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Annales De L Institut Henri Poincare-physique Theorique

A note on quantising Kolmogorov systems

Lyn C. Thomas

Published Jan 1, 1974• 1 citation
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Second quantised information distance

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Abstract The Kolmogorov complexity of a string is the minimum length of a programme that can produce that string. Information distance between two strings based on Kolmogorov complexity is defined as the minimum length of a programme that can transform either string into the other one, both ways. The second quantised Kolmogorov complexity of a quantum state is the minimum average length of a quantum programme that can reproduce that state. In this paper, a second quantised information distance is defined based on the second quantised Kolmogorov complexity. It is described as the minimum average length of a transformation quantum programme between two quantum states. This distance's basic properties are discussed. A practical analogue of quantum information distance is also developed based on quantum data compression.

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AN OBSERVER-BASED DE-QUANTISATION OF DEUTSCH'S ALGORITHM

Cristian S. Calude, Matteo Cavaliere, Radu Mardare

Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".

International Journal of Foundations of Computer Science•Jan 1, 2011•1 citation
Abstract The Kolmogorov complexity of a string is the minimum length of a programme that can produce that string. Information distance between two strings based on Kolmogorov complexity is defined as the minimum length of a programme that can transform either string into the other one, both ways. The second quantised Kolmogorov complexity of a quantum state is the minimum average length of a quantum programme that can reproduce that state. In this paper, a second quantised information distance is defined based on the second quantised Kolmogorov complexity. It is described as the minimum average length of a transformation quantum programme between two quantum states. This distance's basic properties are discussed. A practical analogue of quantum information distance is also developed based on quantum data compression.
Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".