Universal control induced by noise

On the basis of the quantum Zeno effect, it has been recently shown [D. K. Burgarth et al., Nat. Commun. 5, 5173 (2014)] that a strong-amplitude-damping process applied locally on a part of a quantum system can have a beneficial effect on the dynamics of the remaining part of the system. Quantum operations that cannot be implemented without the dissipation become achievable by the action of the strong dissipative process. Here we generalize this idea by identifying decoherence-free subspaces (DFSs) as the subspaces in which the dynamics becomes more complex. Applying methods from quantum control theory, we characterize the set of reachable operations within the DFSs. We provide three examples that become fully controllable within the DFSs while the control over the original Hilbert space in the absence of dissipation is trivial. In particular, we show that the (classical) Ising Hamiltonian is turned into a Heisenberg Hamiltonian by strong collective decoherence, which provides universal quantum computation within the DFSs. Moreover, we perform numerical gate optimization to study how the process fidelity scales with the noise strength. As a by-product, a subsystem fidelity that can be applied in other optimization problems for open quantum systems is developed.

Figure 1

Schematic representation of an $N$-level atom with a higher-lying unstable level $|e\rangle$ that decays with rates ${\gamma }_1,...,{\gamma }_N$ to the lower-lying levels $|1\rangle ,...,|N\rangle$ spanning a DFS.

Figure 2

Numerical gate optimization for the two-qubit model in Sec. 4a with the amplitude damping (16) for different values of $\gamma$. The gate error between the reduced dynamics ${\mathcal{E}}_T^{\left(1\right)}$ and the Hadamard gate on qubit 1 obtained from the numerical minimizations of ${\varepsilon }_1$ (green triangles) and ${\varepsilon }_2$ (blue points) for different values of $\gamma$ with gate time $T=1$. Qubit 2 is initially prepared in the totally mixed state, and for ${\varepsilon }_1, \tilde{\mathcal{E}}$ is chosen to be the superprojection $\mathcal{P}$ that brings qubit 2 into the ground state $|0\rangle$. To reduce the effect of local minima in the minimum value 100 randomly chosen initial pulses are taken.