Algorithmic Cooling

This project is currently under development.

Reference

Moussa, Osama. "On heat-bath algorithmic cooling and its implementation in solid-state NMR." PhD diss., University of Waterloo, 2005.
Experiment: Baugh, Jonathan, Osama Moussa, Colm A. Ryan, Ashwin Nayak, and Raymond Laflamme. "Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance." Nature 438, no. 7067 (2005): 470-473.

Simple Demo¶

In the below figure, the left-most bar, is the probability of this 6-qubit system being in the state $\boxed{\rho=\ket{000000}\bra{000000}}$, as it undergoes the cooling process.

Physical limitations¶

Check if the following statement is true or not:

Given any finite temperature reservoir, we can use algorithmic cooling to cool a qubit to sufficiently low entropy state (e.g., $<10^{-4}$ bit flip error), maybe only limited by the gate error.

short answer: False.

For example, the green curve in the below plot saturates near $0.4$, which means that even if we take infinite steps to cool down a 3-qubit system, its first qubit's polarization would only reach near $0.4$, which is quite far away from our goal of $1.0$.

We can see from the plot as well that similar circumstances exist in 4-qubit system and 5-qubit system (saturates near $0.66$ and $0.91$ respectively).

How to cool it down more?

Or in other words, given:

an $n$-qubit system, and
a large heatbath with polarization $\epsilon_b$,

what's the lowest entropy state reachable by the first qubit in that system? (assume the system starts from the maximally mixed state)

For example, in the case above, the "achievable maximum one qubit polarization" under the three configurations:

$\{n=3, \epsilon_b=0.2\}$ is $0.40$,
$\{n=4, \epsilon_b=0.2\}$ is $0.66$,
$\{n=5, \epsilon_b=0.2\}$ is $0.91$.

We can generalize the case, and see how the combination of different $n$ and different $\epsilon_b$ affects the result:

In the plot below, we let $n\in\{3, 4, 5, 6, 7\}$ and $\epsilon_b\in[0.001, 1]$.

Observations

We can get a higher first qubit's polarization at the end if we:

use a cooler reservoir, or
start from a system with more qubits.

Consider the above plot, if we change the unit of the horizontal axis from "Heatbath Polarization $(\epsilon_b)$" to "Scaled Heatbath Polarization $(2^n\times\epsilon_b)$", and switch the vertical axis to log-scaled, then all curves would overlap on top of each other:

Observations

If the heatbath polarization is higher than $2^{-n}$, then we can cool the first qubit down to almost arbitrarily low temperature.
If the heatbath polarization is lower than $2^{-n}$, then there is a threshold to which we can cool the first qubit down (see the fitting line below): $$ \text{cooling threshold} = \epsilon_b\times 2^{n-2} $$

This threshold is theoretically deduced in Schulman, Leonard J., Tal Mor, and Yossi Weinstein. "Physical limits of heat-bath algorithmic cooling." Physical review letters 94.12 (2005): 120501.

Simple Comparison of Cooling Performances¶

Source codes¶

open in Google Colab | Algorithmic Cooling

open in Github | algorithmic-cooling