arXiv subjects

Markus Grassl

Publications and source records attributed to Markus Grassl.

77 records · Page 5Linked to original sources

arXiv1998

Computing Local Invariants of Qubit Systems

We investigate means to describe the non-local properties of quantum systems and to test if two quantum systems are locally equivalent. For this we consider quantum systems that consist of several subsystems, especially multiple qubits. We compute invariant polynomials, i. e., polynomial functions of the entries of the density operator which are invariant under local unitary operations. As an example, we consider a system of two qubits. We compute the Molien series for the corresponding representation which gives information about the number of linearly independent invariants. Furthermore, we present a set of polynomials which generate all invariants (at least) up to degree 23. Finally, the use of invariants to check whether two density operators are locally equivalent is demonstrated.

quant-ph↗

arXiv1998

Generalized Grover Search Algorithm for Arbitrary Initial Amplitude Distribution

Grover's algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T \sim O(\sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.

quant-ph↗

arXiv1997

On the Complexity of Quantum Searching Using Complex Queries

We discuss the quantum search algorithm using complex queries that has recently been published by Grover (quant-ph/9706005). We recall the algorithm adding some details showing which complex query has to be evaluated. Based on this version of the algorithm we discuss its complexity.

quant-ph↗

arXiv1997

A Note on Non-Additive Quantum Codes

A method to combine two quantum error-correcting codes is presented. Even when starting with additive codes, the resulting code might be non-additive. Furthermore, the notion of the erasure space is introduced which gives a full characterisation of the erasure-correcting capabilities of the codes. For the special case that the two codes are unitary images of each other, the erasure space and the pure erasure space of the resulting code can be calculated.

quant-ph↗

arXiv1997

Codes for the Quantum Erasure Channel

The quantum erasure channel (QEC) is considered. Codes for the QEC have to correct for erasures, i. e., arbitrary errors at known positions. We show that four qubits are necessary and sufficient to encode one qubit and correct one erasure, in contrast to five qubits for unknown positions. Moreover, a family of quantum codes for the QEC, the quantum BCH codes, that can be efficiently decoded is introduced.

quant-ph↗