Abstract This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC'2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves certainty of success, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it is impossible to have a deterministic quantum algorithm with exponential speedups for the welded tree problem.
Abstract This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC'2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves certainty of success, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it is impossible to have a deterministic quantum algorithm with exponential speedups for the welded tree problem.
This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves certainty of success, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the welded tree problem.
In the welded tree problem, an algorithm is tasked with navigating a graph formed from two binary trees joined at the leaves through a ``weld'' of connecting edges. A quantum walk can navigate from root to root exponentially faster than any classical algorithm. However, known efficient quantum algorithms cannot find a path between the roots, as recording the path destroys constructive interference and thus the speedup. We prove that this is inherent: any quantum algorithm needs exponentially many queries to find a path between the roots of an independently matched welded tree graph. This provides an example of a problem that a quantum computer can solve exponentially faster than any classical algorithm by exploring exponentially many paths in superposition, but where it is provably intractable to find any such path. The proof uses compressed permutation oracles to record the progress of a quantum algorithm as it queries the graph. We show that the compressed database remains path-free up to a small error. By controlling such errors and bounding the progress of the algorithm with each compressed oracle query, we show that Ω(2n/12) queries are required to find a path in a height-n tree with constant success probability.
This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC'2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves zero-error, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the weled tree problem.
Welded Trees are a class of graphs designed to exhibit exponential algorithmic speedup for certain search problems via quantum walks, outperforming any classical algorithm. A Welded Tree of depth d consists of two complete binary trees, each of depth d, whose leaves are interconnected by an additional cycle or permutation alternating between the two trees. Access to this graph is provided only through queries to an oracle, which, given a vertex, returns its neighbors. In 2021, Aaronson suggested the FINDING-PATH-TO-EXIT problem, which consists of finding a path to the root of the opposite tree given the root of one tree. This can be seen as an extension of the FINDING-EXIT problem, where the task is simply to find the root of the opposite tree, for which Childs et al. demonstrated an exponential speed-up in 2003. We provide a quantum algorithm for solving the Finding Path To Exit problem that performs Õ(2d/3) queries and also formalize the folklore lower bound that any classical algorithm requires at least Ω̃(2d/2 ) queries. This is the first proof of a polynomial speed-up for this problem over any classical algorithm. In addition, we present and analyze several other algorithms, focusing on the trade-offs between queries and classical or quantum space.
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.
We give a quantum algorithm for a novel type of black-box problem: identifying a hidden d-regular base graph G on n vertices from oracle access to an obfuscated version of it, rather than traversing it. From G we build the spired graph Gspire in three steps: each vertex is lifted into an exponentially large cluster, with adjacent clusters joined by a random bipartite graph; each cluster is then crowned with a balanced spire; finally, all vertices are randomly relabelled. Specializing to G=K2 recovers the welded-trees graph.
Our algorithm is conceptually simple: a continuous-time quantum walk on Gspire, followed by a single Hadamard test at a classically precomputed time t∗; the algorithm returns the candidate whose predicted amplitude is closest to the measurement. The design rests on a rigorous spectral theory: from the apex of any spire, the walk is confined to a polynomial-dimensional invariant subspace evolving under the adjacency matrix of a simpler towered graph Gtower; that matrix block-diagonalizes into n independent tridiagonal systems of size n, each solved in closed form by a Chebyshev secular equation. Efficient numerics enabled by this decomposition supply t∗ and the predicted amplitudes.
On the prism graphs Ym versus the Möbius ladders Mm (each on n=2m vertices), the numerical study supports a precise conjecture that O(n2/logn) measurements at evolution time of order m2 suffice to distinguish the two families; we have tested 4≤m≤5121 (n up to 10242). By analogy with the welded-trees lower bounds, we further conjecture that any classical algorithm requires queries exponential in n. Together these conjectures point to an exponential quantum speedup for the identification of an obfuscated base graph.