Quantum 1-PCA with Pauli Measurements in Nearly Linear Time

Abstract

We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $\lambda$ and spectral gap at least $\Delta>0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \varepsilon$ with high probability using $\tilde{O}\left( {2^n \cdot \eta^2} / {\Delta^3 \varepsilon^3}\right)$ copies and $\tilde{O}((2^n/\Delta \varepsilon) \cdot \operatorname{poly}(\eta/\Delta\varepsilon))$ time, where $\eta = \max (1 - \lambda, \varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\varepsilon = \Omega (\eta)$, our runtime and copy complexity become $\tilde{O} (2^n / \varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $\eta = 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.