On estimating Schatten norm and power distances between quantum states

Abstract

We study the computational complexity of estimating the quantum Schatten $\hat{I}\pm$-norm distance ${\mathrm{T}}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$, given $\mathrm{poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $\ddot{I}{}_0$ and $\ddot{I}{}_1$. This quantity serves as a lower bound on the trace distance and, for $\hat{I}\pm >1$, is interchangeable with its powered version $\hat{I}{›}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$. For any constant $\hat{I}\pm >1$, we develop an efficient rank-independent quantum estimator for ${\mathrm{T}}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ with time complexity $\mathrm{poly}(n)$, achieving an exponential speedup over the prior best results of $\exp (n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). When $0<\hat{I}\pm <1$ is a constant, the quantum Schatten $\hat{I}\pm$-power distance $\hat{I}{›}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ becomes a distance metric. Accordingly, we provide a rank-efficient quantum estimator for this quantity. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $\hat{I}\pm$-norm (QSD ${}_{\hat{I}}\pm$), which involves deciding whether ${\mathrm{T}}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of $\hat{I}\pm >1$ and $0<\hat{I}\pm \le 1$: 1. For any constant $\hat{I}\pm >1$, QSD${}_{\hat{I}}\pm$ is $\mathsf{BQP}$-complete. 2. For any $1\le \hat{I}\pm (n)\le 1+\mathrm{negl}(n)$, QSD${}_{\hat{I}}\pm$ is $\mathsf{QSZK}$-complete, implying that no efficient quantum estimator for ${\mathrm{T}}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ exists unless $\mathsf{BQP}=\mathsf{QSZK}$. This $\mathsf{QSZK}$-hardness result also extends to the promise problem defined by $\hat{I}{›}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ for constant $0<\hat{I}\pm <1$. The hardness results follow from reductions based on new rank-dependent inequalities for ${\mathrm{T}}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ when $1\le \hat{I}\pm \le \infty$ and for $\hat{I}{›}_{\hat{I}}\pm (\ddot{I}{}_0,\ddot{I}{}_1)$ when $0<\hat{I}\pm <1$, which are of independent interest.

50 pages, 1 table, 4 algorithms. v3: Added results on estimating the Schatten power distance of order between 0 and 1; reorganized sections and made minor changes. v2: Minor changes; corrected parameters in the proofs of Theorems 4.5 and A.1