Pauli measurements are not optimal for single-copy tomography
Jayadev Acharya, Abhilash Dharmavarapu, Yuhan Liu, Nengkun Yu
TL;DR
This work analyzes quantum state tomography under measurement constraints, focusing on Pauli measurements. It introduces a measurement information channel (MIC) framework to quantify how well a given measurement set distinguishes quantum states, and uses Assouad's method to derive a lower bound via a measurement-dependent hard instance. The authors establish a first nontrivial adaptive lower bound for Pauli tomography, showing $n = \Omega\left(\frac{9.118^{N}}{\varepsilon^{2}}\right)$ copies are necessary, and they provide a refined upper bound of $n = O\left(\frac{10^{N}}{\varepsilon^{2}}\right)$, yielding a separation from structured POVMs that achieve $n = \tilde{O}\left(\frac{8^{N}}{\varepsilon^{2}}\right)$. The framework also yields plug-and-play lower bounds for general $k$-outcome measurements and extends to adaptive tomography with measurement constraints, underscoring the broad applicability and practical implications for near-term quantum devices.
Abstract
Quantum state tomography is a fundamental problem in quantum computing. Given $n$ copies of an unknown $N$-qubit state $ρ\in \mathbb{C}^{d \times d},d=2^N$, the goal is to learn the state up to an accuracy $ε$ in trace distance, with at least probability 0.99. We are interested in the copy complexity, the minimum number of copies of $ρ$ needed to fulfill the task. Pauli measurements have attracted significant attention due to their ease of implementation in limited settings. The best-known upper bound is $O(\frac{N \cdot 12^N}{ε^2})$, and no non-trivial lower bound is known besides the general single-copy lower bound $Ω(\frac{8^n}{ε^2})$, achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of $O(\frac{10^N}{ε^2})$. To complement it with a lower bound of $Ω(\frac{9.118^N}{ε^2})$, which holds under adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage over prior methods is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements. Moreover, we connect the copy-complexity lower bound to the eigenvalues of the measurement information channel, which governs the measurement's capacity to distinguish states. To demonstrate the generality of the new framework, we obtain tight-bounds for adaptive quantum tomography with $k$-outcome measurements, where we recover existing results and establish new ones.