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.
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.
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.
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.
In unitary property testing a quantum algorithm, also known as a tester, is given query access to a black-box unitary and has to decide whether it satisfies some property. We propose a new technique for proving lower bounds on the quantum query complexity of unitary property testing and related problems, which utilises its connection to unitary channel discrimination. The main advantage of this technique is that all obtained lower bounds hold for any C -tester with C⊆QMA(2)/qpoly , showing that even having access to both (unentangled) quantum proofs and advice does not help for many unitary property testing problems. We apply our technique to prove lower bounds for problems like quantum phase estimation, the entanglement entropy problem, quantum Gibbs sampling and more, removing all logarithmic factors in the lower bounds obtained by the sample-to-query lifting theorem of Wang and Zhang (2023). As a direct corollary, we show that there exist quantum oracles relative to which QMA(2)⊅SBQP and QMA/qpoly⊅SBQP . The former shows that, at least in a black-box way, having unentangled quantum proofs does not help in solving problems that require high precision.
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) and the inner product 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).
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.
In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that Θ(N) queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where N denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform O(N1/3) queries, showing a polynomial speedup. They also proved that Ω(N1/4) queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight Ω(N1/3) quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.
Property testing of distributions is a central topic in information theory, learning theory, and statistics. While quantum algorithms are known to offer significant speedups for property testing of a single distribution or a pair of distributions, it is unclear whether quantum algorithms provide speedups for property testing of m (m≥3) distributions. This work focuses on quantum algorithms for testing whether m distributions have similar means or are ε-far from mean similarity under two models. In the query model, the algorithm can choose which distribution to sample from, whereas in the sampling model, the distributions are selected uniformly. We design quantum algorithms with complexities O~(1/ε) (the O~ notation hides poly-logarithmic factors) and O~(m/ε) in the query and sampling models, respectively, achieving quadratic speedups over the classical counterparts. We further establish quantum lower bounds of Ω(1/ε) and $Ω\rbra{m^{1/3}+\frac{m^{1/4}}ε}$ for the query model and the sampling model, demonstrating the optimality of our quantum algorithms in terms of the dependence on ε up to logarithmic factors.