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arXiv:2503.11888 · Physical review. D/Physical review. D.

Efficient truncations of SU ( N c ) lattice gauge theory for quantum simulation

Anthony N. Ciavarella, Iván Mauricio Burbano, C. Bauer

Published Aug 26, 2025• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/ylqb-phv5

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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.
Quantum simulations of quantum chromodynamics (QCD) require a representation of gauge fields and fermions on the finitely many degrees of freedom available on a quantum computer. We introduce a truncation of lattice QCD coupled to staggered fermions that includes (i) a local Krylov truncation that generates allowed basis states; (ii) a maximum allowed electric energy per link; (iii) a limit on the number of fermions per site; and (iv) a truncation in the large N_c scaling of Hamiltonian matrix elements. Explicit truncated Hamiltonians for 1+1D and 2+1D lattices are given, and numerical simulations of string-breaking dynamics are performed.
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 simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. 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_{c}. 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 energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.
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.
Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely-used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories.
Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories. Published by the American Physical Society 2024