We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between $\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum complexity theory.
We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory.
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.
Algebraic methods have become a powerful tool for analyzing the complexity of various computational models, including low-depth circuits, algebraic proofs, and quantum query algorithms. In particular, the complexity of computing a function in these models is related to whether or not the function admits a low-degree polynomial approximation. In this thesis, we present two novel applications of algebraic methods in computational complexity theory. In the first part of the thesis, we study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether or not it satisfies some property. In addition to containing the classical query complexity model as a special case, this model also contains "inherently quantum" problems that have no classical analogue. Our main contribution is a generalized polynomial method for analyzing the complexity of unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a subspace, and approximating the entanglement entropy of a state. We also present a candidate problem towards an oracle separation of QMA and QMA(2), a long standing open question in quantum complexity theory. In the second part of the thesis, we study the tensor isomorphism problem (TI), which has recently emerged as having connections to multiple areas of research, including quantum information theory, post-quantum cryptography, and computational algebra. However, the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, the study of tensor isomorphism naturally lends itself to algebraic and semi-algebraic proof systems such as the polynomial calculus (PC) and sum-of-squares (SoS). We show a linear degree lower bound on PC degree or SoS degree for tensor isomorphism and a non-trivial upper bound for testing isomorphism of tensors of bounded rank. Along the way, we also show that PC cannot perform basic linear algebra in sublinear degree, such as comparing the rank of two matrices. We introduce a strictly stronger proof system, called PC + Inv, which enables linear algebra to be done in low degree. We conjecture that even PC + Inv cannot solve TI in polynomial time either, and highlight many other open questions about proof complexity approaches to TI.
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.
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.
Query complexity studies the amount of information about an input that is needed to solve a computational problem. This thesis investigates query complexity in several settings, with applications to cryptography and property testing, as well as connections to communication complexity and other computational models. In property testing, we study tolerant junta testing and establish the first superpolynomial quantum advantage for this problem: within a certain parameter regime, a quantum algorithm uses polynomially many queries, whereas every classical randomised algorithm requires quasipolynomially many queries. In cryptography, we study the average-case complexity of search problems. We show that verifiable delay functions do not exist in the random oracle model, and strengthen known barriers to constructing average-case hard TFNP problems from one-way functions. Finally, we study lower bounds and their behaviour across computational models. We prove new direct sum results for randomised parity decision trees. We also establish a new separation between query complexity measures of total Boolean functions and, through query-to-communication lifting, derive new communication lower bounds and lower bounds for unambiguous finite automata.
We study the query complexity of testing quantum juntas. Given an unknown n-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most k qubits or is ε-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to U, but not to U† or controlled-U. We give tight bounds for both unitary and channel junta testing. For sufficiently large k, n≥2k, and a broad range of ε, both problems have query complexity Θ(k/(ε2logk)). Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an O(k) upper bound and a Ω(k) lower bound at constant ε, establishing the optimal Θ(k/logk) query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant ε, they also give a separation between Θ(k/logk) forward queries and O(k) queries with inverse access.