First-quantized quantum simulation with non-local potentials by matrix-product-state encoding

arXiv:2610.00521v1 Announce Type: new Abstract: Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of $O(\mathrm{deg}(V_{\mathrm{loc}}) n^{\mathrm{deg}(V_{\mathrm{loc}})} + 8Lnn_{\mathrm{atom}}\chi^2 + 2n^2)$, where $\mathrm{deg}(V_{\mathrm{loc}})$ is the degree of the polynomial local potential, $n_{\mathrm{atom}}$ is the number of atoms in the system, $L$ is the maximum number of projector functions across all atoms, and $n$ is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.

Abstract:Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of $O(\mathrm{deg}(V_{\mathrm{loc}}) n^{\mathrm{deg}(V_{\mathrm{loc}})} + 8Lnn_{\mathrm{atom}}\chi^2 + 2n^2)$, where $\mathrm{deg}(V_{\mathrm{loc}})$ is the degree of the polynomial local potential, $n_{\mathrm{atom}}$ is the number of atoms in the system, $L$ is the maximum number of projector functions across all atoms, and $n$ is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.

arXiv:2610.00521 [quant-ph]

(or arXiv:2610.00521v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00521 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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First-quantized quantum simulation with non-local potentials by matrix-product-state encoding

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