Low-depth simulations of fermionic systems on square-grid quantum hardware

Manuel G. Algaba, P. V. Sriluckshmy, Martin Leib, and Fedor Šimkovic IV

IQM, Nymphenburgerstr. 86, 80636 Munich, Germany

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We present a general strategy for mapping fermionic systems to quantum hardware with square qubit connectivity which yields low-depth quantum circuits, counted in the number of native two-qubit fSIM gates. We achieve this by leveraging novel operator decomposition and circuit compression techniques paired with specifically chosen low-depth fermion-to-qubit mappings and allow for a high degree of gate cancellations and parallelism. Our mappings retain the flexibility to simultaneously optimize for qubit counts or qubit operator weights and can be used to investigate arbitrary fermionic lattice geometries. We showcase our approach by investigating the tight-binding model, the Fermi-Hubbard model as well as the multi-orbital Hubbard-Kanamori model. We report unprecedentedly low circuit depths per single Trotter layer with up to a $70 \%$ improvement upon previous state-of-the-art. Our compression technique also results in significant reduction of two-qubit gates. We find the lowest gate-counts when applying the XYZ-formalism to the DK mapping. Additionally, we show that our decomposition and compression formalism produces favourable circuits even when no native parameterized two-qubit gates are available.

Simulating fermions like electrons or nucleons using qubits, which show different commutation relations require a transformation between these two kind of entities called fermion-to-qubit mappings. These mappings may transform local fermionic operators into non-local qubit gates. For avoiding this issue different local fermion-to-qubit mappings have been created. In this work, we describe a novel family of fermion-to-qubit mappings that intertwines very well with the XYZ decomposition and we show that the latter can optimally decompose fermionic hopping operators into qubit gates. These improvements upon previous literature allow us to find the shallowest quantum circuits ever described for simulating a Trotter step of the Fermi-Hubbard model, which is a key model in condensed matter physics and high temperature superconductivity. We extend this work to other non-trivial models that allow richer physics, i.e. the Hubbard-Kanamori Hamiltonian.

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[3] Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, "Quantum Simulations of Hadron Dynamics in the Schwinger Model using 112 Qubits", arXiv:2401.08044, (2024).

[4] P. V. Sriluckshmy, Vicente Pina-Canelles, Mario Ponce, Manuel G. Algaba, IV Fedor Šimkovic, and Martin Leib, "Optimal, hardware native decomposition of parameterized multi-qubit Pauli gates", Quantum Science and Technology 8 4, 045029 (2023).

[5] Yu-An Chen, Alexey V. Gorshkov, and Yijia Xu, "Error-correcting codes for fermionic quantum simulation", SciPost Physics 16 1, 033 (2024).

[6] Georg Bergner, Masanori Hanada, Enrico Rinaldi, and Andreas Schafer, "Toward QCD on Quantum Computer: Orbifold Lattice Approach", arXiv:2401.12045, (2024).

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