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arXiv:2606.26034

Estimating Fidelity to a Reference Quantum State

Qisheng Wang

quant-phcs.CCcs.ITSubmitted Jun 24, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error $\varepsilon$. We show that the sample complexity is $O(r^2/\varepsilon^2)$ with optimal $\varepsilon$-dependence when the reference state is of rank $r$, improving the previous best $O(r^2\log^2(1/\varepsilon)/\varepsilon^4)$ due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of $Ω(r/\varepsilon^2)$, improving the previous best $Ω(r/\varepsilon+1/\varepsilon^2)$, with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most $r$ while the reference state can be arbitrary, for which the sample complexity is shown to be $O(r^2/\varepsilon^4)$. As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in Bădescu, O'Donnell, and Wright (STOC 2019).

Comments
20 pages, 1 table, 1 figure, 4 algorithms

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We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error ε\varepsilonε. We show that the sample complexity is O(r2/ε2)O(r^2/\varepsilon^2)O(r2/ε2) with optimal ε\varepsilonε-dependence when the reference state is of rank rrr, improving the previous best O(r2log⁡2(1/ε)/ε4)O(r^2\log^2(1/\varepsilon)/\varepsilon^4)O(r2log2(1/ε)/ε4) due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of Ω(r/ε2)Ω(r/\varepsilon^2)Ω(r/ε2), improving the previous best Ω(r/ε+1/ε2)Ω(r/\varepsilon+1/\varepsilon^2)Ω(r/ε+1/ε2), with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most rrr while the reference state can be arbitrary, for which the sample complexity is shown to be O(r2/ε4)O(r^2/\varepsilon^4)O(r2/ε4). As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in Bădescu, O'Donnell, and Wright (STOC 2019).
The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣{\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|F(ρ0​,ρ1​)=tr∣ρ0​​ρ1​​∣ 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\varepsilonε 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 ρ0ρ_0ρ0​ and ρ1ρ_1ρ1​. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε1/\varepsilon1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩ ⁣⟨ψ∣){\rm F}(ρ,|ψ\rangle\!\langleψ|)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∣){\rm F}(ρ, |ψ\rangle\!\langleψ|)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∣){\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)trA​(∣ψ0​⟩⟨ψ1​∣) (or its inverse) to the reference register R\sf RR of the purification ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩ (or ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩) on the registers A\sf AA and R\sf RR, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most sss, we give an estimator using O(s2/ε2)O(s^2/\varepsilon^2)O(s2/ε2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,sr,sr,s as min⁡{r,s}2/ε2\min\{r,s\}^2/\varepsilon^2min{r,s}2/ε2 up to logarithmic factors in the smaller rank. The same estimator gives the upper bound O(r(tr⁡σ)2/ε2)O(r(\operatorname{tr}\sqrtσ)^2/\varepsilon^2)O(r(trσ​)2/ε2) for a rank-rrr reference σσσ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λi∝i−αλ_i\propto i^{-α}λi​∝i−α with fixed 1<α≤21<α\le21<α≤2, as ε↓0\varepsilon\downarrow0ε↓0 with r≥Cαε−2/(α−1)r\ge C_α\varepsilon^{-2/(α-1)}r≥Cα​ε−2/(α−1), the bounds determine the complexity as Θ~α(ε−4/(α−1))\widetildeΘ_α(\varepsilon^{-4/(α-1)})Θα​(ε−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
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We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρρρ and the description of a state σσσ, the goal is to test whether ρ=σρ=σρ=σ, or whether ρρρ and σσσ are far in a given distance measure. In equivalence testing, σσσ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρCρ_{AC}=ρ_A\otimesρ_CρAC​=ρA​⊗ρC​, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε\varepsilonε in trace distance: in the single-copy measurement setting with ddd-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2)Θ(d^{3/2}/\varepsilon^2)Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity. In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σσσ of rank rrr does not benefit from adaptivity and requires Θ~(r3/2/ε)\widetildeΘ(r^{3/2}/\varepsilon)Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O~(min⁡{d3/2/ε2,d9/4/ε})\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})O(min{d3/2/ε2,d9/4/ε}) and O~(min⁡{(dAdC)3/2/ε2,dA9/4dC3/4/ε})\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})O(min{(dA​dC​)3/2/ε2,dA9/4​dC3/4​/ε}) samples, for dA≥dCd_A\geq d_CdA​≥dC​, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2\ell_2ℓ2​-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω~(1/ε2)\widetildeΩ(1/\varepsilon^2)Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.
We investigate the computational complexity of estimating the operator norm distance T∞(ρ0,ρ1){\rm T}_{\infty}(ρ_0,ρ_1)T∞​(ρ0​,ρ1​), defined via the operator norm ∥A∥∞=σmax⁡(A)\|A\|_{\infty} = σ_{\max}(A)∥A∥∞​=σmax​(A), given poly(n){\rm poly}(n)poly(n)-size state-preparation circuits of nnn-qubit quantum states ρ0ρ_0ρ0​ and ρ1ρ_1ρ1​. 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)Θ(1/ε)Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε=1/5ε=1/5ε=1/5, our estimator runs in poly(n){\rm poly}(n)poly(n) time. Since the operator norm distance T∞(∣ψ⟩ ⁣⟨ψ∣,ρ){\rm T}_{\infty}(|ψ\rangle\!\langleψ|,ρ)T∞​(∣ψ⟩⟨ψ∣,ρ) is exactly half of the trace distance T(∣ψ⟩ ⁣⟨ψ∣,ρ){\rm T}(|ψ\rangle\!\langleψ|,ρ)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){\rm rank}(ρ)rank(ρ), which can be exp⁡(n)\exp(n)exp(n) in general. 2. For general quantum states, we also provide a quantum estimator using O~(1/ε3/2)\widetilde{O}(1/ε^{3/2})O(1/ε3/2) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP{\sf BQP}BQP-complete and improves the QMA{\sf QMA}QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε)Ω(1/ε)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⟩|ψ\rangle∣ψ⟩ has overlap at least 1/21/21/2 with the top unit eigenvector of ∣ψ⟩ ⁣⟨ψ∣−ρ|ψ\rangle\!\langleψ|-ρ∣ψ⟩⟨ψ∣−ρ, reflecting a structural feature specific to the operator norm distance.
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