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International Conference on Micro- and Nano-Electronics (ICMNE)

Descriptional complexity of unitary transformations

Alexei Kaltchenko

Published Jan 31, 2022• 0 citations
Publisher

Abstract

Weintroduce the notion of descriptional complexity of nite-dimensional unitary transformations, which is closely related to the notion of Kolmogorov complexity of a quantum state. In quantum computing, quantum states are associated with quantum data, and unitary transformations of quantum states are associated with quantum computer programs.

DOI
10.1117/12.2620457

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Weintroduce the notion of descriptional complexity of nite-dimensional unitary transformations, which is closely related to the notion of Kolmogorov complexity of a quantum state. In quantum computing, quantum states are associated with quantum data, and unitary transformations of quantum states are associated with quantum computer programs.
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to positive operators, which are subsequently "purified". We discuss the optimality of our notion of Kolmogorov complexity in a broad sense and obtain a simple complexity bound.
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to unnormalized density operators, which are subsequently "purified" into unnormalized vectors in Hilbert space. We discuss the optimality of our notion of Kolmogorov complexity in a broad sense.
Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.
In analogy of classical Kolmogorov complexity we develop a theory of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above. With high probability a quantum object is incompressible. There are two alternative approaches possible: to define the complexity as the length of the shortest qubit program that effectively describes the object, and to use classical descriptions with computable real parameters.