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($N$) 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.

• The paper presents streamlined Hamiltonians and an R^4 embedding that significantly reduce quantum resource requirements for SU(2) gauge simulations.
• It introduces non-compact variable representations that map SU(2) link variables into R^4, preserving canonical commutation relations and achieving the KS limit.
• Monte Carlo benchmarks validate convergence to Wilson-action values and demonstrate reduced scalar masses through an effective counter-term tuning.

Toward Quantum Simulation of SU(2) Gauge Theory Using Non-Compact Variables: An Expert Overview

Introduction

The quantum simulation of non-Abelian lattice gauge theories (LGTs), notably SU(2) Yang-Mills theory, represents a critical challenge due to the intrinsic complexity of mapping unitary group-valued variables onto quantum hardware. This work addresses key bottlenecks in resource-scaling and circuit synthesis through the adoption of non-compact variable representations inspired by the orbifold lattice construction. Three principal contributions are presented: the derivation of two computationally streamlined Hamiltonians, a compact encoding of SU(2) link variables into $\mathbb{R}^4$, and the introduction of a Hamiltonian term that permits convergence to the Kogut-Susskind (KS) limit at significantly lower scalar masses.

Orbifold Lattice Formalism and Its Quantum Encoding

A central advance is the orbifold lattice formalism, which circumvents the necessity for compact variables by parametrizing SU(N) link variables as complex matrices in $R^{2N^2}$. Each link is decomposed into a product of a positive-definite Hermitian scalar (W) and a unitary gauge field (U), representing a Yang-Mills system coupled to scalars. The canonical commutation relations are maintained, and in the infinite scalar mass limit, known as the KS limit, the theory rigorously reduces to that described by the Kogut-Susskind Hamiltonian.

The mapping to quantum hardware exploits a bosonic encoding: each of the $2N^2$ components per link is discretized with $Q$ qubits per bosonic mode, giving a scalable, polynomial overhead in both qubit count and gate complexity. The interaction term scales as $N^4Q^4$, substantially ameliorating the circuit complexity relative to compact-link constructions, whose resource requirements typically grow exponentially with system size.

Streamlined Hamiltonians and SU(2) Embedding

To further reduce computational cost, the authors systematically remove terms from the orbifold Hamiltonian that are strictly negligible in the KS limit. Two "orbifold-ish" Hamiltonians, $H_1$ and $H_2$, are proposed. Both eliminate higher-order corrections whose contributions vanish as the scalar fields decouple ($W \to 1$).

A significant technical innovation is the $\mathbb{R}^4$-embedding for SU(2), leveraging the isomorphism $SU(2) \cong S^3$. This embedding halves the bosonic degrees of freedom per link compared to the original $R^{2N^2}$0 representation, producing a substantial reduction in required quantum resources while maintaining exactness in the group manifold representation.

Benchmarking via Monte Carlo Methods

The efficacy of these new Hamiltonians and embeddings is validated via extensive Hybrid Monte Carlo simulations in $R^{2N^2}$1 dimensions. Observables including plaquette traces and deviations of the scalar field from the identity are tracked as functions of the inverse scalar mass $R^{2N^2}$2. All three Hamiltonians ($R^{2N^2}$3, $R^{2N^2}$4, $R^{2N^2}$5) exhibit smooth, monotonic convergence of Wilson loop and plaquette observables to their Wilson-action values as $R^{2N^2}$6 increases. In the infinite-mass (KS) limit, scalar field contributions decouple as required, quantitatively confirming the theoretical expectation.

Mass Reduction and Counter-Term Tuning

An outstanding issue is the requirement of large scalar masses (on the order of thousands) for convergence to the Wilson action, which is impractical for near-term hardware. The source is identified as a residual linear term in the effective scalar potential, necessitating a large mass to suppress vacuum expectation value shifts. The introduction of a bare action counter-term $R^{2N^2}$7 cancels this shift, allowing for order-of-magnitude smaller scalar masses ($R^{2N^2}$8–$R^{2N^2}$9) without loss of fidelity with Wilson action results. Monte Carlo benchmarks demonstrate that, after appropriate tuning of $2N^2$0, all relevant observables for $2N^2$1, $2N^2$2, and $2N^2$3 agree with standard Wilson gauge theory at substantially reduced quantum simulation cost.

Implications and Outlook

This framework achieves a convergence of physical accuracy, gate, and qubit efficiency currently unmatched in quantum LGT simulation paradigms. By enabling the simulation of non-Abelian SU(2) gauge theory with polynomial resource requirements using non-compact Cartesian encodings, the formalism provides a path for scaling to physically relevant lattice sizes on both NISQ devices and, prospectively, fault-tolerant quantum computers.

The $2N^2$4 encoding and counter-term methodology are generalizable to higher gauge groups (e.g., SU(3)), suggesting straightforward scalability to quantum chromodynamics (QCD) and more complex theories. The elimination of the sign problem and the intrinsic suitability for Hamiltonian real-time evolution further underscores the potential impact for quantum simulations in high-energy and nuclear physics.

Conclusion

The orbifold lattice construction, especially when augmented with the technical innovations presented—simplified Hamiltonians, $2N^2$5 embedding for SU(2), and scalar mass counter-terms—constitutes an effective and scalable route to quantum simulation of Yang-Mills theories. The demonstrated agreement with standard Wilson lattice results alongside dramatic reductions in qubit and gate counts positions this approach as a viable candidate for the future simulation of non-Abelian gauge theories on quantum hardware.

Further extensions to $2N^2$6 dimensions, explicit gate decomposition, and real-time dynamics studies are immediate next steps. This work paves the way for practical quantum simulation of fundamental gauge theories, with immediate applicability in high-energy theory and quantum computing.