Exponential separation in quantum query complexity of the quantum switch with respect to simulations with standard quantum circuits

TL;DR

This paper proves that the quantum switch, which allows indefinite causal order, cannot be simulated by standard quantum circuits with fewer than an exponential number of queries, demonstrating a fundamental complexity separation.

Contribution

It establishes an exponential separation in quantum query complexity between indefinite causal order and standard quantum circuits for simulating the quantum switch.

Findings

Quantum switch cannot be simulated deterministically by standard circuits with limited queries.

Exponential separation in query complexity between indefinite causal order and standard circuits.

Demonstrates fundamental limits of standard quantum circuit simulation.

Abstract

Quantum theory is consistent with a computational model permitting black-box operations to be applied in an indefinite causal order, going beyond the standard circuit model of computation. The quantum switch -- the simplest such example -- has been shown to provide numerous information-processing advantages. Here, we prove that the action of the quantum switch on two $n$-qubit quantum channels cannot be simulated deterministically and exactly by any causally ordered quantum circuit that uses $M$ calls to one channel and one call to the other, if $M \leq \max(2, 2^n-1)$. This demonstrates an exponential separation in quantum query complexity of indefinite causal order compared to standard quantum circuits.