We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in Õ(√{nh}) time, w.h.p., where h is the size of the output.
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in Õ(√{nh}) time, w.h.p., where h is the size of the output.
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in O~(nh) time, w.h.p., where h is the size of the output.
The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size n into smaller subproblems (say, a copies of size n/b each), along with some auxiliary work of cost Caux(n) , to give a recurrence relation \(\begin{equation*} C(n) \le a \, C(n/b) + C^{\mathrm{aux}}(n) \end{equation*}\) for the classical complexity C(n) . We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation \(\begin{equation*} C_Q(n) \le \sqrt {a} \, C_Q(n/b) + O(C^{\mathrm{aux}}_Q(n)) \end{equation*}\) that characterizes the quantum query complexity. We apply this framework to obtain near-optimal quantum query complexities for various string problems, such as (i) recognizing the regular language Σ∗20∗2Σ∗ over the alphabet Σ={0,1,2} ; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.
Abstract Despite the rapid development of quantum research in recent years, there is very little research in computational geometry. In this paper, to achieve the convex hull of a point set in a quantum system, a quantum convex hull algorithm based on the quantum maximum or minimum searching algorithm (QUSSMA) is proposed. Firstly, the novel enhanced quantum representation of digital images is employed to represent a group of point set, and then the QUSSMA algorithm and vector operation are used to search the convex hull of the point set. In addition, the algorithm is simulated and compared with the classical algorithm. It is concluded that the quantum algorithm accelerates the classical algorithm when the M p value of the convex hull point is under a certain condition.
We discuss applications of quantum computation to geometric data processing. These applications include problems on convex hulls, minimum enclosing balls, linear programming, and intersection problems. Technically, we apply well-known Grover’s algorithm (and its variants) combined with geometric algorithms, and no further knowledge of quantum computing is required. However, revealing these applications and emphasizing potential usefulness of quantum computation in geometric data processing will promote research and development of quantum computers and algorithms.
In this work, we initiate a systematic study of the time complexity of quantum divide and conquer (QD&C) algorithms for classical problems, and propose a general framework for their analysis. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup, and apply these theorems to various problems involving strings, integers, and geometric objects. These include Longest Distinct Substring, Klee's Coverage, several optimization problems on stock transactions, and k-Increasing Subsequence. For most of these problems our quantum time upper bounds match the quantum query lower bounds, up to polylogarithmic factors. We give a structured framework for describing and classifying a wide variety of QD&C algorithms so that quantum speedups can be more easily identified and applied, and prove general statements on QD&C time complexity covering a range of cases, accounting for the time required for all operations. In particular, we explicitly account for memory access operations in the commonly used QRAM (read-only) and QRAG (read-write) models, which are assumed to take unit time in the query model, and which require careful analysis when involved in recursion. Our generic QD&C theorems have several nice features. 1) To apply them, it suffices to come up with a classical divide and conquer algorithm satisfying the conditions of the theorem. The quantization of the algorithm is then completely handled by the theorem. This can make it easier to find applications which admit a quantum speedup, and contrast with dynamic programming algorithms which can be difficult to quantize due to their highly sequential nature. 2) As these theorems give bounds on time complexity, they can be applied to a greater range of problems than those based on query complexity, e.g., where the best-known quantum algorithms require super-linear time. 3) It can handle minimization problems as well as boolean functions, which allows us to improve on the query complexity result of Childs et al. [Childs et al., 2025] for k-Increasing Subsequence by a logarithmic factor.
One of the challenges of quantum computers in the near- and mid- term is the limited number of qubits we can use for computations. Finding methods that achieve useful quantum improvements under size limitations is thus a key question in the field. In this vein, it was recently shown that a hybrid classical-quantum method can help provide polynomial speed-ups to classical divide-and-conquer algorithms, even when only given access to a quantum computer much smaller than the problem itself. In this work, we study the hybrid divide-and-conquer method in the context of tree search algorithms, and extend it by including quantum backtracking, which allows better results than previous Grover-based methods. Further, we provide general criteria for polynomial speed-ups in the tree search context, and provide a number of examples where polynomial speed ups, using arbitrarily smaller quantum computers, can be obtained. We provide conditions for speedups for the well known algorithm of DPLL, and we prove threshold-free speed-ups for the PPSZ algorithm (the core of the fastest exact Boolean satisfiability solver) for well-behaved classes of formulas. We also provide a simple example where speed-ups can be obtained in an algorithm-independent fashion, under certain well-studied complexity-theoretical assumptions. Finally, we briefly discuss the fundamental limitations of hybrid methods in providing speed-ups for larger problems.