### Abstract

Higher-order transformations that act on a certain number of input quantum channels in an indefinite causal order - such as the quantum switch - cannot be described by standard quantum circuits that use the same number of calls of the input quantum channels. However, the question remains whether they can be simulated, i.e., whether their action on their input channels can be deterministically reproduced, for all arbitrary inputs, by a quantum circuit that uses a larger number of calls of the input channels. Here, we prove that when only one extra call of each input channel is available, the quantum switch cannot be simulated by any quantum circuit. We demonstrate that this result is robust by showing that, even when probabilistic and approximate simulations are considered, higher-order transformations that are close to the quantum switch can be at best simulated with a probability strictly less than one. This result stands in stark contrast with the known fact that, when the quantum switch acts exclusively on unitary channels, its action can be simulated.

Publication

Can the quantum switch be deterministically simulated?

Higher-order transformations that act on a certain number of input quantum channels in an indefinite causal order - such as the quantum switch - cannot be described by standard quantum circuits that use the same number of calls of the input quantum channels. However, the question remains whether they can be simulated, i.e., whether their action on their input channels can be deterministically reproduced, for all arbitrary inputs, by a quantum circuit that uses a larger number of calls of the input channels. Here, we prove that when only one extra call of each input channel is available, the quantum switch cannot be simulated by any quantum circuit. We demonstrate that this result is robust by showing that, even when probabilistic and approximate simulations are considered, higher-order transformations that are close to the quantum switch can be at best simulated with a probability strictly less than one. This result stands in stark contrast with the known fact that, when the quantum switch acts exclusively on unitary channels, its action can be simulated.