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Quantum Kolmogorov complexity and duality of unitary transformations and quantum states.

Alexei Kaltchenko

Published Oct 13, 2021• 0 citations
OpenAlex

Abstract

The Kolmogorov complexity of a quantum state is known as quantum Kolmogorov complexity. For every unitary transformation, we construct a corresponding quantum state. Then, we define the Kolmogorov complexity of the unitary transformation via the Kolmogorov complexity of the constructed quantum state and show the consistency of such an approach. Thus, we establish the duality of unitary evolutions and quantum states in quantum computing and quantum mechanics.

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The Kolmogorov complexity of a quantum state is known as quantum Kolmogorov complexity. For every unitary transformation, we construct a corresponding quantum state. Then, we define the Kolmogorov complexity of the unitary transformation via the Kolmogorov complexity of the constructed quantum state and show the consistency of such an approach. Thus, we establish the duality of unitary evolutions and quantum states in quantum computing and quantum mechanics.
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.
We show that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding string. It follows that quantum complexity is an extension of classical complexity to the domain of quantum states. This is true even if we allow a small probabilistic error in the quantum computer's output. We outline a mathematical proof of this statement, based on an inequality for outputs of quantum operations and a classical program for the simulation of a universal quantum computer.
The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach — our measure of the second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities, it is continuous. We give examples where the second quantized and quantum Kolmogorov complexity differ.
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.