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arXiv:2402.10265 · Physical Review Letters

Quantum Simulation of SU(3) Lattice Yang-Mills Theory at Leading Order in Large- N c Expansion

Anthony N. Ciavarella, C. Bauer

Published Sep 9, 2024• 55 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.133.111901

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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.
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.
We develop a quantum algorithmic framework for the efficient simulation of Yang--Mills theories, including the SU(3)\mathrm{SU}(3)SU(3) gauge theory in Quantum Chromodynamics (QCD). The framework uses maximal-tree gauge in terms of gauge field variables that removes all local gauge redundancies. In the resulting gauge-fixed formulation and digitization in the field-amplitude basis, we show that Hamiltonian time evolution admits an efficient implementation based on quantum singular value transformation (QSVT). We derive upper bounds on the total number of qubits and gate complexity, finding polynomial scaling with the inverse simulation precision 1/εs1/\varepsilon_s1/εs​, lattice volume V\mathcal{V}V, gauge coupling ggg, and target energy scale EEE. Our results provide a rigorous complexity-theoretic demonstration that non-Abelian Yang--Mills theories can be simulated efficiently on quantum computers, paving the way toward first-principles quantum simulations of non-perturbative QCD dynamics.
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 many-body systems offer novel methods for probing the dynamics of the Standard Model and its constituent gauge theories. Extracting low-energy predictions from such simulations rely on formulating systematically-improvable representations of lattice gauge theory Hamiltonians that are efficient at all values of the gauge coupling. One such candidate representation for SU(2) is the fully gauge-fixed Hamiltonian defined in the mixed basis. This work focuses on the quantum simulation of the smallest non-trivial system: two plaquettes with open boundary conditions. A mapping of the continuous gauge field degrees of freedom to qubit-based representations is developed. It is found that as few as three qubits per plaquette is sufficient to reach per-mille level precision on predictions for observables. Two distinct algorithms for implementing time evolution in the mixed basis are developed and analyzed in terms of quantum resource estimates. One algorithm has favorable scaling in circuit depth for large numbers of qubits, while the other is more practical when qubit count is limited. The latter algorithm is used in the measurement of a real-time observable on IBM's Heron superconducting quantum processor, ibm_fez. The quantum results match classical predictions at the percent-level. This work lays out a path forward for two- and three-dimensional simulations of larger systems, as well as demonstrating the viability of mixed-basis formulations for studying the properties of SU(2) gauge theories at all values of the gauge coupling.
The development of quantum technology and the advent of quantum computation have renewed interest in Hamiltonian formulations of lattice gauge theories. In contrast to Euclidean Monte Carlo calculations, Hamiltonian simulation requires a direct representation of the physical Hilbert space, an efficient treatment of enforcing Gauss' law as constraints, and a formulation of real-time dynamics in terms of degrees of freedom that can be encoded on quantum or quantum-inspired classical platforms. The starting point corresponds to finding a suitable basis for Hamiltonian lattice gauge theories. The existing literature focuses on several complementary approaches: electric basis, magnetic basis, gauge-fixed, quantum-link, qubit-regularised, finite-group, and orbifold formulations. This article primarily focuses on the prepotential and loop--string--hadron formulations, where non-Abelian Gauss laws are solved locally and the dynamics is written directly in terms of gauge-invariant loop, string and hadron excitations. The discussion covers the SU(2) and SU(3) constructions in one spatial dimension, the role of residual Abelian Gauss laws, and the extension to higher dimensions through point splitting. We conclude by highlighting its direct applications: eliminating redundant gauge degrees of freedom makes exact diagonalisation and tensor network calculations feasible for studying real-time dynamics, provides insights into thermalisation properties, enables mapping of physical degrees of freedom onto analog and digital quantum hardware, and enables quantum simulation of dynamics.
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.