- #1

James1238765

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- TL;DR Summary
- How to do algebra on the Kitaev toric code grid?

The toric code is a basic computational model as follows:

There are 2 operations that can be performed, A and B, on this grid.

To compute the value A at each point on the grid, we transform the raw values at each dot (located in between two vertices) according to some predefined operators ##f_a## and ##f_b##, and multiply the four surrounding values from the operations:

$$ A = f_a(left) f_a(up) f_a(right) f_a(down) $$

$$ B = f_b(left) f_b(up) f_b(right) f_b(down) $$

This seems to be a straightforward question, but I could not find a clear answer anywhere.

A commonly used toric code in physics is the model by A. Kitaev (https://www.physics.rutgers.edu/grad/602/Lectures/JC_Presentations/0419/Intro_Toric_Code.pdf)

What confuses me is what type of *quantity* is prescribed at each dot, be it a scalar ##0.24##, or a complex number ##0.24 + 0.12 i##, or a vector ##[0.24, 0.12]## in this Kitaev toric code model?

Also what is the form of the operator function ##f_a## and ##f_b##, be they 2x2 matrices, or something else?

There are 2 operations that can be performed, A and B, on this grid.

To compute the value A at each point on the grid, we transform the raw values at each dot (located in between two vertices) according to some predefined operators ##f_a## and ##f_b##, and multiply the four surrounding values from the operations:

$$ A = f_a(left) f_a(up) f_a(right) f_a(down) $$

$$ B = f_b(left) f_b(up) f_b(right) f_b(down) $$

This seems to be a straightforward question, but I could not find a clear answer anywhere.

A commonly used toric code in physics is the model by A. Kitaev (https://www.physics.rutgers.edu/grad/602/Lectures/JC_Presentations/0419/Intro_Toric_Code.pdf)

What confuses me is what type of *quantity* is prescribed at each dot, be it a scalar ##0.24##, or a complex number ##0.24 + 0.12 i##, or a vector ##[0.24, 0.12]## in this Kitaev toric code model?

Also what is the form of the operator function ##f_a## and ##f_b##, be they 2x2 matrices, or something else?