Succinct Fermion Data Structures

Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of $F$ fermions in $M$ modes. An information theoretic minimum of $\mathcal{I}:=\lceil\log \binom{M}{F}\rceil$ qubits is required for such systems, a bound we nearly match over the entire parameter regime. (1) Our first construction uses $\mathcal I+o(\mathcal I)$ qubits when $F=o(M)$, and allows rotations generated by creation-annihilation operators in $O(\mathcal I)$ gates and $O(\log M \log \log M)$ depth. (2) Our second construction uses $\mathcal I+O(1)$ qubits when $F=\Theta(M)$, and allows rotations generated by creation-annihilation operators in $O(\mathcal I^3)$ gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.