Dequantizing read-once quantum formulas

TL;DR

This paper demonstrates that read-once quantum formulas are equivalent to classical formulas of the same size and depth, but this equivalence breaks down without the read-once restriction, leading to a new computational model.

Contribution

It proves the equivalence of read-once quantum and classical formulas over certain gates and introduces a new one-qubit model that can compute all boolean functions.

Findings

Read-once quantum formulas are equivalent to classical formulas over similar gates.

Removing the read-once restriction allows quantum formulas to surpass classical limitations.

The one-qubit model can compute all boolean functions.

Abstract

Quantum formulas, defined by Yao [FOCS '93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any read-once quantum formula over a gate set that contains all single-qubit gates is equivalent to a read-once classical formula of the same size and depth over an analogous classical gate set. For example, any read-once quantum formula over Toffoli and single-qubit gates is equivalent to a read-once classical formula over Toffoli and NOT gates. We then show that the equivalence does not hold if the read-once restriction is removed. To show the power of quantum formulas without the read-once restriction, we define a new model of computation called the one-qubit model and show that it can compute all boolean functions. This model may also be of independent interest.