Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping

Abstract

We devise a deterministic quantum algorithm to produce antisymmetric states of single-particle orbitals in the first quantization mapping. Unlike sorting-based antisymmetrization algorithms, which require ordered input states and high Clifford-gate overhead, our approach initializes the state of each particle independently. For a system of $\hat{I}\cdot$ particles and $N$ single-particle states, our algorithm prepares antisymmetrized states of non-trivial localized (e.g., Hartree-Fock) orbitals using $O(\hat{I}{\cdot }^2\sqrt{N})$ $T$-gates, outperforming alternative algorithms when $\hat{I}\cdot \lesssim \sqrt{N}$. To achieve such scaling, we require $O(\sqrt{N})$ dirty ancilla qubits for intermediate calculations. Knowledge of the single-particle states to be antisymmetrized can be leveraged to further improve the efficiency of the circuit, and a measurement-based variant reduces gate cost by roughly a factor of two. We show example circuits for two- and three-particle systems and discuss the generalization to an arbitrary number of particles. For a specific three-particle example, we decompose the circuit into Clifford$+T$ gates and study the impact of noise on the prepared state.

Updated cost analysis, accepted at Quantum, 18 pages, 13 figures, 4 tables