Quantum algorithms for simulating the Lattice Schwinger Model

The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories.

Given an elementary or high-energy quantum field theory, the best demonstration of its efficacy is showing that it correctly accounts for physics observed at longer length scales. Progress on extracting long-wavelength properties has indeed followed. For example, in Quantum ChromoDynamics (QCD), the proton is a complex hadronic bound state that is three orders of magnitude heavier than any of its constituent quarks. However, methods have been found to derive its static properties directly from the theory of quarks and gluons; using lattice QCD, calculations of the static properties of hadrons and nuclei with controlled uncertainties at precisions relevant to current experimental programs have been achieved. It is even possible with lattice QCD to study reactions of nucleons.

Particle physics, like other disciplines of physics, generically suffers from a quantum many-body problem. The Hilbert space grows exponentially with the system size to accommodate the entanglement structure of states and the vast amount of information that can be stored in them. This easily overwhelms any realistic classical-computing architecture.

Quantum computing offers an alternative approach to simulation that should avoid such sign problems by returning to a Hamiltonian point of view and leveraging controllable quantum systems to simulate the degrees of freedom we are interested in. The idea of simulation is to map the time-evolution operator of the target quantum system into a sequence of unitary operations that can be executed on the controllable quantum device: the quantum computer.

A team of researchers has given scalable, explicit digital quantum algorithms to simulate the Lattice Schwinger Model in both NISQ and fault-tolerant settings.

They provided scalable measurement schemes and algorithms to estimate observables which the cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. They 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.

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