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.
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.
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.
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 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.
In this thesis, Quantum computing, norms, and polynomials, we investigate the interplay between quantum mechanics, complexity theory, and functional analysis, three central areas of physics, computer science, and mathematics, respectively. The unifying theme throughout the thesis is the dynamic exchange between quantum computing and functional analysis: we explore new applications of functional inequalities in quantum computing, and, in the process, establish novel results in functional analysis itself. In the first part, we study quantum query algorithms through their correspondence with completely bounded polynomials, as established in earlier work. We begin by revisiting this correspondence, extending it, and presenting it in a new form. Building on this foundation, we draw an analogy between quantum query algorithms and the Grothendieck inequality. Finally, we conclude this part by employing completely bounded polynomials to solve a special case of one of the main open problems in quantum query complexity, the Aaronson–Ambainis conjecture. In the second part, we turn to quantum learning theory, which seeks to determine how much information must be extracted from a quantum system to fully characterize it. We begin by applying existing versions of the Bohnenblust–Hille inequalities and deriving new ones to obtain results in the learning of low-degree quantum objects. We conclude by presenting some of the first results in the emerging area of Hamiltonian testing and learning. We also include a third part, as a bonus, where we gather three new proofs, that we find elegant and concise, of known results related to the analysis of Boolean functions.
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.
The ability (or inability) to represent or approximate Boolean functions by polynomials is a central concept in complexity theory, underlying interactive and probabilistically checkable proof systems, circuit lower bounds, quantum complexity theory, and more. In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over This book covers recent progress on proving approximate degree lower and upper bounds and describes some applications of the new bounds to oracle separations, quantum query and communication complexity, and circuit complexity. The authors explain how several of these advances have been unlocked by a particularly simple and elegant technique, called dual block composition, for constructing solutions to this dual linear program. They also provide concise coverage of even more recent lower bound techniques based on a new complexity measure called spectral sensitivity. Finally, they show how explicit constructions of approximating polynomials have been inspired by quantum query algorithms. This book provides a comprehensive review of the foundational and recent developments of an important topic in both classical and quantum computing. The reader has a considerable body of knowledge condensed in an accessible form to quickly understand the principles and further their own research.