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.
- Received 29 January 2021
- Accepted 29 June 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.030335
Published 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 Society
Physics Subject Headings (PhySH)
Erratum
Erratum: Quantum Circuits with Classical Versus Quantum Control of Causal Order [PRX Quantum 2, 030335 (2021)]
Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard
PRX Quantum 4, 030901 (2023)
Popular Summary
The usual framework in quantum computing is that of quantum circuits, where quantum operations are applied to physical systems in a definite, fixed causal order. Recently, it has been found that quantum operations can be wired together in more exotic ways, e.g., with no well-defined causal order. Such situations are of interest for our fundamental understanding of causality, but they also open up prospects for quantum information processing. A central question is how such possibilities beyond the quantum circuit paradigm can be realized in practice. In our paper, we introduce and fully characterize two types of physically realizable circuits that go beyond the fixed-order case.
We first consider circuits where the causal order is still well defined, but the order of future operations can depend on previous ones. On that basis, we then introduce circuits in which the order is controlled by a quantum system, which can evolve throughout the circuit and whose quantum indeterminate nature can make the whole process incompatible with any well-defined causal order. We show how these circuits can be physically implemented and illustrate their features with some examples. We furthermore demonstrate how to compare the performance of these different types of circuits in quantum information processing tasks, and we present an example for which circuits with quantum control outperform those with definite causal order.
Our work opens the possibility of more extensive studies of such applications for physically realizable circuits with no well-defined causal order. This will shed light on the practical usefulness of indefinite causal order for quantum information processing.
