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arXiv:2607.29680

Spectrum Estimation is Almost as Hard as Tomography

Marco Fanizza, Ryan O'Donnell, Chirag Wadhwa

quant-phcs.DSSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations• Revised Sep 14, 2026 (v2)
v1 · Jul 31, 2026v2 · Sep 14, 2026
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Abstract

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 $d$-dimensional states, and for every $γ>0$, we prove a sample complexity lower bound of $Ω(d^{2-γ})$ 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 $f$-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.

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48 pages

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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.
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form Lφ(ρ)=tr⁡(φ(dρ))/d\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/dLφ​(ρ)=tr(φ(dρ))/d, where ρρρ is an unknown ddd-dimensional quantum state and φφφ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating Lφ(ρ)\mathcal{L}_φ(ρ)Lφ​(ρ) from properties of φφφ, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is Θ ⁣(d2εlog⁡2(d)max⁡{1,εlog⁡2(d)}+log⁡2(d)ε2), Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), Θ(εlog2(d)max{1,εlog2(d)}d2​+ε2log2(d)​), and the sample complexity of spectrum estimation is Θ ⁣(d2ε2log⁡2(d)max⁡{1,ε2log⁡2(d)}). Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). Θ(ε2log2(d)max{1,ε2log2(d)}d2​). In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.
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​.
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We study estimation of the von Neumann entropy of a ddd-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let κν∈[1,d+1]κ_ν\in[1,d+1]κν​∈[1,d+1] denote the normalized trace-zero frame spread; κν=1κ_ν=1κν​=1 for exact complex projective 222-designs. For any fixed rank-one measurement and n≤c2d2/κνn\le c_2d^2/κ_νn≤c2​d2/κν​, the minimax mean-squared error is at least c1log⁡2{d2/(κνn)}c_1\log^2\{d^2/(κ_νn)\}c1​log2{d2/(κν​n)}. For bounded frame spread and every fixed 0<a<20<a<20<a<2, the minimax risk is of order log⁡2d\log^2 dlog2d throughout n≲d2−an\lesssim d^{2-a}n≲d2−a; hence fixed uniform accuracy requires Ω(d2)Ω(d^2)Ω(d2) observations. Existing tomography bounds give bounded risk from O(d3log⁡3d)O(d^3\log^3 d)O(d3log3d) observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.
The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error ε\varepsilonε, rank upper bound RRR, and fixed order ααα. Tsallis entropy estimation has near-optimal rank-independent query complexity O~(ε−1/(α−1))\widetilde O(\varepsilon^{-1/(α-1)})O(ε−1/(α−1)) for 1<α<21<α<21<α<2, when the allowed dimension and rank accommodate the lower-bound instances, and O~(1/ε)\widetilde O(1/\varepsilon)O(1/ε) for α≥2α\ge2α≥2, where O~\widetilde OO suppresses logarithmic factors. The first saves a factor 1/ε1/\varepsilon1/ε; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from O~(R/ε2)\widetilde O(R/\varepsilon^2)O(R/ε2) to O~(R/ε)\widetilde O(R/\varepsilon)O(R/ε) and obtain O~(R/ε)\widetilde O(R/\varepsilon)O(R/ε) for noninteger Rényi orders 1<α<31<α<31<α<3, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using O~(R/ε)\widetilde O(R/\varepsilon)O(R/ε) queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.