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 $η$ particles and $N$ single-particle states, our algorithm prepares antisymmetrized states of non-trivial localized (e.g., Hartree-Fock) orbitals using $O(η^2\sqrt{N})$ $T$-gates, outperforming alternative algorithms when $η\lesssim \sqrt{N}$. 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.

Figures (13)

• Figure 1: Quantum circuit that prepares the antisymmetric state on $\eta$ particles from the antisymmetric state on $\eta-1$ particles. $\eta-1$ ancilla qubits are prepared in the $Y_{\eta-1}$ state (see Appendix \ref{['app:ancilla_prep']}). Controlling on the state of the ancillae, we perform swap operations between each of the previous $\eta-1$ particles and the $\eta$-th particle (blue shaded region). To uncompute the ancillae, we detect if a swap occurred by acting the operator $U_\eta^\dagger$ on each of the $\eta-1$ previous particles and controlling on the particle being in the state $\ket{0}^{\otimes k}$. The multi-controlled $X$-gates that uncompute the ancilla qubits (red shaded region) can be performed in parallel. Finally, we restore the state of the first $\eta-1$ particles by acting $U_\eta$ on each.
• Figure 2: Quantum circuit for generating the antisymmetric state of two particles in single-particle states.
• Figure 3: Same as in Fig. \ref{['fig:asymN']}, but for three particles starting with the previously computed antisymmetric state of two particles (see Fig. \ref{['fig:asym2']}). The swaps are controlled on either ancilla $a_1$ or $a_2$ being in state $\ket{1}$, depending on whether the swap involves particle $p_1$ or $p_2$, respectively.
• Figure 4: Measurement-based antisymmetrization of two particles: (a) general circuit, which requires a phase gate in case the ancilla $a_0$ is measured in state $\ket{1}$. The requisite phase gate $\mathcal{P}(U)$ is defined in panel (b).
• Figure 5: Bottom: Total number of $T$-gates required to antisymmetrize a system of $\eta$ particles with $k=19$ qubits per particle. The solid black points correspond to the method of Berry et al.Berry2018a with odd-even mergesort while the red points correspond to the non-measurement-based implementation of our algorithm. Top: The percentage of the total $T$-count associated with preparing the ancilla qubits (see text). We assume that the additional $O(1)$$T$-gate cost per integer comparator required by Berry et al.'s algorithm is zero.