Measuring the largest coefficients of a quantum state

TL;DR

This paper presents a hierarchical algorithm for efficiently identifying the largest Pauli coefficients of an unknown quantum state, enabling targeted characterization without full tomography.

Contribution

The authors introduce a novel tree-based method for estimating dominant Pauli coefficients, improving efficiency for structured quantum states.

Findings

Algorithm effectively reconstructs dominant Pauli components in sparse states.

Sample complexity bounds are derived for the coefficient estimation process.

Numerical simulations show competitive performance with existing methods.

Abstract

We introduce a hierarchical algorithm for identifying the largest Pauli coefficients of an unknown $n$-qubit quantum state. The algorithm traverses a prefix-based tree whose nodes represent partial sums of squared Pauli coefficients, always expanding branches with the largest estimated weight and discarding the rest. Node weights are estimated using Bell sampling on two copies of the state, or alternatively via SWAP tests on subsystems. We analyze the sample complexity of each node estimation and derive bounds on the total number of nodes expanded as a function of the desired number of coefficients and the state's purity. For states admitting a sparse representation in the Pauli basis, the algorithm achieves a good reconstruction of the dominant components without requiring full state tomography. We validate the method with numerical simulations on Pauli-singleton states and random stabilizer states, showing that the algorithm's performance is competitive with other methods for structured states. Our work addresses an open problem in Pauli sampling and provides a practical tool for the targeted characterization of structured quantum states.