Pseudo-Random Operators for Quantum Information Processing: Theory and Applications

Chiu Fan Lee, Neil F. Johnson

Joseph Emerson, Yaakov S. Weinstein, Marcos Saraceno +2 more

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 pseudorandom unitary operators that reproduce the expected random distribution of matrix elements.

Khole Timothy Poumai, Nikhil Khanna, Abdulla M. Jarrah +1 more

This paper develops a comprehensive framework for quantum information theory within the paradigm of pseudo-Hermitian quantum mechanics. We establish necessary and sufficient conditions for the existence of biorthogonal Riesz bases, a fundamental component in the formulation of pseudo-Hermitian quantum information. Additionally, we propose a spectral theorem for pseudo-Hermitian operators and explore their decomposition. We introduce key postulates of pseudo-Hermitian quantum mechanics and independently derive the time-evolution operator for pseudo-Hermitian systems. Furthermore, we extend the classical notion of quantum channels to pseudo-Hermitian quantum information theory and provide various characterizations, including the Kraus, Stinespring, and Choi representations. Our findings contribute to the mathematical foundations of non-Hermitian quantum mechanics and its applications in quantum information processing.

Osamu Hirota

Joseph Emerson

Pseudo‐random operators consist of sets of operators that exhibit important statistical features of uniformly distributed operators although their parametrization and generation requires exponentially fewer resources. Pseudo‐random operators can be applied for quantum communication tasks such as data‐hiding and super‐dense coding of quantum states, and enable efficient stochastic methods for noise estimation on prototype quantum processors. This paper summarizes published work by Emerson et al. [Science 302:2098 (2003)] demonstrating a random circuit method for the implementation of pseudo‐random unitary operators on a quantum processor and some more recent related results.

Apostol Vourdas