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In close analogy to the fundamental role of random numbers in classical information theory, random operators are a basic component of quantum information theory. Unfortunately, the implementation of random unitary operators on a quantum processor is exponentially hard. Here we introduce a method for generating pseudo-random unitary operators that can reproduce those statistical properties of random unitary operators most relevant to quantum information tasks. This method requires exponentially fewer resources, and hence enables the practical application of random unitary operators in quantum communication and information processing protocols. Using a nuclear magnetic resonance quantum processor, we were able to realize pseudo-random unitary operators that reproduce the expected random distribution of matrix elements.
Random numbers are a fundamental component of classical information theory, with practical applications including stochastic estimation, system identification, and cryptographic protocols. For example, in the important case of Monte Carlo simulation, sequences of random numbers permit an unbiased statistical estimation of quantities that are impractical to evaluate by exact methods. It is now clear that quantum theory provides a more general framework for information theory, one that offers important advantages over classical methods of computation and communication [for an introduction, see (1)]. In the quantum information paradigm, the basic elements are quantum state vectors [describing the possible states of the quantum bits (qubits)] and unitary operators (describing the desired transformations); information is encoded in a quantum state, and the computational algorithm, or communication protocol, is implemented via a sequence of unitary operators acting on that quantum state.
Quantum analogs of random numbers (i.e., random unitary operators and random quantum states) provide an equally useful and fundamental component to this emerging theory of quantum information. In the case of quantum communication, random quantum states are known to saturate the classical communication capacity of a noisy quantum channel (2). Moreover, sets of randomizing unitary operators enable the superdense coding of arbitrary quantum states (3) and lead to a decrease in the classical communication cost (in bits) for remote state preparation (4). Random unitary operators also allow the construction of more efficient data-hiding schemes and provide a means to reduce the key length required for the (approximate) encryption of quantum states (5). In the case of quantum processing, random unitary operators enable methods for characterizing the...