Quantum Circuits with Classical Versus Quantum Control of Causal Order

Abstract

Quantum supermaps are transformations that map quantum operations to quantum operations. It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits, also called quantum combs. A systematic understanding of the physical interpretation of more general types of quantum supermaps—in particular, those incompatible with a definite causal structure—is however lacking. In this paper, we identify two types of circuits that naturally generalize the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterize. We first introduce “quantum circuits with classical control of causal order,” in which the order of operations is still well defined, but not necessarily fixed in advance: it can, in particular, be established dynamically, in a classically controlled manner, as the circuit is being used. We then consider “quantum circuits with quantum control of causal order,” in which the order of operations is controlled coherently. The supermaps described by these classes of circuits are physically realizable, and the latter encompasses all known examples of physically realizable processes with indefinite causal order, including the celebrated “quantum switch.” Interestingly, it also contains other examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process. Nevertheless, we show that quantum circuits with quantum control of causal order can only generate “causal” correlations, compatible with a well-defined causal order. We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterizations derived in this work allow us to identify advantages for quantum information processing tasks that could be demonstrated in practice.6 MoreReceived 29 January 2021Accepted 29 June 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030335Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum channelsQuantum computationQuantum controlQuantum correlations in quantum informationQuantum networksQuantum Information