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

TL;DR

This paper presents an optimal fermion-to-qubit mapping based on ternary trees, enabling efficient learning of reduced quantum states with fewer repetitions, advancing quantum simulation techniques.

Contribution

It introduces a novel ternary-tree fermion-to-qubit mapping that is proven to be optimal and applies it to improve methods for learning reduced density matrices.

Findings

Mapping acts on rom 1 to log_3(2n+1) qubits per Majorana operator.

Allows determination of all k-fermion RDM elements with rom (2n+1)^k to \u00a0rom 3^k circuit repetitions.

Improves efficiency over existing schemes for qubit RDM determination.

Abstract

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.