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 $\operatorname{tr}(ρ^2)$ and the inner product $\operatorname{tr}(ρσ)$.
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 $\operatorname{tr}(ρ^k)$.
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.
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:
The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity tr(ρ2) and the inner product tr(ρσ).
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).
The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal.
The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state.
The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states.
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.
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form Lφ(ρ)=tr(φ(dρ))/d, where ρ is an unknown d-dimensional quantum state and φ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating Lφ(ρ) from properties of φ, 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(εlog2(d)max{1,εlog2(d)}d2+ε2log2(d)), and the sample complexity of spectrum estimation is 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.
Abstract. The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998; [ J. ACM, 48 (2001), pp. 778–797]) and the adversary method by Ambainis (STOC 2000; [ J. Comput. System Sci., 64 (2002), pp. 750–767]) and the compressed oracle method by Zhandry [ Proceedings of the 39 th CRYPTO, 2019, pp. 239–268] have been shown to be powerful in proving quantum query lower bounds for a wide variety of problems. In this paper, we propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem, which is from an information theory perspective. Using this method, we obtain the following new results: (1) A quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Here, the sample complexity is measured given sample access to the quantum state to be tested, while the query complexity is measured given query access to an oracle that block-encodes the quantum state. (2) A matching lower bound [Formula: see text] for quantum Gibbs sampling at inverse temperature [Formula: see text] ([Formula: see text] suppresses logarithmic factors), showing that the quantum Gibbs sampler by Gilyén, Su, Low, and Wiebe [ Proceedings of the 51 st STOC, 2019, pp. 193–204] is optimal. (3) A new lower bound [Formula: see text] for the entanglement entropy problem with gap [Formula: see text], which was recently studied by She and Yuen [ Proceedings of the 14 th ITCS, 2023, pp. 96:1–96:17]. (4) A series of quantum query lower bounds for matrix spectrum testing, based on the sample lower bounds for quantum state spectrum testing by O’Donnell and Wright (STOC 2015; [ Comm. Math. Phys., 387 (2021), pp. 1–95]). In addition, we provide unified proofs for some known lower bounds that have been proven previously via different techniques, including those for phase/amplitude estimation and Hamiltonian simulation.
Quantum sample-to-query lifting, a relation between quantum sample complexity and quantum query complexity presented in Wang and Zhang (SIAM J. Comput. 2025), was significantly strengthened by Tang, Wright, and Zhandry (2025) to the case of state-preparation oracles. In this paper, we compile a list of quantum lower and upper bounds for property testing that are obtained by quantum sample-to-query lifting. The problems of interest include testing properties of probability distributions and quantum states, such as entropy and closeness. This collection contains new results, as well as new proofs of known bounds. In total, we present 49 complexity bounds, where 41 are new and 18 are (near-)optimal.
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 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-d phase states over Fq in [Alrabiah et al., 2026] and generalized Boolean phase states over Zq [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.
We study the power of local test for bipartite quantum states. Our central result is that, for properties of bipartite pure states, unitary invariance on one part implies anoptimal(over all global testers) local tester acting only on the other part. As an application, we demonstrate • Purified samples offer no advantage in property testing of mixed states. • A matching lower bound Ω(r2/ε2) for testing the Schmidt rank of bipartite states with perfect completeness, settling an open question raised in the survey of Montanaro and de Wolf (ToC 2016). • A lower bound Ω(( √n+ √r)· √r/ε2) for testing whether ann-partite state is a matrix product state of bond dimensionror ε-far, improving the prior lower bounds Ω( √n/ε2) by Soleimanifar and Wright (SODA 2022) and Ω( √r) by Aaronson et al. (ITCS 2024). • A matching lower bound Ω(d/ε2) for testing whether ad-dimensional bipartite state is maximally entangled or ε-far, showing that the algorithm of O’Donnell and Wright (STOC 2015) is optimal for this task. • A query lower bound Ω(d/ √ Δ) for thed-dimensional entanglement entropy problem with gap Δ, improving the prior lower bounds Ω(4√d) by She and Yuen (ITCS 2023) and Ω(1/ √ Δ) by Wang and Zhang (SICOMP 2025) and Weggemans (Quantum 2025). Moreover, we extend our central result to a robust version where the tested states are subject to noise and are not guaranteed to be pure: in this case, one-way LOCC is sufficient to realize the optimal tester.
We study the following quantum state identity problem: given access to copies of n unknown pure quantum states, the goal is to determine whether every pair has fidelity at least 1−ε or some pair has fidelity at most ε. This problem relaxes the identical-or-orthogonal promise considered in previous work. For any fixed overlap parameter ε∈(0,1/4), we present two quantum algorithms with complementary guarantees on circuit depth and sample complexity. The first runs in constant depth using parallel swap tests and takes O(log(n/δ)log(2/δ)) copies of each input state to succeed with probability at least 1−δ. The second uses the Schur transform to reduce the sample complexity to O(log(n/δ)), through an analysis relating Jucys--Murphy elements to Schur sampling. We also prove a matching sample complexity lower bound of Ω(log(n/δ)) for any quantum algorithm with failure probability at most δ≤1/2−c, for any constant c>0, establishing that the second algorithm achieves the optimal dependence on both n and δ for fixed ε.