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arXiv:2609.40182

A mixing time method for estimating the sample complexity of quantum state discrimination

Juntai Zhou, Felix Leditzky

quant-phcs.CCcs.ITmath.PRmath.RTSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group $G$ forms a Gelfand pair with the stabilizer subgroup $H$ of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra $\operatorname{End}_G(\mathbb C[G/H])$. In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-$d$ phase states over $\mathbb F_q$ in [Alrabiah et al., 2026] and generalized Boolean phase states over $\mathbb Z_q$ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouz{é}, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.

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We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group GGG forms a Gelfand pair with the stabilizer subgroup HHH of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra End⁡G(C[G/H])\operatorname{End}_G(\mathbb C[G/H])EndG​(C[G/H]). In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-ddd phase states over Fq\mathbb F_qFq​ in [Alrabiah et al., 2026] and generalized Boolean phase states over Zq\mathbb Z_qZq​ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouz{é}, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.
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Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. Lower bounds on sample complexity in these models were previously established via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. The quantum sample complexity of this problem has been characterised up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, whereas the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem via the generalised Holevo-Curlander bounds. All the aspects of the problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.
Tight bounds on quantum sample complexity and quantum query complexity have been known for various computational problems in the literature, whereas tight bounds on quantum time complexity (i.e., the size of quantum circuits) remain unresolved. In this paper, we provide a framework to establish lower bounds on the quantum time complexity for testing permutation-invariant properties of quantum states, via a reduction from quantum sample complexity. As an application, we obtain a series of matching lower bounds when given sample access to the input quantum states, including: 1. The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity tr⁡(ρ2)\operatorname{tr}(ρ^2)tr(ρ2) and the inner product tr⁡(ρσ)\operatorname{tr}(ρσ)tr(ρσ). 2. The Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki, and Kwek (Phys. Rev. Lett. 2002) is time-optimal to estimate the high-order functionals tr⁡(ρk)\operatorname{tr}(ρ^k)tr(ρk). 3. The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal. 4. The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state. 5. The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states. 6. The estimator for pure-state trace distance and fidelity due to Wang and Zhang (ICALP 2026) is time-optimal. To the best of our knowledge, this is the first method that allows us to systematically establish tight lower bounds on quantum time complexity.
Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-kkk, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order kkk increases. For pure-state ensembles of size NNN, estimating MMD-kkk with arbitrary measurement schemes requires Θ(N1−1/k)Θ(N^{1-1/k})Θ(N1−1/k) samples for constant kkk. At the same time, we prove that any stable distance metric with full discriminative power admits an O(Nlog⁡N)O(N\log N)O(NlogN) upper bound and an Ω(N)Ω(N)Ω(N) instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an O(Nlog⁡N)O(N\log N)O(NlogN) upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.
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 analyze the complexity of learning nnn-qubit quantum phase states. A degree-ddd phase state is defined as a superposition of all 2n2^n2n basis vectors xxx with amplitudes proportional to (−1)f(x)(-1)^{f(x)}(−1)f(x), where fff is a degree-ddd Boolean polynomial over nnn variables. We show that the sample complexity of learning an unknown degree-ddd phase state is Θ(nd)Θ(n^d)Θ(nd) if we allow separable measurements and Θ(nd−1)Θ(n^{d-1})Θ(nd−1) if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime poly(n)\textsf{poly}(n)poly(n) (for constant ddd) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli XXX and ZZZ bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when fff has sparsity-sss, degree-ddd in its F2\mathbb{F}_2F2​ representation (with sample complexity O(2dsn)O(2^d sn)O(2dsn)), fff has Fourier-degree-ttt (with sample complexity O(22t)O(2^{2t})O(22t)), and learning quadratic phase states with ε\varepsilonε-global depolarizing noise (with sample complexity O(n1+ε)O(n^{1+\varepsilon})O(n1+ε)). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP~circuits.