Simulating the quantum switch with quantum circuits is impossible

arXiv:2611.10615v1 Announce Type: new Abstract: The quantum switch is a higher-order transformation that combines input quantum channels in an indefinite causal order. Whether quantum circuits can simulate its action, and how many calls to the input channels such a simulation requires, are natural questions about the power of quantum control of causal order. In this work, we prove that no quantum circuit with classically controlled causal order can exactly and deterministically simulate the quantum switch for arbitrary input channels using any finite number of calls. This impossibility holds already for qubit channels and extends to the complete $r$-switch, which coherently controls all permutations of $r$ input channels.