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arXiv:2110.05937

Kolmogorov complexity of unitary transformations in quantum computing

Alexei Kaltchenko

Published Oct 10, 2021• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2110.05937

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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.
The purpose of this thesis is to give a formal definition of quantum Kolmogorov complexity (QC), and rigorous mathematical proofs of its basic properties. The definition used here is similar to that by Berthiaume, van Dam, and Laplante. It defines the QC of some indeterminate-length qubit string ρas the minimum base length of any quantum input which makes a quantum Turing machine (QTM) halt and output ρ, up to some error tolerance. First, we prove that there is a QTM which is universal in the sense of input base length. Furthermore, we show several general properties of QTMs, including a result on mutually orthogonal ``halting spaces'', and a way to transform every almost-halting input into a deterministically-halting input by adding at most a constant number of qubits. Afterwards, we apply these results to QC. In particular, we show that QC is invariant, incompressible, agrees with classical Kolmogorov complexity for classical strings, and is closely related to von Neumann entropy for ergodic quantum information sources.
In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the shortest quantum input to a universal quantum Turing machine that produces the initial qubit string with high fidelity. The definition of P. Vitanyi (2000) measures the amount of classical information, whereas we consider the amount of quantum information in a qubit string. We argue that our definition is natural and is an accurate representation of the amount of quantum information contained in a quantum state.