Exponentially faster implementations of Select(H) for fermionic Hamiltonians

Kianna Wan

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 $\text{Select}(H) := \sum_\ell |\ell\rangle\langle\ell|\otimes H_\ell$, 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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Cited by

[1] John S. Van Dyke, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preparing Bethe Ansatz Eigenstates on a Quantum Computer", PRX Quantum 2 4, 040329 (2021).

[2] Andrew J. Landahl and Benjamin C. A. Morrison, "Logical Majorana fermions for fault-tolerant quantum simulation", arXiv:2110.10280.

The above citations are from Crossref's cited-by service (last updated successfully 2021-12-07 23:17:18) and SAO/NASA ADS (last updated successfully 2021-12-07 23:17:18). The list may be incomplete as not all publishers provide suitable and complete citation data.