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Decoupling with random diagonal unitaries

Yoshifumi Nakata, Christoph Hirche, Ciara Morgan, Andreas Winter

Published Jan 1, 2017• 14 citations
Publisher

Abstract

We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-Z and -X bases. This strategy was recently shown to achieve an approximate unitary 2-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary 2-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.

DOI
10.15488/4802

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We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-Z and -X bases. This strategy was recently shown to achieve an approximate unitary 2-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary 2-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.
We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli- Z and - X bases. This strategy was recently shown to achieve an approximate unitary 2 -design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary 2 -designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.
Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after ℓ repetitions achieves a Θ(d−ℓ)-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.
We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large ttt, applying a unitary chosen uniformly at random from an approximate ttt-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary ttt-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.
Consider a bipartite system, of which one subsystem, A , undergoes a physical evolution separated from the other subsystem, R . One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R , i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems , which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate two-designs are appropriate for decoupling even if the dimension of this system is large.
Proving achievability of protocols in quantum Shannon theory usually does not consider the efficiency at which the goal of the protocol can be achieved. Nevertheless it is known that protocols such as coherent state merging are efficiently achievable at optimal rate.We aim to investigate this fact further in a general one-shot setting, by considering certain classes of decoupling theorems and give exact rates for these classes. Moreover we compare results of general decoupling theorems using Haar distributed unitaries with those using smaller sets of operators, in particular ε-approximate 2-designs. We also observe the behavior of our rates in special cases such as ε approaching zero and the asymptotic limit.