Algorithmic Complexity of Quantum States

Physical Review Letters, 2005

We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.

RAIRO - Theoretical Informatics and Applications, 2014

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum finite automata with very little quantumness (so-called semi-quantum one-and two-way finite automata); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.

2001

2011

This handout was created in the context of a talk we gave at the \Graduate Seminar on Topics in Quantum Computation" by Prof. Nitin Saxena at the University of Bonn in the Summer Semester 2011. It is heavily based on the lecture notes [AAR] and the book [NC].

Advanced Quantum Technologies

The creation complexity of a quantum state is the minimum number of elementary gates required to create it from a basic initial state. The creation complexity of quantum states is closely related to the complexity of quantum circuits, which is crucial in developing efficient quantum algorithms that can outperform classical algorithms. A major question unanswered so far is what quantum states can be created with a number of elementary gates that scales polynomially with the number of qubits. In this work, it is first shown that for an entirely general quantum state it is exponentially hard (requires a number of steps that scales exponentially with the number of qubits) to determine if the creation complexity is polynomial. Then, it is shown that it is possible for a large class of quantum states with polynomial creation complexity to have common coefficient features such that, given any candidate quantum state, an efficient coefficient sampling procedure can be designed to determine if the state belongs to the class or not with arbitrarily high success probability. Consequently, partial knowledge of a quantum state's creation complexity is obtained, which can be useful for designing quantum circuits and algorithms involving such a state.

Mathematics, 2021

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Probably the first algebraic algorithm is the famous result of Lovasz, who proposed a Monte Carlo algorithm for finding the size of a maximum matching based on computing the rank of the Tutte matrix. During the talk we will cover some classical algebraic algorithms, the heart of which is finding rank or determinant of some matrix. Next we will investigate another algebraic tool, i.e. the Baur-Strassen’s theorem, where partial derivatives provide us with much more information within asymptotically same running time. We will show simple algebraic algorithms for problems such as computing the diameter, finding the shortest cycle or maximum weight perfect matching, where instances have integral weights from the range [−W,W ]. The talk is based on a joint work with Harold N. Gabow and Piotr Sankowski. Quantum Complexity of Matrix Multiplication

Journal of Mathematical Physics, 2009

A two-parameter family of complexity measures C ͑␣,␤͒ based on the Rényi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the Lopez-Ruiz-Mancini-Calbet complexity, which is recovered for ␣ = 1 and ␤ = 2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, ␣ or ␤, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator, and square well.

Physica A: Statistical Mechanics and its Applications, 2013

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