We investigate the computational complexity of estimating the operator norm distance ${\rm T}_{\infty}(ρ_0,ρ_1)$, defined via the operator norm $\|A\|_{\infty} = σ_{\max}(A)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
1. When one state is pure, we establish an optimal quantum estimator using $Θ(1/ε)$ queries to the state-preparation circuits. Consequently, for constant additive error, say $ε=1/5$, our estimator runs in ${\rm poly}(n)$ time. Since the operator norm distance ${\rm T}_{\infty}(|ψ\rangle\!\langleψ|,ρ)$ is exactly half of the trace distance ${\rm T}(|ψ\rangle\!\langleψ|,ρ)$, our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with ${\rm rank}(ρ)$, which can be $\exp(n)$ in general.
2. For general quantum states, we also provide a quantum estimator using $\widetilde{O}(1/ε^{3/2})$ queries to the state-preparation circuits, which shows that the corresponding promise problem is ${\sf BQP}$-complete and improves the ${\sf QMA}$ upper bound sketched by Liu and Wang (ESA 2025). Together with an $Ω(1/ε)$ quantum query complexity lower bound, this leaves only square-root room for improvement.
The key intuition behind our estimators is that, when one state is pure, the pure state $|ψ\rangle$ has overlap at least $1/2$ with the top unit eigenvector of $|ψ\rangle\!\langleψ|-ρ$, reflecting a structural feature specific to the operator norm distance.
Journal reference
Proceedings of the 34th Annual European Symposium on Algorithms (ESA 2026), pp. 71:1-71:18, 2026
We investigate the computational complexity of estimating the operator norm distance T∞(ρ0,ρ1), defined via the operator norm ∥A∥∞=σmax(A), given poly(n)-size state-preparation circuits of n-qubit quantum states ρ0 and ρ1. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
When one state is pure, we establish an optimal quantum estimator using Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε=1/5, our estimator runs in poly(n) time. Since the operator norm distance T∞(∣ψ⟩⟨ψ∣,ρ) is exactly half of the trace distance T(∣ψ⟩⟨ψ∣,ρ), our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general.
For general quantum states, we also provide a quantum estimator using O(1/ε3/2) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε) quantum query complexity lower bound, this leaves only square-root room for improvement.
The key intuition behind our estimators is that, when one state is pure, the pure state ∣ψ⟩ has overlap at least 1/2 with the top unit eigenvector of ∣ψ⟩⟨ψ∣−ρ, reflecting a structural feature specific to the operator norm distance.
The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error ε requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of ρ0 and ρ1. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure.
In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of trA(∣ψ0⟩⟨ψ1∣) (or its inverse) to the reference register R of the purification ∣ψ1⟩ (or ∣ψ0⟩) on the registers A and R, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.
We settle the problem of estimating the trace distance and (square root) fidelity between n-qubit pure quantum states to within additive error ε, given their independent samples, which was raised as an open question by Wang (IEEE Trans. Inf. Theory 2024). This is achieved by a quantum algorithm with optimal sample complexity Θ(1/ε2), improving the long-standing folklore with sample complexity O(1/ε4). At the heart of our algorithm is a samplized phase estimation of the product of two Householder reflections. This is realized by an improved (multi-)samplizer for pure states, through which any quantum query algorithm using Q queries to the reflection operator I−2∣ψ⟩⟨ψ∣ can be converted to a δ-close (in the diamond norm distance) quantum sample algorithm using Θ(Q2/δ) samples of the state ∣ψ⟩. This samplizer for pure states is also shown to be optimal.
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq ) for an n-qubit mixed quantum state ρ, given its state-preparation circuit of size poly(n ). This quantity is closely related to and often interchangeable with the Tsallis entropywhere q = 1 corresponds to the von Neumann entropy. For any non-integer q ≥ 1 + Ω(1), we provide a quantum estimator for Sq (ρ ) with time complexity poly(n), exponentially improving the prior best results of exp(n ) due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (ESA 2024). Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation.
We determine the optimal query complexity of ground-state preparation to trace-distance error ε when an energy threshold in the spectral gap is known. Let UH be an α-block-encoding of a Hamiltonian with unique ground state ∣ψ0⟩, and suppose ∣⟨ψ0∣UI∣0⟩∣≥γ for a state-preparation oracle UI. The threshold lies at least Δ/2 above the ground-state energy and at least Δ/2 below every excited-state energy. We give two algorithms that prepare a state within trace distance ε of the ground state. One uses O((α/Δ)(γ−1+log(1/ε))) calls to UH in expectation; the other uses O((α/(γΔ))log(1/ε)) calls to UH in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to UI are O(1/γ) in expectation and O(γ−1log(1/ε)) in the worst case. On (N+1)-dimensional systems, these UI bounds are also optimal when the expected or worst-case count of UH calls, respectively, is o((α/Δ)N). Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq) for annqubit mixed quantum state ρ, given its state-preparation circuit of size poly(n). This quantity is closely related to and often interchangeable with the Tsallis entropy Sq(ρ) = 1—tr(ρq)/q—1, whereq= 1 corresponds to the von Neumann entropy. For any non-integerq≥ 1 + Ω(1), we provide a quantum estimator for Sq(ρ) with time complexity poly(n), exponentially improving the prior best results of exp(n)due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024)| Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (TIT 2025). Our speedup is achieved by introducing efficiently computableuniform approximationsof positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case ofq= 1 and constantq> 1 in the computational complexity of the Quantumq-Tsallis Entropy Difference Problem (TsallisQEDq), particularly deciding whether the difference Sq(ρ0)−Sq(ρ1) is at least 0.001 or at most −0.001: • For any 1 + Ω (1) ≤q≤ 2, TsallisQEDqis BQP-complete, which implies that Purity Estimation is also BQP-complete. • For any 1≤q≤1+1/n— 1, TsallisQED q is QSZK-hard, leading to hardness of approximating the von Neumann entropy because Sq(ρ) ≤ S (ρ), as long as BQPQSZK. The hardness results are derived from reductions based on new inequalities for the quantumq-Jensen–(Shannon–)Tsallis divergence with 1 ≤q≤ 2, which are of independent interest.
Measuring the distinguishability between quantum states is a basic problem in quantum information theory. In this paper, we develop optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure states to within additive error$\varepsilon using\Theta (1/\varepsilon)queriestotheirstate−preparationcircuits,quadraticallyimprovingthelong−standingfolkloreO(1/\varepsilon ^{2}) $. At the heart of our construction, is an algorithmic tool for quantum square root amplitude estimation, which generalizes the well-known quantum amplitude estimation.