Abstract
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 onN/2physical sites with coupling constantx−1/2and electric field cutoffx−1/2Λcan be simulated on a quantum computer for time2xTusing a number ofT-gates or CNOTs inO~(N3/2T3/2xΛ)for fixed operator error. This scaling with the truncationΛis 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.
Highlights
The 20th century saw tremendous success in discovering and explaining the properties of Nature at the smallest accessible length scales, with just one of the crowning achievements being the formulation of the Standard Model
We report explicit algorithms that could be used for time evolution in the lattice Schwinger model, a gauge theory in one spatial dimension that shares qualitative features with quantum chromodynamics (QCD) [66]— making it a testbed for the gauge theories that are so central to the Standard Model
Due to the need to work with hardware constraints that will change over time, we provide algorithms that could be realized in the Noisy Intermediate Scale Quantum (NISQ) era, followed by efficient algorithms suitable for fault-tolerant computers where the most costly operation has transitioned from entangling gates (CNOT gates), to non-Clifford operations (T -gates)
Summary
The 20th century saw tremendous success in discovering and explaining the properties of Nature at the smallest accessible length scales, with just one of the crowning achievements being the formulation of the Standard Model. While strides are being made in developing the theory for quantum simulation of gauge field theories [3, 4, 6, 8, 19, 27, 34, 44, 46, 53,54,55,56, 60, 62, 64, 65, 71, 78, 79, 86, 87], much remains to be explored in terms of developing explicit gate implementations that are scalable and can surpass what is possible with classical machines. We discuss how to take advantage of the geometric locality of the lattice Schwinger model Hamiltonian may be used in simulating time evolution or in estimating expectation values of observables in Section 8 before concluding
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