Quantum Query Complexity of Boolean Functions under Indefinite Causal Order

TL;DR

This paper investigates the query complexity of Boolean functions within quantum computations that allow indefinite causal order, showing limitations of advantages but identifying specific functions with improved computation accuracy.

Contribution

It extends query complexity analysis to causally indefinite quantum supermaps, demonstrating bounds and identifying functions with lower error rates using indefinite causal order.

Findings

Causally indefinite supermaps do not reduce query complexity universally.

Quantum circuits with indefinite causal order cannot outperform fixed-order circuits in general.

Some functions can be computed with lower error using indefinite causal order with only two queries.

Abstract

The standard model of quantum circuits assumes operations are applied in a fixed sequential "causal" order. In recent years, the possibility of relaxing this constraint to obtain causally indefinite computations has received significant attention. The quantum switch, for example, uses a quantum system to coherently control the order of operations. Several ad hoc computational and information-theoretical advantages have been demonstrated, raising questions as to whether advantages can be obtained in a more unified complexity theoretic framework. In this paper, we approach this problem by studying the query complexity of Boolean functions under general higher order quantum computations. To this end, we generalise the framework of query complexity from quantum circuits to quantum supermaps to compare different models on an equal footing. We show that the recently introduced class of quantum circuits with quantum control of causal order cannot lead to any reduction in query complexity, and that any potential advantage arising from causally indefinite supermaps can be bounded by the polynomial method, as is the case with quantum circuits. Nevertheless, we find some functions for which the minimum error with which they can be computed using two queries is strictly lower when exploiting causally indefinite supermaps.