Exponentially faster implementations of Select(H) for fermionic Hamiltonians

Abstract

We present a simple but general framework for constructing quantum circuits that implement the multiply-controlled unitary Select(H) := Σ l |lihl| Hl, where H = P l Hl is the Jordan-Wigner transform of an arbitrary second-quantised fermionic Hamiltonian. Select(H) is one of the main subroutines of several quantum algorithms, including stateof- the-art techniques for Hamiltonian simulation. If each term in the second-quantised Hamiltonian involves at most k spin-orbitals and k is a constant independent of the total number of spin-orbitals n (as is the case for the majority of quantum chemistry and condensed matter models considered in the literature, for which k is typically 2 or 4), our implementation of Select(H) requires no ancilla qubits and uses O(n) Clifford+T gates, with the Clifford gates applied in O(log2 n) layers and the T gates in O(log n) layers. This achieves an exponential improvement in both Clifford- and T-depth over previous work, while maintaining linear gate count and reducing the number of ancillae to zero.