Joint EigenValue Decomposition for Quantum Information Theory and Processing

The interest in quantum information processing has given rise to the development of programming languages and tools that facilitate the design and simulation of quantum circuits. However, since the quantum theory is fundamentally based on linear algebra, these high-level languages partially hide the underlying structure of quantum systems. We show that in certain cases of practical interest, keeping a handle on the matrix representation of the quantum systems is a fruitful approach because it allows the use of powerful tools of linear algebra to better understand their behavior and to better implement simulation programs. We especially focus on the Joint EigenValue Decomposition (JEVD). After giving a theoretical description of this method, which aims at finding a common basis of eigenvectors of a set of matrices, we show how it can easily be implemented on a Matrix-oriented programming language, such as Matlab (or, equivalently, Octave). Then, through two examples taken from the quantum information domain (quantum search based on a quantum walk and quantum coding), we show that JEVD is a powerful tool both for elaborating new theoretical developments and for simulation

Identification Of Quantum Encoder Matrix From A Collection Of Pauli Errors

International audienceQuantum information processing is a rapidly evolving field, due to promising applications in communications, cryptography, and computing. In this framework, there is a need to protect quantum information against errors, using quantum error-correcting codes. Efficient quantum codes, based on the stabilizer formalism (that exploits elements of the Pauli group), have been proposed. The stabilizer formalism allows one to simulate quantum codes and quantum errors using operations inside the Pauli group only, leading to huge gains in simulation time. However, to deeply study and simulate unconventional quantum errors and devices, there is a need to know the true quantum operator (represented by a unitary matrix). In this paper, we propose an algorithm, based on linear algebra, to identify the quantum encoder matrix from a collection of Pauli errors. The approach is two-steps. First, from a collection a Pauli errors whose matrix representation is diagonal, a search of common eigenvectors identifies the encoder matrix up to phase indeterminates. Second, additional Pauli errors with nondiagonal matrix representations are used to eliminate the remaining in-determinations. Simulation results are also provided to illustrate and validate the approach. Index Terms-quantum information, quantum error correction , Pauli errors, physical layer integrit

Hypercube quantum search: exact computation of the probability of success in polynomial time

The version of record of this article, first published in Quantum Information Processing, is available online at Publisher’s website: https://doi.org/10.1007/s11128-023-03883-9International audienceIn the emerging domain of quantum algorithms, Grover’s quantum search is certainly one of the most significant. It is relatively simple, performs a useful task and more importantly, does it in an optimal way. However, due to the success of quantum walks in the field, it is logical to study quantum search variants over several kinds of walks. In this paper, we propose an in-depth study of the quantum search over a hypercube layout. First, through the analysis of elementary walk operators restricted to suitable eigenspaces, we show that the acting component of the search algorithm takes place in a small subspace of the Hilbert workspace that grows linearly with the problem size. Subsequently, we exploit this property to predict the exact evolution of the probability of success of the quantum search in polynomial time