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arXiv:2503.08866 · Physical review. D/Physical review. D.

Quantum circuits for SU(3) lattice gauge theory

Praveen Balaji, Cianán Conefrey-Shinozaki, Patrick Draper, Jason K. Elhaderi, Drishti Gupta, Luis Hidalgo, Andrew Lytle, Enrico Rinaldi

Published Aug 26, 2025• 12 citations
arXivHTMLPDF

Abstract

Lattice gauge theories in varying dimensions, lattice volumes, and truncations offer a rich family of targets for Hamiltonian simulation on quantum devices. In return, formulating quantum simulations can provide new ways of thinking about the quantum structure of gauge theories. In this work, we consider pure S U ( 3 ) gauge theory in two and three spatial dimensions in a streamlined version of the electric basis. We use a formulation of the theory that balances locality of the Hamiltonian and size of the gauge-invariant state space, and we classically pre-compute dictionaries of plaquette operator matrix elements for use in circuit construction. We build circuits for simulating time evolution on arbitrary lattice volumes, spanning circuits suitable for Noisy Intermediate-Scale Quantum era hardware to future fault-tolerant devices. Relative to spin models, time evolution in lattice gauge theories involves more complex local unitaries, and the Hilbert space of all quantum registers may have large unphysical subspaces. Based on these features, we develop general, volume-scalable tools for optimizing circuit depth, including pruning and fusion algorithms for collections of large multicontrolled unitaries. We describe scalings of quantum resources needed to simulate larger circuits and some directions for future algorithmic development.

