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 and exactly reproduced, for all arbitrary inputs, by a quantum circuit that uses a larger number of calls of the input channels.

Here, we show 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. Moreover, we prove that the action of the quantum switch on two n-qubit quantum channels cannot be simulated by any causally ordered quantum circuit that uses M calls to one channel and one call to the other, if M <= max(2, 2^n-1). This result implies the first exponential separation in quantum query complexity for a process with indefinite causal order with respect to standard quantum circuits. Finally, we discuss the potential experimental implication of these results.
This talk will be based on arXiv:2409.18202 [quant-ph] and arXiv:2409.18420 [quant-ph].