Stabilizer codes for Heisenberg-limited many-body Hamiltonian estimation

TL;DR

This paper investigates how stabilizer quantum error correcting codes can enable Heisenberg-limited many-body Hamiltonian estimation under noise, demonstrating specific code families that achieve optimal scaling with probe number and time.

Contribution

It introduces and analyzes stabilizer codes that attain optimal scaling in noisy quantum Hamiltonian estimation, including new code families and no-go theorems for certain stabilizer structures.

Findings

Thin surface codes achieve (nt)^{-1} scaling.

Quantum Reed--Muller codes achieve (n^2 t)^{-1} scaling.

Shor codes achieve (n^3 t)^{-1} scaling.

Abstract

Estimating many-body Hamiltonians has wide applications in quantum technology. By allowing coherent evolution of quantum systems and entanglement across multiple probes, the precision of estimating a fully connected $k$-body interaction can scale up to $(n^kt)^{-1}$, where $n$ is the number of probes and $t$ is the probing time. However, the optimal scaling may no longer be achievable under quantum noise, and it is important to apply quantum error correction in order to recover this limit. In this work, we study the performance of stabilizer quantum error correcting codes in estimating many-body Hamiltonians under noise. When estimating a fully connected $ZZZ$ interaction under single-qubit noise, we showcase three families of stabilizer codes -- thin surface codes, quantum Reed--Muller codes and Shor codes -- that achieve the scalings of $(nt)^{-1}$, $(n^2t)^{-1}$ and $(n^3t)^{-1}$, respectively, all of which are optimal with $t$. We further discuss the relation between stabilizer structure and the scaling with $n$, and identify several no-go theorems. For instance, we find codes with constant-weight stabilizer generators can at most achieve the $n^{-1}$ scaling, while the optimal $n^{-3}$ scaling is achievable if and only if the code bears a repetition code substructure, like in Shor code.