DOI
10.1103/k8f6-yft8

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Lattice gauge theories in varying dimensions, lattice volumes, and truncations offer a rich family of targets for Hamiltonian simulation on quantum devices. In return, formulating quantum simulations can provide new ways of thinking about the quantum structure of gauge theories. In this work, we consider pure S U ( 3 ) gauge theory in two and three spatial dimensions in a streamlined version of the electric basis. We use a formulation of the theory that balances locality of the Hamiltonian and size of the gauge-invariant state space, and we classically pre-compute dictionaries of plaquette operator matrix elements for use in circuit construction. We build circuits for simulating time evolution on arbitrary lattice volumes, spanning circuits suitable for Noisy Intermediate-Scale Quantum era hardware to future fault-tolerant devices. Relative to spin models, time evolution in lattice gauge theories involves more complex local unitaries, and the Hilbert space of all quantum registers may have large unphysical subspaces. Based on these features, we develop general, volume-scalable tools for optimizing circuit depth, including pruning and fusion algorithms for collections of large multicontrolled unitaries. We describe scalings of quantum resources needed to simulate larger circuits and some directions for future algorithmic development.
Quantum simulations of lattice gauge theories offer the potential to directly study the nonperturbative dynamics of quantum chromodynamics, but naive analyses suggest that they require large computational resources. Large N c expansions are performed to order 1 / N c to simplify the Hamiltonian of pure SU ( N c ) lattice gauge theories. A reformulation of the electric basis is introduced with a truncation strategy based on the construction of local Krylov subspaces with plaquette operators. Numerical simulations show that these truncated Hamiltonians are consistent with traditional lattice calculations at relatively small couplings. It is shown that the computational resources required for quantum simulation of time evolution generated by these Hamiltonians is 17–19 orders of magnitude smaller than previous approaches, provided that the truncations in this work can reach lattice spacings in three-dimensional simulations comparable to the two-dimensional simulations performed.
Deforming the Trail: Baseline Quantum Circuitry for SU(2)k\text{SU(2)}_kSU(2)k​ Lattice Gauge Theory
Quantifying quantum resources for simulating the fundamental forces of Nature is sensitive to the mapping of gauge fields onto finite quantum computational architectures. When locally truncating lattice gauge theories in the irreducible representation basis, it has been proposed to further deform the theory via quantum groups. The purpose of this deformation is (1) to provide an infinite tower of finite-dimensional (d=k+1d = k+1d=k+1) groups systematically approximating the infinite-dimensional gauge links and (2) to restore the physical unitarity of a plaquette operator diagonalization procedure analytically derived from the field continuum by recontracting vertex pairs. For the SU(2)k_kk​ Yang-Mills pure-gauge theory, we provide a constructive strategy of gauge-variant completions to extend this unitarity to the entire computational Hilbert space, leading to well-defined time evolution unitaries as targets for optimized circuit synthesis. Leveraging basic circuit decompositions and symmetries of the diagonalized plaquette operator, we report resource upper-bounds on the generalized-controlled-X two-qudit gates for arbitrary local truncation ddd, reducing estimates and scaling relative to the non-deformed theory by three polynomial powers from O(d8)O(d^8)O(d8) to O(d5)O(d^5)O(d5). Examining the stronger q-deformed gauge constraint, which softens the total flux at vertices, we show that the physical Hilbert space dimension of the deformed plaquette operator scales equivalently to its non-deformed counterpart with a constant factor 0.2563(5)0.2563(5)0.2563(5). Thus, despite affecting interactions at all scales as exemplified by the observed flux hierarchy inversion symmetry, q-deformation continues to pass scrutiny as a reliable truncation offering advantages in quantum circuit synthesis.
A Hamiltonian lattice formulation of lattice gauge theories opens the possibility for quantum simulations of the non-perturbative dynamics of QCD. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local electric energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits to leading order in large N.
Quantum simulation provides a promising framework for studying non-perturbative phenomena in lattice gauge theories that are difficult to access using classical computational methods. In particular, classical Monte Carlo approaches suffer from the sign problem, limiting their applicability to real-time dynamics and finite-density regimes. Quantum computers, by directly encoding quantum dynamics, offer a natural pathway to overcoming these limitations. In this thesis, we study quantum simulations of a truncated SU(2) lattice gauge theory using a Hamiltonian formulation, representing an important non-Abelian step toward quantum chromodynamics. We focus on three-dimensional gauge-theory simulations in the noisy intermediate-scale quantum (NISQ) era, where limited qubit resources pose a significant challenge. To address this, we introduce the triamond lattice, a three-dimensional lattice geometry that enforces local gauge invariance while providing an economical encoding of gauge degrees of freedom, thereby reducing qubit overhead compared to conventional lattice constructions. We derive the corresponding Hamiltonian, construct physical gauge-invariant states, and implement quantum simulations using quantum imaginary time evolution to prepare ground states, alongside real-time evolution methods to study dynamical behavior. Error-mitigation techniques are employed to extract reliable physical observables from noisy quantum hardware. Our results demonstrate that the triamond lattice enables resource-efficient three-dimensional quantum gauge simulations and represents a practical step toward scalable quantum simulations of non-perturbative gauge theories.
The simulation of real-time dynamics in lattice gauge theories is particularly hard for classical computing due to the exponential scaling of the required resources. On the other hand, quantum algorithms can potentially perform the same calculation with a polynomial dependence on the number of degrees of freedom. A precise estimation is however particularly challenging for the simulation of lattice gauge theories in arbitrary dimensions, where gauge fields are dynamical variables, in addition to the particle fields. Moreover, there exist several choices for discretizing particles and gauge fields on a lattice, each of them coming at different prices in terms of qubit register size and circuit depth. Here we provide a resource counting for real-time evolution of U(1)U(1)U(1) gauge theories, such as quantum electrodynamics, on arbitrary dimension using the Wilson fermion representation for the particles, and the quantum link model approach for the gauge fields. We study the phenomena of flux-string breaking up to a genuine bidimensional model using classical simulations of the quantum circuits and discuss the advantages of our discretization choice in simulation of more challenging SU(N)SU(N)SU(N) gauge theories such as quantum chromodynamics.
The quantum simulation of fermionic gauge field theories is one of the anticipated uses of quantum computers in the noisy intermediate-scale quantum (NISQ) era. Recently work has been done to simulate properties of the fermionic Z2{\mathbb{Z}}_{2}Z2​ gauge field theory in (1+1)D(1+1)\mathrm{D}(1+1)D and the pure gauge theory in (2+1)D(2+1)\mathrm{D}(2+1)D. In this work, we investigate various options for simulating the fermionic Z2{\mathbb{Z}}_{2}Z2​ gauge field theory in (2+1)D(2+1)\mathrm{D}(2+1)D. To simulate the theory on a NISQ device it is vital to minimize both the number of qubits used and the circuit depth. In this work we propose ways to optimize both criteria for simulating time dynamics. In particular, we develop a new way to simulate this theory on a quantum computer, with minimal qubit requirements. We provide a quantum circuit for simulating a single first-order Trotter step that minimizes the number of 2-qubit gates needed and gives comparable results to methods requiring more qubits. Furthermore, we investigate variational Trotterization approaches that allow us to further decrease the circuit depth.