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A formalism for quantum computing in decoherence-free subspaces is presented. The constructed subspaces are partial triangulated to an index related to environment. The quantum states in the subspaces are just projected states which are ruled by a subdynamic kinetic equation. These projected states can be used to perform ideal quantum logical operations without decoherence.

Recent publications have formulated a remarkable theory for Decoherence-Free (DF) subspaces and subsystems in which quantum computing is performed in a DF subspace although the total space is still subject to decoherence [

Consider a general quantum open system

Moreover, by replacing the Hamiltonian

The creation (destruction) correlation operator can be available by means of subdynamics theory. In fact, by introducing the eigen-projectors of the total Hamiltonian

This construction of SSKE or LSKE in subspace can be related to the original Schrödinger or Liouville equation. For instance, using the intertwining relations from subdynamics theory [

The second-order approximation for the LSKE also corresponds to the Master, Boltzmann, Pauli, and Fokker-Plank equations of kinetic theory and Brownian motion. For example, from the LSKE for a time-independent open system

An interesting advantage using the above formalism is to construct a precise decoherence-free (DF) subspace. It is remarkable that the projected space on which subdynamics operates is a kind of DF subspace that occurs naturally by choosing a suitable basis to expand that subspace. In fact, if decoherence exists in a system arising from interactions with its environment, then the spectral decomposition of the Hamiltonian can be expressed using the subdynamic formalism as

For example, let us consider a typical two-qubit quantum computing system

In the presence of the environment, the nonideal action of the swap operation must be considered because it may introduce decoherence in the ideal swap operation. Here the environment is assumed to consist of a set of two-level particles randomly embedded in an environment

To control the induced decoherence, we choose the time-independent eigen-projectors of

For canceling this phase shift, we consider the triangular decomposition of the

In the upper-triangular subspace

Different models treat the interaction of the system with its environments quite differently; here, we propose a general procedure to construct a DF subspace by triangulation. Suppose that the states used in quantum computing system are the eigenvectors of the free Hamiltonian

Now, if the system is subject to decoherence induced from the environment by (a general)

It may be necessary to emphasize that the states required to perform quantum computing in the DF (triangular) subspace are just the projected states

now cannot influence the original quantum Controlled-Not logic operation

The entanglement between the system and environment is canceled in the constructed partial triangular subspace although it exists indeed in the original total space or even in the subspace with the normal inner product. The role of upper-triangular inner product with respect to the index from environment in the subspace is only to cancel the decoherence (phase shift) from the environment. Then, how can one realize the above procedure in the practical procedure for the quantum computing? We suppose to establish an additional measure or count system to read or calculate some dates, such as the eigenvalues and eigenvectors from the original system, and transfer the relevant dates to the expression in the frame of the partial triangular subspace based on (

A scheme for quantum computing in the DF triangular subspaces is presented. The DF subspaces are ruled by the subdynamic kinetic equation (SKE). The used quantum computing states in the DF subspaces are just the projected states

This work was supported by the grants from the National Natural Science Foundation of China (Grant no. 60874087) as well as Canadian NSERC, CIPI, MMO, and CITO.