Non-unitary Evolution of Quantum Logics

Kendon, V

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

Antonio Zecca A

Bogdan Mielnik

Viv M. Kendon

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

David Finkelstein

Xiaodong Ding

In quantum mechanics,the evolution of a quantum state is a unitary evolution,the evolution process can be described by an action of a unitary operator on the quantum state and the process is time reversible.Based on this fundamental fact,Gerard t Hooft has introduced the concept of equivalence classes of quantum states and described the unitary evolutions of quantum states by unitary transformation between two equivalence classes.In the present paper,we will use the concept of equivalence classes and their transformations to construct universal quantum logic gates in the quantum information theory and give their generalized forms.In the end of this paper we will show that these quantum logic gates can be realized by some intrinsic interaction Hamiltonian of the system.