This thesis studies space-bounded quantum computation, focusing on quantum algorithms in settings where memory is limited. The central goal is to understand which quantum advantages persist under such constraints and how quantum resources affect classical time-space tradeoffs. The thesis approaches these questions through st-connectivity, a central problem in space complexity theory. We develop quantum algorithms and tradeoff results for both undirected and directed st-connectivity. In the undirected case, we show that quantum algorithms can achieve near-optimal time using only logarithmic space, in contrast to classical algorithms, which exhibit nontrivial time-space tradeoffs. This demonstrates a fundamental difference between classical and quantum computation in memory-constrained regimes. For directed st-connectivity, we establish the first nontrivial quantum time–space tradeoff. This problem is more challenging, as techniques such as quantum walks and reversibility do not directly apply. We show that quantum computation can improve upon known classical tradeoff bounds even when only a small quantum workspace is available. A key contribution of the thesis is a compositional framework for quantum algorithms based on multidimensional quantum walks. We introduce subspace graphs, an abstract model that enables modular design and efficient recursion without the overheads that typically slow down recursive quantum algorithms. Finally, we extend undirected st-connectivity to unitary-labeled graphs, where edges represent quantum transformations. We present an efficient quantum algorithm for this generalized connectivity problem. Together, these results advance the understanding of space-bounded quantum computation and quantum time-space tradeoffs.
This thesis studies space-bounded quantum computation, focusing on quantum algorithms in settings where memory is limited. The central goal is to understand which quantum advantages persist under such constraints and how quantum resources affect classical time-space tradeoffs. The thesis approaches these questions through st-connectivity, a central problem in space complexity theory. We develop quantum algorithms and tradeoff results for both undirected and directed st-connectivity. In the undirected case, we show that quantum algorithms can achieve near-optimal time using only logarithmic space, in contrast to classical algorithms, which exhibit nontrivial time-space tradeoffs. This demonstrates a fundamental difference between classical and quantum computation in memory-constrained regimes. For directed st-connectivity, we establish the first nontrivial quantum time–space tradeoff. This problem is more challenging, as techniques such as quantum walks and reversibility do not directly apply. We show that quantum computation can improve upon known classical tradeoff bounds even when only a small quantum workspace is available. A key contribution of the thesis is a compositional framework for quantum algorithms based on multidimensional quantum walks. We introduce subspace graphs, an abstract model that enables modular design and efficient recursion without the overheads that typically slow down recursive quantum algorithms. Finally, we extend undirected st-connectivity to unitary-labeled graphs, where edges represent quantum transformations. We present an efficient quantum algorithm for this generalized connectivity problem. Together, these results advance the understanding of space-bounded quantum computation and quantum time-space tradeoffs.
Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T=O~(n2/S) for any S such that S=Ω(log(n)) and S=O(n2/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time O~(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and O~(n1.5) time, or O~(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.
Quantum computing offers the potential to solve problems more efficiently than classical methods, yet understanding its capabilities requires new conceptual frameworks. This dissertation explores quantum algorithms with a focus on maintaining connections to classical intuition, clarifying both their strengths and limitations. In Part I, we explore the capabilities of a class of quantum algorithms known as quantum walks. These quantum walks, the quantum analogs of classical random walks, serve as powerful yet accessible tools for solving a variety of computational problems. Our key contribution is the development of a novel way to construct quantum walks, resulting in multidimensional quantum walks. By applying these walks to the k-distinctness problem, we achieve a time-efficient algorithm matching the best-known query upper bound up to polylogarithmic factors. Additionally, applying them to the welded tree problem results in exponential speedups, marking the first instance of a discrete quantum walk to do so. Furthermore, we extend the well-known link between quantum walks and electrical networks to the multidimensional quantum walks. In Part II, we turn to the limitations of quantum algorithms by examining techniques for establishing quantum query lower bounds, representing the minimum computational cost required for any quantum algorithm to solve a given computational problem, regardless of the quantum algorithm used. To this goal, we introduce the multiplicative ladder adversary method, a simplified version of the multiplicative adversary method which unifies the compressed oracle technique within the broader adversary framework while extending its applicability, offering a more intuitive approach to establishing quantum lower bounds.
A Quantum Time-Space Tradeoff for Directed st-Connectivity
Directed st-connectivity (DSTCON) is the problem of deciding if there exists a directed path between a pair of distinguished vertices s and t in an input directed graph. This problem appears in many algorithmic applications, and is also a fundamental problem in complexity theory, due to its NL-completeness. We show that for any S≥log2(n), there is a quantum algorithm for DSTCON using space S and time T≤221log(n)log(n/S)+o(log2(n)), which is an (up to quadratic) improvement over the best classical algorithm for any S=o(n). Of the S total space used by our algorithm, only O(log2(n)) is quantum space - the rest is classical. This effectively means that we can trade off classical space for quantum time.
Abstract. We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices [Formula: see text], including the discrete Fourier transform matrix, we prove that quantum circuits with at most [Formula: see text] input queries and [Formula: see text] qubits of memory require [Formula: see text] to compute matrix-vector product [Formula: see text] for [Formula: see text]. We similarly prove that matrix multiplication for [Formula: see text] binary matrices requires [Formula: see text]. Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for [Formula: see text] Boolean matrix multiplication to [Formula: see text] from [Formula: see text]. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.
We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n2/S) to compute matrix-vector product Ax for x ∈ {0,1}n. We similarly prove that matrix multiplication for n× n binary matrices requires T=Ω(n3 / √S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n2.5/S1/2) to T=Ω(n2.5/S1/4) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms.
Time and space (memory) are two of the most important measures of cost in computation, even more so for quantum computation. Yet, our tools for proving unconditional quantum tradeoffs between time and space are surprisingly limited.
The first quantum time-space tradeoff lower bounds were proven for sorting by Klauck, Špalek and de Wolf. Unfortunately, their method is limited to proving output-oblivious lower bounds (i.e. the lower bounds only apply to algorithms with a non-adaptive output schedule) and other methods have yielded nothing beyond output-oblivious lower bounds for sorting. Here, we prove the first fully general quantum time-space tradeoff lower bound for sorting.
We do so by introducing a novel method based on the noise operator to add to the analysis toolkit for proving quantum query and time-space tradeoff lower bounds. By combining our resulting quantum noise stability bound with quantum recording query methods, we prove an Ω(n4/3(loglogn)/(S1/3logn)) lower bound on the number of queries that a fully general quantum algorithm with at most S qubits of memory requires to sort n numbers from [n2]. Applying our noise operator argument involves purely classical reasoning, which makes it particularly simple to use.
We also use it to prove that, for any strongly universal (pairwise independent) hash function family H from n bits to m bits, almost all hash functions in H require any algorithm with at most S qubits of memory to make Ω(nm/S) quantum queries to input x in order to compute h(x), even with very small success probability. Previously, Mansour, Nisan, and Tiwari had shown a matching classical lower bound for computing h(x) with both h and x as inputs using their hash mixing lemma. Our noise operator method allows us to use a related property of hash functions to prove our quantum lower bounds.