Quantum Algorithms for Simulating the Lattice Schwinger Model

Alexander F. Shaw1,5, Pavel Lougovski1, Jesse R. Stryker2, and Nathan Wiebe3,4

1Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.
2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, U.S.A.
3Department of Physics, University of Washington, Seattle, WA 98195, U.S.A.
4Pacific Northwest National Laboratory, Richland, WA 99354, U.S.A.
5Department of Physics, University of Maryland, College Park, Maryland 20742, U.S.A.

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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 $x^{-1/2}\Lambda$ can be simulated on a quantum computer for time $2xT$ using a number of $T$-gates or CNOTs in $\widetilde{O}( N^{3/2} T^{3/2} \sqrt{x} \Lambda )$ for fixed operator error. This scaling with the truncation $\Lambda$ 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.

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[26] Bárbara Andrade, Zohreh Davoudi, Tobias Graß, Mohammad Hafezi, Guido Pagano, and Alireza Seif, "Engineering an Effective Three-spin Hamiltonian in Trapped-ion Systems for Applications in Quantum Simulation", arXiv:2108.01022.

[27] Ronak Desai, Yuan Feng, Mohammad Hassan, Abhishek Kodumagulla, and Michael McGuigan, "Z3 gauge theory coupled to fermions and quantum computing", arXiv:2106.00549.

[28] Ying Chen, Yunheng Ma, and Shun Zhou, "Quantum Simulations of the Non-Unitary Time Evolution and Applications to Neutral-Kaon Oscillations", arXiv:2105.04765.

[29] M. Sohaib Alam, Stuart Hadfield, Henry Lamm, and Andy C. Y. Li, "Quantum Simulation of Dihedral Gauge Theories", arXiv:2108.13305.

The above citations are from Crossref's cited-by service (last updated successfully 2021-09-23 06:54:58) and SAO/NASA ADS (last updated successfully 2021-09-23 06:54:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

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