Quantum algorithms through graph composition

TL;DR

We develop the graph composition framework, a span-program-based generalization of $st$-connectivity for quantum query algorithms, and provide an exact witness-size characterization in terms of effective resistances on related graphs. The framework supports time-efficient, QROM-based implementations and unifies span-program, dual adversary, learning graphs, and multidimensional quantum-walk perspectives. It is applied to string-search problems including the Dyck language, 3-increasing subsequences, and OR ∘ pSEARCH, while also simplifying pattern matching and directed $st$-connectivity algorithms. Overall, the work introduces modular graph-based composition for quantum algorithms, enabling cross-framework insights and practical improvements for a broad class of problems.

Abstract

In this work, we introduce the graph composition framework, a generalization of the st-connectivity framework for generating quantum algorithms, where the availability of each of the graph's edges is computed by a span program. We provide an exact characterization of the resulting witness sizes in terms of effective resistances of related graphs. We also provide less-powerful, but easier-to-use upper bounds on these witness sizes. We give generic time-efficient implementations of algorithms generated through the graph composition framework, in the quantum read-only memory model, which is a weaker assumption than the more common quantum random-access model. Along the way, we simplify the span program algorithm, and remove the dependence of its analysis on the effective spectral gap lemma. We unify the quantum algorithmic frameworks that are based on span programs or the quantum adversary bound. In particular, we show how the st-connectivity framework subsumes the learning graph framework, the weighted-decision-tree framework, and a zero-error version of the latter. We show that the graph composition framework subsumes part of the quantum divide and conquer framework, and that it is itself subsumed by the multidimensional quantum walk framework. Moreover, we show that the weighted-decision-tree complexity is quadratically related to deterministic query complexity, and to the GT-bound with polynomial exponent 3/2. For the latter, we also provide a matching separation. We apply our techniques to give improved algorithms for various string-search problems, namely the Dyck-language recognition problem of depth 3, the 3-increasing subsequence problem, and the OR $\circ$ pSEARCH problem. We also simplify existing quantum algorithms for the space-efficient directed st-connectivity problem, the pattern-matching problem and the infix-search problem.

Figures (18)

• Figure 1.1: Examples of the tree decomposition (left), and the parallel decomposition (right). The dashed, dotted and solid sets of edges represent the disjoint edge sets $E_1, \dots, E_k$.
• Figure 1.2: Relations between quantum algorithmic frameworks. Framework A points to B, if it is generically possible to turn an instance of framework A into one of B. The results in bold are new in this work. The frameworks below the dashed line are all complete in terms of the quantum query complexity of $\mathsf{Q}$, i.e., one can always devise query-optimal algorithms in all these frameworks. Above the dashed line, it is not (known to be) possible to generically devise query-optimal algorithms, and thus it makes sense to define a complexity measure as the minimal number of queries made by a quantum query algorithm designed through the framework. These complexity measures are denoted in parenthesis.
• Figure 1.3: Hasse diagram of the relationships between complexity measures of boolean functions. When two measures are connected by a solid black line, then the upper complexity measure is bigger than the lower one, for every total boolean function $f : \{0,1\}^n \to \{0,1\}$. On the other hand, if two measures are connected by a dashed red line, then there exists a total boolean function $f : \{0,1\}^n \to \{0,1\}$ for which the upper measure is bigger than the lower one. This picture is valid for any $\varepsilon > 0$. The bold connections are proved in this work.
• Figure 1.4: A graph composition that computes the $\Sigma^*20^*2\Sigma^*$-problem. The graph composition $\mathcal{P}_j$ computes whether the $20^*2$-pattern exists starting at position $j$. We then generate the full graph composition by composing the $\mathcal{P}_j$'s for $j \in [n-1]$ in parallel.
• Figure 4.1: Examples of the tree decomposition (left), and the parallel decomposition (right). The dashed, dotted and solid sets of edges represent the disjoint edge sets $E_1, \dots, E_k$.

Theorems & Definitions (111)

• Theorem 1.1: Graph composition witness sizes (informal version of \ref{['thm:graph-composition']})
• Theorem 1.2: Upper bound on graph composition witness sizes (informal version of \ref{['thm:witness-upper-bounds']})
• Theorem 1.3: Informal version of \ref{['thm:circulation-space-reflection-implementation']}
• Theorem 2.1: Quantum state preparation (see, e.g., prakash2014quantum)
• Theorem 2.2: Reflection through a one-dimensional subspace
• proof
• Theorem 2.3: Uniform state preparation (see, e.g., shukla2024efficient)
• Definition 2.4: Span programs
• Definition 2.5: Span program witnesses
• Definition 2.6: Scalar multiplication of span programs