Cyclic cooling algorithm

We introduce a scheme to perform the cooling algorithm, first presented by Boykin et al. in 2002, for an arbitrary number of times on the same set of qbits. We achieve this goal by adding an additional SWAP gate and a bath contact to the algorithm. This way one qbit may repeatedly be cooled without adding additional qbits to the system. By using a product Liouville space to model the bath contact we calculate the density matrix of the system after a given number of applications of the algorithm.

Figure 1

Cyclical cooling algorithm. Three quantum gates are applied: First a SWAP gate, then a CNOT gate, and finally a CSWAP gate. Bath contact at inverse temperature $\beta \left(0\right)$ is symbolized by the boxes on qbit 2 and 3.

Figure 2

Trend of $\beta \left(n\right)$ as a function of cycles $n$ for short bath contact time $(\tau =1∕50T1)$. The three curves together describe the temperature evolution of qbit 1: $\beta \left(n\right)$ jumps from the upper curve to the lowest, then to the middle one, and up again.

Figure 3

$\beta \left(n\right)$ as a function of cycles $n$ for long bath contact time $(\tau =4T1)$. Here the final temperature is reached approximately after a few steps.

Figure 4

The final inverse temperature $\beta \left(300\right)$ as a function of different bath contact times $\tau$. The line at $\beta \left(300\right)∕\beta \left(0\right)=2$ represents the upper limit for infinite bath contact; the line at $\beta \left(300\right)∕\beta \left(0\right)=1,53$ marks the inverse temperature of qbit 1 after one application of the algorithm.

Figure 5

The efficiency ${\eta }_1$ of the first step $n=1$ of the algorithm.