Simulating the quantum switch with quantum circuits is computationally hard

TL;DR

This paper proves that simulating the quantum switch with standard quantum circuits requires exponentially more calls to input channels, establishing a fundamental complexity separation in quantum information processing.

Contribution

It demonstrates an exponential lower bound on simulating the quantum switch with quantum circuits, showing the intrinsic complexity of indefinite causal order.

Findings

Quantum switch cannot be simulated with fewer than 2^n calls.

Simulation remains impossible even with one extra call to each channel.

The separation holds for probabilistic and approximate simulations.

Abstract

Higher-order transformations acting on input quantum channels in an indefinite causal order, such as the quantum switch, cannot be described by quantum circuits using the same number of calls to the input channels. A natural question is whether they can be simulated, i.e., whether their action can be exactly and deterministically reproduced by a quantum circuit with more calls to the input channels. Here, we prove that the quantum switch acting on two $n$-qubit channels cannot be simulated by any quantum circuit using $k$ calls to one channel and one to the other, if $k<2^n$. This establishes an exponential separation in quantum query complexity between processes with indefinite causal order and quantum circuits. Moreover, even with one extra call to both input channels, such a simulation remains impossible. We further demonstrate the robustness of this separation by extending the result to probabilistic and approximate simulations scenarios.