Abstract. A sequential network of quantum operations is efficiently described by its quantum comb [1], a non-negative operator with suitable normalization constraints. Here we analyze the case of networks enjoying symmetry with respect to the action of a given group of physical transformations, introducing the notion of covariant combs and testers, and proving the basic structure theorems for these objects. As an application, we discuss the optimal alignment of reference frames (without preestablished common references) with multiple rounds of quantum communication, showing that i) allowing an arbitrary amount of classical communication does not improve the alignment, and ii) a single round of quantum communication is sufficient. A quantum comb [1] describes a quantum network with N open slots in which an ordered sequence of variable quantum devices can be inserted, thus programming the quantum operation of the resulting circuit. Mathematically, a comb implements an admissible supermap [2, 3], that transforms an input network of N quantum operations into an output quantum operation. Having at disposal a suitable formalism opens the possibility of optimizing the architecture of quantum circuits for a large number of computational, cryptographic, and game-theoretical tasks, such as discrimination of
Abstract. A sequential network of quantum operations is efficiently described by its quantum comb [1], a non-negative operator with suitable normalization constraints. Here we analyze the case of networks enjoying symmetry with respect to the action of a given group of physical transformations, introducing the notion of covariant combs and testers, and proving the basic structure theorems for these objects. As an application, we discuss the optimal alignment of reference frames (without preestablished common references) with multiple rounds of quantum communication, showing that i) allowing an arbitrary amount of classical communication does not improve the alignment, and ii) a single round of quantum communication is sufficient. A quantum comb [1] describes a quantum network with N open slots in which an ordered sequence of variable quantum devices can be inserted, thus programming the quantum operation of the resulting circuit. Mathematically, a comb implements an admissible supermap [2, 3], that transforms an input network of N quantum operations into an output quantum operation. Having at disposal a suitable formalism opens the possibility of optimizing the architecture of quantum circuits for a large number of computational, cryptographic, and game-theoretical tasks, such as discrimination of
A sequential network of quantum operations is efficiently described by its quantum comb [1], a non‐negative operator with suitable normalization constraints. Here we analyze the case of networks enjoying symmetry with respect to the action of a given group of physical transformations, introducing the notion of covariant combs and testers, and proving the basic structure theorems for these objects. As an application, we discuss the optimal alignment of reference frames (without pre‐established common references) with multiple rounds of quantum communication, showing that i) allowing an arbitrary amount of classical communication does not improve the alignment, and ii) a single round of quantum communication is sufficient.
We present a framework to treat quantum networks and all possible transformations thereof, including as special cases all possible manipulations of quantum states, measurements, and channels, such as, e.g., cloning, discrimination, estimation, and tomography. Our framework is based on the concepts of quantum comb---which describes all transformations achievable by a given quantum network---and link product---the operation of connecting two quantum networks. Quantum networks are treated both from a constructive point of view---based on connections of elementary circuits---and from an axiomatic one---based on a hierarchy of admissible quantum maps. In the axiomatic context a fundamental property is shown, which we call universality of quantum memory channels: any admissible transformation of quantum networks can be realized by a suitable sequence of memory channels. The open problem whether this property fails for some nonquantum theory, e.g., for no-signaling boxes, is posed.
Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a (G×H)-invariant quantum comb for compact groups G and H using the corresponding Clebsch--Gordan transforms, which naturally extends to the G-covariant quantum comb. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation U∈SU(d) into its inverse U† or transpose U⊤. From numerics, we find a deterministic and exact unitary transposition protocol for d=3 with 7 queries to U. This protocol improves upon the protocol shown in the previous work, which requires 13 queries to U.
We introduce the concept of quantum supermap , describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.
Given black-box access to the input and output systems, we develop the first efficient quantum causal order discovery algorithm with polynomial query complexity with respect to the number of systems. We model the causal order with quantum combs, and our algorithms output the order of inputs and outputs that the given process is compatible with. Our algorithm searches for the last input and the last output in the causal order, removes them, and iteratively repeats the above procedure until we get the order of all inputs and outputs. Our method guarantees a polynomial running time for quantum combs with a low Kraus rank, namely processes with low noise and little information loss. For special cases where the causal order can be inferred from local observations, we also propose algorithms that have lower query complexity and only require local state preparation and local measurements. Our algorithms will provide efficient ways to detect and optimize available transmission paths in quantum communication networks, as well as methods to verify quantum circuits and to discover the latent structure of multipartite quantum systems.
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.