Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA
PsiQuantum, Palo Alto, CA 94304, USA
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Abstract
We present a simple but general framework for constructing quantum circuits that implement the multiply-controlled unitary $text{Select}(H) := sum_ell |ellranglelangleell|otimes H_ell$, where $H = sum_ell H_ell$ is the Jordan-Wigner transform of an arbitrary second-quantised fermionic Hamiltonian. $text{Select}(H)$ is one of the main subroutines of several quantum algorithms, including state-of-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 $text{Select}(H)$ requires no ancilla qubits and uses $mathcal{O}(n)$ Clifford+T gates, with the Clifford gates applied in $mathcal{O}(log^2 n)$ 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.
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Source: https://quantum-journal.org/papers/q-2021-01-12-380/
