On estimating the quantum $\ell_α$ distance

TL;DR

This work analyzes the computational complexity of estimating the quantum ℓ_α distance ${\mathrm{T}_α}(\rho_0,\rho_1)$, a Schatten-based generalization linking to the trace distance for $α\!=1$. It provides a rank-independent quantum estimator for constant $α>1$, achieving poly$(n)$ time and poly$(1/ε)$ query complexity by leveraging efficiently computable uniform polynomial approximations of signed powers within Quantum Singular Value Transformation. A central insight is the dichotomy in QSD_α: for constant $α>1$ with the constant-above regime, the problem is BQP-complete, while for near-unity $α$ it is QSZK-complete, implying a fundamental hardness gap and connecting to polarization techniques. The paper also develops rank-dependent inequalities between ${\mathrm{T}_α}$ and the trace distance, enabling reductions that establish hardness and lower bounds. Collectively, these results advance both algorithmic capabilities for quantum state distance estimation and a nuanced understanding of the complexity landscape across Schatten-parameterized distances, with implications for quantum state discrimination and subroutines in quantum information processing.

Abstract

We study the computational complexity of estimating the quantum $\ell_α$ distance ${\mathrm{T}_α}(ρ_0,ρ_1)$, defined via the Schatten $α$-norm $\|A\|_α = \mathrm{tr}(|A|^α)^{1/α}$, given $\operatorname{poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. This quantity serves as a lower bound on the trace distance for $α> 1$. For any constant $α> 1$, we develop an efficient rank-independent quantum estimator for ${\mathrm{T}_α}(ρ_0,ρ_1)$ with time complexity $\operatorname{poly}(n)$, achieving an exponential speedup over the prior best results of $\exp(n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). Our improvement leverages efficiently computable uniform polynomial approximations of signed positive power functions within quantum singular value transformation, thereby eliminating the dependence on the rank of the quantum states. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $α$-norm (QSD$_α$), which involves deciding whether ${\mathrm{T}_α}(ρ_0,ρ_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of constant $α> 1$ and $α=1$: - For any $1+Ω(1) \leq α\leq O(1)$, QSD$_α$ is $\mathsf{BQP}$-complete. - For any $1 \leq α\leq 1+\frac{1}{n}$, QSD$_α$ is $\mathsf{QSZK}$-complete, implying that no efficient quantum estimator for $\mathrm{T}_α(ρ_0,ρ_1)$ exists unless $\mathsf{BQP} = \mathsf{QSZK}$. The hardness results follow from reductions based on new rank-dependent inequalities for the quantum $\ell_α$ distance with $1\leq α\leq \infty$, which are of independent interest.

Theorems & Definitions (64)

• theorem 1.1: Quantum estimator for ${\mathrm{T}_\alpha}$, informal
• theorem 1.2: Computational hardness of
• theorem 1.3: ${\mathrm{T}_\alpha}$ vs. $\mathrm{T}$, informal
• definition 2.1: Trace distance
• lemma 1: Trace distance vs. Uhlmann fidelity, adapted from FvdG99
• definition 2.2: Quantum $\ell_\alpha$ distance and its powered version
• lemma 2: Triangle inequality for ${\mathrm{T}_\alpha}$, adapted from AS17
• proposition 2.3: ${\mathrm{T}_\alpha}$ vs. powered ${\mathrm{T}_\alpha}$
• proof
• definition 2.4: Quantum State Distinguishability Problem, QSD, adapted from Wat02