Quantum Error Correction

Summary and Intro

These are scripts that simulate quantum circuits in Qiskit across a depolarizing channel of:

$$ \mathcal{E}(\rho) = (1 - p)\rho + \frac{p}{3}\left( X\rho X^\dagger + Y\rho Y^\dagger + Z\rho Z^\dagger \right) $$

This means that theres a probability, $1-p$, of no error occuring and a probability of $p/3$ of either a Pauli-X ,Y or Z occuring.

We iterate through different values of $p$ and measure the errorate against the depolarizing channel which is plotted.

All codes used are of minimum distance, $d$=3, and are only able to correct up to one error.

Files

my_surface.py

This is simulates the [[13,1,3]] surface code with stabilizers:

X Stabilizers	Z Stabilizers
s₀ = X₀X₁X₃	s₆ = Z₀Z₃Z₅
s₁ = X₁X₂X₄	s₇ = Z₁Z₃Z₄Z₆
s₂ = X₃X₅X₆X₈	s₈ = Z₂Z₄Z₇
s₃ = X₄X₆X₇X₉	s₉ = Z₅Z₈Z₁₀
s₄ = X₈X₁₀X₁₁	s₁₀ = Z₆Z₈Z₉Z₁₁
s₅ = X₉X₁₁X₁₂	s₁₁ = Z₇Z₉Z₁₂

my_color.py

[[7,1,3]] color AKA Steane Code

X-Stabilizer	Z-Stabilizer
s₀ = X₀X₁X₂X₃	s₃ = Z₀Z₁Z₂Z₃
s₁ = X₂X₃X₄X₅	s₄ = Z₂Z₃Z₄Z₅
s₂ = X₁X₃X₅X₆	s₅ = Z₁Z₃Z₅Z₆

Rotated

[[9,1,3]] rotated surface code

X-Stabilizers	Z-Stabilizers
s₀ = X₁X₂	s₄ = Z₀Z₃
s₁ = X₀X₁X₃X₄	s₅ = Z₁Z₂Z₄Z₅
s₂ = X₄X₅X₇X₈	s₆ = Z₃Z₄Z₆Z₇
s₃ = X₆X₇	s₇ = Z₅Z₈

How to run files

please make sure each file is is in the same directory before running, as some functions are shared across them. as long as you have qiskit and matplotlib installed you should be able to run them. I do utilize multiprocessing libraries (concurrent futures) to parrallelize the code.

When you run them it may take a while as i do ALOT of runs (n), if you wish to change this go ahead but be aware of really inconsistent outputs.

Under the main heading, where the code is executed, the graphs should be displayed and the numerical outputs should be saved in a .npy file, once you run and save each file you can go to plots.py and run either of the functions with input parameters as the .npy files and some colors. The variables you want to save are highlighted in the code but to be clear they are:

qbers - array that stores the QBER values (indexed across probability)

degen_ratio - Array that stores degeneracy ratios (0 when disabled, again, indexed across probability)

If you change probability values (p_values) be sure to change it in the rest of the files and plots.py (that is, if you want to visualise all of them together)

I have tried to make this as simple and as customizable as possible, but as one can imagine, they're kind of a trade-off with eachother.

Feel free to mess around with it, just make sure file names and variable names are kept consistent

To enable degenracy just uncomment the code within the simulate_circuit function.

test.py contains the raw surface code except only with a single run, run it to understand what the functions and how the surface code corrects errors (the circuit is written out in full in this script). This is essential to understand before running other scripts (alot of the functions here are implemented in other files as well).

How a typical run should go

Run test.py to understand everything (if you want) OR

1. run each file my_color, my_surface and rotated_surface - each should have their own respective output plot as well

3. save outputs from each file
4. run a function from "plots.py" to visualise outputs