Holonomic Quantum Computation by Time dependent Decoherence Free Subspaces

We show how to realize nonadiabatic holonomic quantum computation in time-dependent decoherence free subspaces (TDFSs). In our scheme, the holonomy is not generated by computational bases in DFSs but time-dependent bases of TDFSs. Therefore, different from the traditional DFSs, the ancillary systems are not necessary in inducing holonomy, which saves qubits used in the holonomic quantum computation. We also analyze the symmetry of the N -qubits system which couples to a common squeezed field. The results show that, there are several independent DFSs presented in Hilbert space, which is determined by eigenvalues of Lindblad operators. Combining the scheme and the model proposed in this paper, we show that, the one-qubit controllable phase gate can be realized by only two physical qubits.