# Quantum algorithms for graph connectivity and formula evaluation

Stacey Jeffery1 and Shelby Kimmel2

1QuSoft and CWI, Amsterdam, the Netherlands
2Middlebury College, Middlebury, VT, USA

### Abstract

We give a new upper bound on the quantum query complexity of deciding $st$-connectivity on certain classes of planar graphs, and show the bound is sometimes exponentially better than previous results. We then show Boolean formula evaluation reduces to deciding connectivity on just such a class of graphs. Applying the algorithm for $st$-connectivity to Boolean formula evaluation problems, we match the $O(\sqrt{N})$ bound on the quantum query complexity of evaluating formulas on $N$ variables, give a quadratic speed-up over the classical query complexity of a certain class of promise Boolean formulas, and show this approach can yield superpolynomial quantum/classical separations. These results indicate that this $st$-connectivity-based approach may be the "right" way of looking at quantum algorithms for formula evaluation.

Note added: The authors have post publication of this paper uploaded version 1704.00765v3, which fixes a bug in statement and proof of Lemma 32 (regarding time complexity of algorithms).

We show a connection between two important classes of problems in computer science: evaluating a logical formula, and deciding if two nodes in a network are connected by a path (called st-connectivity). The latter problem has a particularly elegant quantum algorithm, based on the model of "span programs''. We improve this algorithm for the special case when the network is planar, meaning no two edges of the network cross each other. We show that to evaluate a formula, it is sufficient to solve a related st-connectivity problem on a planar network, and so our new st-connectivity algorithm can also be used to evaluate formulas.

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### Cited by

[1] Michael Jarret and Kianna Wan, "Improved quantum backtracking algorithms using effective resistance estimates", Physical Review A 97 2, 022337 (2018).

[2] Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita, "Span programs and quantum time complexity", arXiv:2005.01323.

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