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Quantum Science and Technology

Simulating lattice quantum electrodynamics on a quantum computer

Angus Kan, Yunseong Nam

Published Nov 7, 2022• 14 citations
Publisher

Abstract

Abstract U(1) lattice gauge theories (LGTs) offer a way to simulate quantum electrodynamics, one of the three forces unified by the Standard Model of particles physics. Here we provide complete, quantum-gate-by-quantum-gate algorithms to simulate U(1) LGTs on a fault-tolerant quantum computer. We further perform rigorous error analysis in order to derive concrete estimates of the quantum computational resources required for an accurate simulation of U(1) LGTs using a second-order product formula. We show that U(1) LGTs in any spatial dimension can be simulated using O ˜ ( T 3 2 N 3 2 Λ ϵ − 1 2 ) non-Clifford T gates, where T is the simulation time, N is the number of lattice sites, Λ is the truncation parameter for the bosonic gauge fields, and ε is the simulation error. This work paves the way towards fault-tolerant quantum simulations of physical models closely related to the Standard Model of particle physics.

DOI
10.1088/2058-9565/aca0b8

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Abstract U(1) lattice gauge theories (LGTs) offer a way to simulate quantum electrodynamics, one of the three forces unified by the Standard Model of particles physics. Here we provide complete, quantum-gate-by-quantum-gate algorithms to simulate U(1) LGTs on a fault-tolerant quantum computer. We further perform rigorous error analysis in order to derive concrete estimates of the quantum computational resources required for an accurate simulation of U(1) LGTs using a second-order product formula. We show that U(1) LGTs in any spatial dimension can be simulated using O ˜ ( T 3 2 N 3 2 Λ ϵ − 1 2 ) non-Clifford T gates, where T is the simulation time, N is the number of lattice sites, Λ is the truncation parameter for the bosonic gauge fields, and ε is the simulation error. This work paves the way towards fault-tolerant quantum simulations of physical models closely related to the Standard Model of particle physics.
It is widely anticipated that a large-scale quantum computer will offer an evermore accurate simulation of nature, opening the floodgates for exciting scientific breakthroughs and technological innovations. Here, we show a complete, instruction-by-instruction rubric to simulate U(1), SU(2), and SU(3) lattice gauge theories on a quantum computer. These theories describe quantum electrodynamics and chromodynamics, the key ingredients that form the fabric of our universe. We further provide a concrete estimate of the quantum computational resources required for an accurate simulation of lattice gauge theories using a second-order product formula. We show that lattice gauge theories in any spatial dimension can be simulated using O~(T3/2N3/2Λ/ε1/2)\tilde{O}(T^{3/2}N^{3/2}Λ/ε^{1/2})O~(T3/2N3/2Λ/ε1/2) T gates, where NNN is the number of lattice sites, ΛΛΛ is the bosonic gauge field truncation, and TTT is the simulation time.
The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on N / 2 physical sites with coupling constant x − 1 / 2 and electric field cutoff x − 1 / 2 Λ can be simulated on a quantum computer for time 2 x T using a number of T -gates or CNOTs in O ~ ( N 3 / 2 T 3 / 2 x Λ ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.
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.
Abstract Quantum simulations of lattice gauge theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss’ law constraint to correct one-qubit errors for a Z2{{\mathbb{Z}}}_{2}Z2​ Z 2 or truncated U(1) LGT in 1+1 and 2+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss’ law, our circuits are fault tolerant and the overall error correction scheme outperforms a naïve application of the [5, 1, 3] code. The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.
In recent years, the quantum computing method has been used to address the sign problem in traditional Monte Carlo lattice gauge theory (LGT) simulations. We propose that the Coulomb gauge (CG) should be used in quantum simulations of LGT. Since the redundant degrees of freedom of gauge fields can be eliminated in CG, the Hamiltonian in CG does not need to be gauge invariant, allowing the gauge field to be discretized naively. Then the discretized gauge fields and fermion fields should be placed on momentum and position lattices, respectively. Under this scheme, the CG condition and Gauss’s law can be conveniently preserved by solving for the polarization vectors from algebraic equations. Furthermore, we discuss the mapping of gauge fields to qubits and evaluate the associated qubit and gate cost of this framework. We point out that this formalism is efficient for simulating hadron scattering processes on future fault-tolerant quantum computers. Finally, we calculate the vacuum expectation value of the U(1) plaquette operator and the Wilson loop on a classical device to test the performance of our discretization scheme.
Quantum simulations of lattice gauge theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian.We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss' law constraint to correct one-qubit errors for a Z 2 or truncated U(1) LGT in 1+1 and 2+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss' law, our circuits are fault tolerant and the overall error correction scheme outperforms a naïve application of the [5, 1, 3] code.The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.