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.
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.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions f and g; a direct sum theorem asserting that computing k copies of a function (or search problem) costs Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k) times the cost of computing one copy to bounded error.
However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing k copies of f when the input comes from the product distribution μk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions.
In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2 norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.
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.
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.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.
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.