Antisymmetrization of composite fermionic states for quantum simulations of nuclear reactions in first-quantization mapping

Abstract

I present a first-quantization deterministic algorithm for antisymmetrizing a spatially separated target-projectile system containing $N_T$ and $N_p$ identical fermions, respectively. The method constructs a fully antisymmetric wavefunction from the product of two independently antisymmetrized many-body states, each of which may be a superposition of Slater determinants. The algorithm uses a Dicke-state ancilla register that coherently encodes all one-particle exchange channels between the two subsystems, and, crucially, requires only single-particle swaps to generate the full antisymmetric structure. A total of $O(N_T N_p)$ single-particle exchanges are needed, with up to $N_p$ of them implemented in parallel, if an additional $N_p$ ancillae are used. The correct fermionic phase is incorporated through application of $Z$ gates on $N_T$ ancillae, after which the ancilla register is efficiently uncomputed using a compact sequence of controlled operations. This construction provides a nontrivial and scalable protocol for preparing fully antisymmetric states in reaction and scattering simulations, significantly expanding the range of systems that can be addressed with first-quantized quantum algorithms.

Figures (2)

• Figure 1: Quantum circuit for antisymmetrizing the two-particle target and projectile states. Starting from independently antisymmetrized two-fermion states for the target and projectile and four ancilla qubits prepared in the Dicke state $D_2^4$, the blue-filled region performs all single-particle exchanges between the subsystems in parallel using two additional ancillae not shown here, with appropriate $Z$ gates providing the correct antisymmetrization phase. The swaps entangle projectile states with ancillae in state $\ket{1}$ and target states with ancillae in state $\ket{0}$, generating all required permutations. The uncomputation stage, shown in the pink-filled region, disentangles the ancilla register using $N_T+N_p$ controlled-not operations.
• Figure 2: Single-particle swap operations generated by the antisymmetrization algorithm for (a) a system with two projectile particles and three target particles, and (b) a system with three particles in both the target and the projectile. Additional ancilla qubits, not shown here, may be introduced to simplify the multi-controlled swaps or to allow more swaps to be executed in parallel.