Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication

Abstract

In communication complexity, a number of distant parties have the task of calculating a distributed function of their inputs, while minimizing the amount of communication between them. It is known that with quantum resources, such as entanglement and quantum channels, one can obtain significant reductions in the communication complexity of some tasks. In this work, we study the role of the quantum superposition of the direction of communication as a resource for communication complexity. We present a tripartite communication task for which such a superposition allows for an exponential saving in communication, compared to one-way quantum (or classical) communication; the advantage also holds when we allow for protocols with bounded error probability. Quantum computation and communication are typically assumed to happen on a definite causal structure, where the order of the operations carried on a quantum system is fixed in advance. However, the interplay between general relativity and quantum mechanics might force us to consider more general situations in which the metric, and hence the causal structure, is indefinite. Recently, a quantum framework has been developed with no assumption of a global causal order [5][6][7]. This framework can also be used to study quantum computation beyond the circuit model, for instance using the "quantum switch" as a resource -a qubit coherently controlling the order of the gates in a quantum circuit [8]. It has recently been realized experimentally [9].It was shown that this new resource provides a reduction in complexity to n black-box queries in a problem for which the optimal quantum algorithm with fixed order between the gates requires a number of queries that scales as n 2 [10]. The quantum switch is also useful in communication complexity; a task has been found for which the quantum switch yields an increase in the success probability, yet no advantage in the asymptotic scaling of the communication complexity was found [11]. Most generally, no information processing task is known for which the quantum switch (or any other causally indefinite resource) would provide an exponential advantage over causal quantum (or classical) algorithms.Here we find a tripartite communication complexity task for which there is an exponential separation in communication complexity between the protocol using the quantum switch and any causally ordered quantum communication scheme. The task requires no promise on inputs and is inspired by the problem of deciding whether a pair of unitary gates commute or anticommute, which can be solved by the quantum switch with only one query of each unitary [12]. If the parties are causally ordered, the number of qubits that needs to be communicated to accomplish the task scales linearly with the number of input bits, whereas the protocol based on the quantum switch only requires logarithmically many communicated qubits. This shows that causally indefinite quantum resources can provide an exponential advantage over causally ordered quantum resources (i.e., ...

Indefinite Causal Order in a Quantum Switch

“…Having transverse spatial modes as the target d.o.f. enables encoding qudits-as opposed to qubits-for investigating quantum communication with indefinite causal order in larger Hilbert space dimensions [5,8]. The benefits of using qudits for quantum information processing applications, such as improved security in quantum cryptography and higher information capacity in quantum communication, are well known [33,34].…”

“…This can also be seen as a particular case of "superposition of time evolution" [11]. The advantages provided by the quantum switch arise from the fact that it cannot be reproduced by an ordinary quantum circuit which uses the same number of black box operations [3][4][5][6][7].Here, we present an optical implementation of the quantum switch where the control system is the photon's polarization and the target is the transverse spatial mode. We verify indefinite causal order by introducing a causal witness [14,15], for which we obtain a value 18 standard deviations beyond the bound for definite ordering.…”

“…This can also be seen as a particular case of "superposition of time evolution" [11]. The advantages provided by the quantum switch arise from the fact that it cannot be reproduced by an ordinary quantum circuit which uses the same number of black box operations [3][4][5][6][7].…”

“…Strikingly, it has been proposed that quantum physics allows for nonclassical causal structures where the order of events is indefinite [1,2]. It has been theoretically shown that such a possibility provides an advantage for computation [3], communication complexity [4,5], and other information processing tasks [6][7][8]. Furthermore, investigations of indefinite causal orders suggest a promising route towards a theory that combines general relativity and quantum mechanics [9,10].…”

Quantum gravity as a promising new information processing resource

“…Suppose a heavy particle stays in a quantum state superposed by two position states, each position component will cause a space‐time bending according to the general relativity, thus the superposition of space‐time emerged. In a further sight, casual order or any other correlations always encoded in certain space‐time structure, this means the abovementioned superposition will induce an indefinite causal order, which has opened a heated research field for its potential applications in information processing in recent years …”