Pauli Measurements Are Near-Optimal for Pure State Tomography

TL;DR

This work shows that pure $n$-qubit state tomography can be achieved with nonadaptive Pauli measurements at near-optimal copy complexity. It introduces a Frobenius-distance estimator that reduces distance estimation to nonadaptive Pauli queries and leverages learning-theoretic tools (Rademacher norms) to control error with $M=\tilde{O}(2^n/\varepsilon^2)$ queries, enabling a bottom-up binary-tree tomography. A key innovation is a Gluing framework that combines leaf-state estimates with optimal coefficients while bounding the Frobenius distance growth, yielding an overall algorithm whose total copy complexity is $\widetilde{O}(2^n\,\mathrm{poly}(n)\,\log(1/\delta)/\varepsilon)$ and runtime polynomial in $2^n$ and $1/\varepsilon$. These results establish that Pauli measurements can be near-optimal for pure-state tomography, matching information-theoretic lower bounds up to polylogarithmic factors and offering a practical, scalable tomography approach for large quantum systems.

Abstract

We give an algorithm for pure state tomography with near-optimal copy complexity using single-qubit measurements. Specifically, given $\widetilde{O}(2^n/ε)$ copies of an unknown pure $n$-qubit state $\lvertψ\rangle$, the algorithm performs only \textit{nonadaptive Pauli measurements}, runs in time $\mathrm{poly}(2^n,1/ε)$, and outputs $\lvert \widehatψ \rangle$ that has fidelity $1-ε$ with $\lvert ψ\rangle$ with high probability. This improves upon the previous best copy complexity bound of $\widetilde{O}(3^n/ε)$.

Theorems & Definitions (30)

• Theorem 1
• Lemma 2: Hoeffding for ${[-1,1]}$ random variables
• Lemma 3: Bernstein for ${[0,B]}$ random variables
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof