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

Quantum 4, 276 (2020).

https://doi.org/10.22331/q-2020-06-04-276

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.

Source Quantum | The open journal for quantum science Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning