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Toward Quantum Simulation of SU(2) Gauge Theory using Non-Compact Variables

Emanuele Mendicelli, Georg Bergner, Masanori Hanada

Abstract

Simulating lattice gauge theories on quantum computers presents unique challenges that drive the development of novel theoretical frameworks. The orbifold lattice approach offers a scalable method for simulating SU($N$) gauge theories in arbitrary dimensions. In this work, we present three improvements: (i) two new simplified Hamiltonians, (ii) an encoding of the SU(2) theory with smaller number of qubits, and (iii) a reduction in the requirement for large scalar masses to reach the Kogut-Susskind limit, achieved via the inclusion of an additional term in the Hamiltonian. These advancements significantly reduce circuit depth and qubit requirements for quantum simulations. We benchmarked these improvements using Monte Carlo simulations of SU(2) in (2+1) dimensions. Preliminary results demonstrate the effectiveness of these developments and further validate the use of noncompact variables as a promising framework for scalable quantum simulations of gauge theories.

Toward Quantum Simulation of SU(2) Gauge Theory using Non-Compact Variables

Abstract

Simulating lattice gauge theories on quantum computers presents unique challenges that drive the development of novel theoretical frameworks. The orbifold lattice approach offers a scalable method for simulating SU() gauge theories in arbitrary dimensions. In this work, we present three improvements: (i) two new simplified Hamiltonians, (ii) an encoding of the SU(2) theory with smaller number of qubits, and (iii) a reduction in the requirement for large scalar masses to reach the Kogut-Susskind limit, achieved via the inclusion of an additional term in the Hamiltonian. These advancements significantly reduce circuit depth and qubit requirements for quantum simulations. We benchmarked these improvements using Monte Carlo simulations of SU(2) in (2+1) dimensions. Preliminary results demonstrate the effectiveness of these developments and further validate the use of noncompact variables as a promising framework for scalable quantum simulations of gauge theories.

Paper Structure

This paper contains 9 sections, 19 equations, 2 figures.

Table of Contents

  1. Introduction and Motivation
  2. Orbifold Lattice Hamiltonian
  3. The gauge invariance of the orbifold lattice
  4. Encoding SU($N$) Orbifold Lattices on Quantum Computers
  5. Orbifold-ish Hamiltonians
  6. Embedding SU(2) into $\mathbb{R}^{4}$
  7. Monte Carlo Simulations
  8. Additional term to reduce necessary $m^2$ and corresponding simulations
  9. Conclusions and future directions

Figures (2)

  • Figure 1: Plot of $\langle\mathrm{Tr}(ZZ\bar{Z}\bar{Z})\rangle$, $\langle\mathrm{Tr}(UU\bar{U} \bar{U})\rangle_{\rm spatial}$, $\langle\mathrm{Tr}(UU\bar{U} \bar{U})\rangle_{\rm temporal}$ and $\langle\mathrm{Tr}(W - \textbf{1}_N)^2\rangle$ as function of $1/m^2$, for $H$, $H_1$, and $H_2$ embedded in $\mathbb{R}^4$ for a lattice size of $8^3$ with two different lattice spacings: $a_t = a_s = 0.1$ [Top] and $a_t = a_s = 0.3$ [Bottom]. The blue circles, red squares, and green hexagons represent measurements for $H$, $H_1$, and $H_2$, respectively. The blue, red, and green solid lines show quadratic fits to these measurements, which are used to extract the infinite-mass values, indicated by blue-down, red-right, and green-up triangles for $H$, $H_1$, and $H_2$, respectively. The horizontal orange dashed line represents the Wilson action value that the observable is expected to approach in the KS limit.
  • Figure 2: [Top] $\mathrm{Tr}(W-\mathbf{1}_N)\rangle$ versus $\gamma$. [Bottom] Corresponding plaquette expectation as a function of $\langle \mathrm{Tr}(W-\mathbf{1}_N)\rangle$. Columns show to different Hamiltonians—$\hat{H}$ [Left], $\hat{H}_1$ [Center], and $\hat{H}_2$ [Right]—with $m^2 = 50$ for the first two and $m^2 = 500$ for the last. The green dashed line marks the target value of zero. The orange dashed line shows the Wilson-action plaquette, with shaded jackknife uncertainty.