Quantum walks through generalized graph composition

TL;DR

This work generalizes the recently-introduced graph composition framework to the non-boolean setting, and provides a novel analysis of irreducible, reversible Markov processes, by linear-algebraically connecting its effective resistance to the random walk operator.

Abstract

In this work, we generalize the recently-introduced graph composition framework to the non-boolean setting. A quantum algorithm in this framework is represented by a hypergraph, where each hyperedge is adjacent to multiple vertices. The input and output to the quantum algorithm is represented by a set of boundary vertices, and the hyperedges act like switches, connecting the input vertex to the output that the algorithm computes. Apart from generalizing the graph composition framework, our new proposed framework unifies the quantum divide and conquer framework, the decision-tree framework, and the unified quantum walk search framework. For the decision trees, we additionally construct a quantum algorithm from an improved weighting scheme in the non-boolean case. For quantum walk search, we show how our techniques naturally allow for amortization of the subroutines' costs. Previous work showed how one can speed up ``detection'' of marked vertices by amortizing the costs of the quantum walk. In this work, we extend these results to the setting of ``finding'' such marked vertices, albeit in some restricted settings. Along the way, we provide a novel analysis of irreducible, reversible Markov processes, by linear-algebraically connecting its effective resistance to the random walk operator. This significantly simplifies the algorithmic implementation of the quantum walk search algorithm, achieves an amortization speed-up for quantum walks over Johnson graphs, avoids the need for quantum fast-forwarding, and removes the log-factors from the query complexity statements.

Figures (7)

• Figure 1.1: Pictorial representation of a state-reflection problem implementing a net-flow $\delta_x^e$ and potential function $U_x^e$, for all $x \in \mathcal{D}$, onto the hyperedge $e$. The black vertices represent $\mathop{\mathrm{supp}}\nolimits(\delta_x^e) \subseteq N(e)$, and so the potential function $U_x^e$ has to be constant on this set, but can be different on the gray vertices. We can think of this state-reflection program $R$ providing a connection between the vertices $v$, $v'$ and $v"$ (the black edges), whilst "cutting" all the connections from this hyperedge towards the other vertices (the gray edges).
• Figure 1.2: Flow and potential function for function evaluation (left) and database update (right). In both cases, the potential function is $1$ and $0$ on all the black and gray vertices, respectively. Intuitively, we can think of the state-reflection problem implementing a switch between an input vertex on the left, and an output vertex on the right.
• Figure 1.3: The hyperedge problem implemented by a span program. On the left-hand side, we have a positive input that implements a unit flow through the edge $e$. This means that the potential function has to be constant on both sides of the edge. On the right-hand side, we have a negative input blocking any flow going through the edge, and thereby allowing there to be a potential difference across the edge.
• Figure 1.4: Overview of quantum algorithmic frameworks. An arrow from $A$ to $B$ represents that every instance of $A$ can be turned into an instance of $B$. The cited resources represent the first reference where these frameworks are considered in the context of quantum algorithms. The new results proved in this work are represented by the bold arrows.
• Figure 5.1: A generalized graph composition for the first marked index finding problem. The span programs $[x_j = b]$, for all $j \in [n]$ and $b \in \{0,1\}$ are trivial span programs that simply make one query, and for all $j \in [n]$, $\alpha_j,\beta_j > 0$.

Theorems & Definitions (86)

• Lemma 1.1: Informal version of \ref{['thm:reformulation-state-conversion']}
• Lemma 1.2: Informal version of \ref{['lem:rescale-state-reflection-problem']}
• Theorem 1.3: Informal version of the resistance-cut theorem, \ref{['thm:path-cut-theorem']}
• Theorem 1.4: Informal version of \ref{['thm:graph-composition']}
• Theorem 1.5: Informal version of \ref{['thm:BT20', 'thm:CMP25']}
• Theorem 1.6: Informal version of \ref{['thm:divide-and-conquer']}
• Theorem 1.7: Informal version of \ref{['thm:quantum-walk-detection']}
• Theorem 1.8: Simplified version of \ref{['thm:quantum-walk-detection']}
• Lemma 1.9: Informal version of \ref{['lem:fast-forwarding']}
• Definition 2.1: levin2017markov