Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang1, Amir Kalev2, Wojciech Mruczkiewicz1, and Hartmut Neven1

1Google Research, Venice, CA 90291
2Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA

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We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an $n$-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on $\lceil \log_3(2n+1)\rceil$ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than $\log_3(2n)$ qubits on average. We apply it to the problem of learning $k$-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all $k$-fermion RDMs in parallel, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim (2n+1)^k \epsilon^{-2}$ times. This result is based on a method we develop here that allows one to determine individual elements of all $k$-qubit RDMs in parallel, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim 3^k \epsilon^{-2}$ times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

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