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Volume 8 (2012) Article 13 pp. 291-319
Span-Program-Based Quantum Algorithm for Evaluating Formulas
Received: September 22, 2008
Revised: October 5, 2011
Published: July 7, 2012
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Keywords: span programs, quantum computation, query complexity, formula evaluation
ACM Classification: F.1.1, F.1.2
AMS Classification: 81P68, 68Q05, 68Q12

Abstract: [Plain Text Version]

We give a quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority). The algorithm is optimal on read-once formulas for which each gate's inputs are balanced in a certain sense.

The main new tool is a correspondence between a classical linear-algebraic model of computation, “span programs,” and weighted bipartite graphs. A span program's evaluation corresponds to an eigenvalue-zero eigenvector of the associated graph. A quantum computer can therefore evaluate the span program by applying spectral estimation to the graph.

For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of randomized alpha-beta pruning is known to be suboptimal. In contrast, our algorithm generalizes the optimal quantum AND-OR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula.