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

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 using the ternary-tree mapping one can determine the elements of all $k$-fermion RDMs, 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 the elements of all $k$-qubit RDMs, 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.