Tensor decomposition technique for qubit encoding of maximal-fidelity Lorentzian orbitals in real-space quantum chemistry

Abstract: To simulate the real- and imaginary-time evolution of a many-electron system on a quantum computer based on the first-quantized formalism, we need to encode molecular orbitals (MOs) into qubit states for typical initial-state preparation. We propose an efficient scheme for encoding an MO as a many-qubit state from a Gaussian-type solution that can be obtained from a tractable solver on a classical computer. We employ the discrete Lorentzian functions (LFs) as a fitting basis set, for which we maximize the fidelity to find the optimal Tucker-form state to represent a target MO. For $n_{\mathrm{prod}}$ three-dimensional LFs, we provide the explicit circuit construction for the state preparation involving $\mathcal{O} (n_{\mathrm{prod}})$ CNOT gates. Furthermore, we introduce a tensor decomposition technique to construct a canonical-form state to approximate the Tucker-form state with controllable accuracy. Rank-$R$ decomposition reduces the CNOT gate count to $\mathcal{O} (R n_{\mathrm{prod}}^{1/3}).$ We demonstrate via numerical simulations that the proposed scheme is a powerful tool for encoding MOs of various quantum chemical systems, paving the way for first-quantized calculations using hundreds or more logical qubits.