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Tailoring non-Abelian lattice gauge theory for quantum simulation

Indrakshi Raychowdhury, Jesse R. Stryker

Published Dec 18, 2018• 4 citations
OpenAlex

Abstract

We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (arXiv:1912.06133) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss's law) exactly. We first discuss the structure of the LSH Hilbert space in $d$ spatial dimensions, its truncation, and its digitization with qubits. Error detection and mitigation in gauge theory simulations would benefit from physicality oracles,'so we decompose circuits that flag gauge invariant wavefunctions. We then analyze the logical qubit costs and entangling gate counts involved with the protocols. The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how the bases are digitized as well as the spatial dimension. The numerous other clear benefits encourage future studies into applying this framework.

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We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (arXiv:1912.06133) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss's law) exactly. We first discuss the structure of the LSH Hilbert space in ddd spatial dimensions, its truncation, and its digitization with qubits. Error detection and mitigation in gauge theory simulations would benefit from physicality oracles,'so we decompose circuits that flag gauge invariant wavefunctions. We then analyze the logical qubit costs and entangling gate counts involved with the protocols. The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how the bases are digitized as well as the spatial dimension. The numerous other clear benefits encourage future studies into applying this framework.
We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (I.Raychowdhury and J. R. Stryker, Phys. Rev. D 101, 114502 (2020)) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss's law) exactly.We first discuss the structure of the LSH Hilbert space in d spatial dimensions, its truncation, and its digitization with qubits.Error detection and mitigation in gauge theory simulations would benefit from physicality "oracles," so we decompose circuits that flag gauge-invariant wave functions.We then analyze the logical qubit costs and entangling gate counts involved with the protocols.The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how the bases are digitized as well as the spatial dimension.The numerous other clear benefits encourage future studies into applying this framework.
The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory—a loop-string-hadron (LSH) formulation—in which the dynamical degrees of freedom are localized pieces of flux loops, meson strings, and hadrons. LSH operators are first derived from Schwinger bosons and used to construct a Hilbert space with the non-Abelian Gauss law built into it. They are subsequently factored into products of “normalized” ladder operators and diagonal matrices, priming them for classical or quantum information processing. The LSH formalism alleviates several disadvantages of quantum simulating the Kogut-Susskind formulation and makes little use of structures specific to SU(2). Its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).
The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory---a loop-string-hadron (LSH) formulation---that describes dynamics directly in terms of its loop, string, and hadron degrees of freedom, while alleviating several disadvantages of quantum simulating the Kogut-Susskind formulation. This LSH formulation transcends the local loop formulation of d+1d+1d+1-dimensional lattice gauge theories by incorporating staggered quarks, furnishing the algebra of gauge-singlet operators, and being used to reconstruct dynamics between states that have Gauss's law built in to them. LSH operators are then factored into products of ``normalized'' ladder operators and diagonal matrices, priming them for classical or quantum information processing. Self-contained expressions of the Hamiltonian are given up to d=3d=3d=3. The LSH formalism makes little use of structures specific to SU(2), and its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).
Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs. Such theories require the imposition of non-Abelian Gauss's laws on the Hilbert space, introducing additional computational complexity. Focusing on the case of SU(2) LGT in 1+11+11+1 dimensions coupled to one flavor of fermionic matter, a number of different formulations of the original Kogut-Susskind framework are analyzed with regard to the dependence of the dimension of the physical Hilbert space on boundary conditions, system size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics obtained from a Hamiltonian computation is examined, and the (classical) computational-resource requirements given these considerations are studied. Besides the well-known angular-momentum formulation of the theory, the cases of purely fermionic and purely bosonic formulations (with open boundary conditions) and the loop-string-hadron formulation are analyzed, along with a brief discussion of a quantum-link-model formulation of the same theory. Clear advantages are found in working with the loop-string-hadron framework which implements non-Abelian Gauss's laws a priori using a complete set of gauge-invariant operators. It is argued that the substantially reduced cost of simulation within the loop-string-hadron formulation compared with the angular-momentum basis has implications for more efficient and robust quantum simulations of the same theory. Future studies will extend this investigation to the analysis of resource requirements for quantum simulations of non-Abelian LGT, with the goal of shedding light on the most efficient Hamiltonian formulation of gauge theories of relevance in nature.
Within the aim of understanding quantum chromodynamics through simulation, an increasingly studied approach is that of quantum computation and simulation. Challenges exist in encoding the minimal and physical degrees of freedom for a non-Abelian gauge theory and maintaining physical or gauge-invariant dynamics in a simulation. In this work, the Loop-String-Hadron (LSH) formulation of the 1+1-dimensional SU(3) lattice gauge theory is used to define an efficient mapping of SU(3) invariant degrees of freedom onto qubits. It is shown that the required number of qubits in the LSH basis is significantly reduced compared to its IRREP basis counterpart. While the non-Abelian Gauss laws of the SU(3) theory are automatically satisfied by the usage of LSH variables, the remnant constraints on the consistency of the flux numbers still exist. During time evolution, the noise can accumulate and take the state out of the sector of the Hilbert space where the constraint is satisfied. With this motivation, an oracle algorithm is constructed to be applied to the qubits for checking the local constraint at a given link. Costs in terms of qubit and gate number, and circuit depth, are found.
We present a quantum simulation of the SU(2) lattice gauge theory in 1+1 dimensions with one flavor of fermions, performed using the trapped-ion Quantinuum System Model H2. This work is based on the loop-string-hadron (LSH) formulation of the gauge theory, and is distinguished by using qubits to represent both bosonic-gauge and fermionic-matter degrees of freedom. Furthermore, we access the j=1j=1j=1 electric-flux sector beyond the jmax⁡=1/2j_{\max}=1/2jmax​=1/2 truncation commonly used in SU(2) quantum simulations. To that end, we digitize an electric-field degree of freedom using a multiqubit register for representing the bosonic quantum number. Our simulation therefore synthesizes several key ingredients relevant to quantum simulations in higher spatial dimensions, in particular explicit local gauge-field degrees of freedom represented here by a multiqubit bosonic register, while relying solely on circuit-construction optimizations that are expected to scale polynomially with lattice size. This work demonstrates that modern quantum hardware is increasingly able to accommodate the features of complicated but physically important lattice gauge theories while successfully reproducing the expected Trotterized dynamics.