space wavefunctions, it is found that n Q ∼ 4(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. 1 1 Only classical computing resources have been used to obtain the results presented in this work.

Unfolding quantum computer readout noise

“…5. This system has been recently studied in the context quantum field theory as a benchmark 0 + 1dimensional noninteracting scalar field theory [43][44][45][46][47][48][49][50] . In practice, all of the qubits of a system would be entangled to achieve the harmonic oscillator wave function.…”

Simulating quantum field theory with a quantum computer

“…(5.1) as a sum of two terms -one is diagonal in the π(x) basis, while the other is diagonal in the φ (x) basis. We can switch back and forth between these two bases by Fourier transforming the field variables at each site [2,3,28]. (I emphasize again that this Fourier transform should not be confused with the spatial Fourier transform.)…”

A tensorial toolkit for quantum computing in lattice gauge theory

“…It would also be very interesting to compare compact and non-compact models. In the non-compact case, a field cut needs to be introduced [7,10]. Note that this makes the perturbative series converge [28].…”

“…[4,5,6]. A complementary approach is to use existing quantum computers [7,8,9,10]. A long term and ambitious goal that could have a significant impact on our understanding of collider physics would be to do jet physics calculations from first principles.…”