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arXiv:2609.30171

Optimal spectrum estimation

Ainesh Bakshi, Apoorv Vikram Singh, Xinyu Tan

quant-phcs.CCcs.DSSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
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Abstract

We prove that the spectrum of an unknown $d$-dimensional quantum state can be estimated to error $\varepsilon$ in total variation distance using \[ O\!\left(d^2\min\left\{ \frac{1}{(\varepsilon\log d)^4},\; \frac{1}{(\varepsilon\log d)^2} \right\}\right) \] copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of $d$ in copy complexity, which we conjecture to be optimal. We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.

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We prove that the spectrum of an unknown ddd-dimensional quantum state can be estimated to error ε\varepsilonε in total variation distance using [
O!\left(d^2\min\left{
\frac{1}{(\varepsilon\log d)^4},;
\frac{1}{(\varepsilon\log d)^2}
\right}\right) ] copies. This matches the recent lower bound of Wang. When restricted to unentangled measurements, we give an algorithm with an additional factor of ddd in copy complexity, which we conjecture to be optimal.
We develop a framework for recovering the small eigenvalues of a quantum state by matching Chebyshev moments. We bound the variance of each Chebyshev moment estimate in terms of scalar derivatives of the corresponding polynomial, using classical and quantum Efron--Stein decompositions. Different rescalings of the Chebyshev polynomials balance approximation error and variance, yielding two regimes in our copy complexity bound.
We show that the von Neumann entropy of an unknown ddd-dimensional quantum state ρρρ can be estimated to within additive error ε\varepsilonε using Θ ⁣(d2εlog⁡2(d)(1+εlog⁡2(d))+log⁡2(d)ε2) Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)(1+\varepsilon\log^2(d))} + \frac{\log^2(d)}{\varepsilon^2}\right) Θ(εlog2(d)(1+εlog2(d))d2​+ε2log2(d)​) samples of ρρρ, matching the lower bound due to Wang (2026). Our estimator is based on a high-degree approximation polynomial and the main technical challenge is bounding its variance despite the large polynomial degree. We overcome this using a quantum Hoeffding decomposition, which preserves cancellations between different polynomial terms and reduces the analysis to derivative bounds for the approximation polynomial.
Optimal single-copy estimation of quantum state moments: why Tr⁡(ρ3)\operatorname{Tr}(ρ^3)Tr(ρ3) and Tr⁡(ρ4)\operatorname{Tr}(ρ^4)Tr(ρ4) are equally hard
Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments Tr⁡(ρt)\operatorname{Tr}(ρ^t)Tr(ρt), allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity Θ(d)Θ(\sqrt d)Θ(d​) at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer t≥2t\ge 2t≥2: for any sufficiently small constant additive error, the optimal sample complexity is Θ ⁣(d⌈log⁡2t⌉1+⌈log⁡2t⌉) Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) Θ(d1+⌈log2​t⌉⌈log2​t⌉​) with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer h≥1h\geq1h≥1, all moment orders 2h−1<t≤2h2^{h-1}<t\leq2^h2h−1<t≤2h share the same exponent. In particular, the third and fourth moments both require Θ(d2/3)Θ(d^{2/3})Θ(d2/3) copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an Ω(d1−1/t)Ω(d^{1-1/t})Ω(d1−1/t) lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every t≥4t\geq4t≥4. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond d\sqrt dd​.
We give an algorithm which, given n=O(d2⋅(log⁡log⁡(d)/log⁡(d))2)n = O(d^2 \cdot (\log\log(d)/\log(d))^2)n=O(d2⋅(loglog(d)/log(d))2) copies of ρρρ, estimates the eigenvalues of ρρρ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the Θ(d2)Θ(d^2)Θ(d2) needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses n=Θ(d2)n = Θ(d^2)n=Θ(d2) copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction ∣w⟩|w\rangle∣w⟩ scales with ⟨w∣ρ∣w⟩\langle w | ρ|w\rangle⟨w∣ρ∣w⟩ for all directions simultaneously. From this stronger "relative-error" bound, we recover better algorithms for principal component analysis in Bures distance and tomography in χ2χ^2χ2-divergence as corollaries.
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.
We study the sample complexity of estimating and testing fundamental unitarily invariant properties of unknown quantum states; namely, the tasks of spectrum estimation, von Neumann entropy estimation, and rank-testing. For ddd-dimensional states, and for every γ>0γ>0γ>0, we prove a sample complexity lower bound of Ω(d2−γ)Ω(d^{2-γ})Ω(d2−γ) for spectrum estimation to constant sorted total-variation error, entropy estimation to constant additive error, and rank-testing to constant trace distance. Our hard instances are constructed from sandwiched products of Haar-random projectors, suitably normalized using a novel technique that lets us derive explicit expressions for high-order tensor moments of the resultant states. These moments can be expressed as symmetric functions of Jucys--Murphy elements of the symmetric group algebra. To show that two such mixtures are indistinguishable, we analyze the log-likelihood ratio and perform moment-matching, i.e., we set its low-order Jucys--Murphy components to zero. Indistinguishability is then obtained by bounding an fff-divergence through the high-order components; the non-zero high-order terms and concentration of functions of Haar-random unitaries also imply separations in typical spectra, entropies, and ranks, proving all our lower bounds.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1M=1M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.