Checkpoint: Toric-Code Logic

This checkpoint consolidates Module 8. You should be able to take an $L \times L$ toric code and, from first principles, state its parameters, identify its stabilizers and logical operators, and explain why each number is what it is. We work the bookkeeping end-to-end and then you reproduce it in code.

The parameter derivation, start to finish

Physical qubits. One qubit per edge of the square lattice on the torus. With $L^2$ vertices and $L^2$ faces, periodic boundary conditions give $2L^2$ edges, so

Stabilizers and their redundancy. There are $L^2$ star operators and $L^2$ plaquette operators, but each family has exactly one relation:

because every edge is counted twice. So there are $L^2 - 1$ independent stars and $L^2 - 1$ independent plaquettes, for

Encoded qubits. Subtracting,

Equivalently $k = 2g$ for a genus-$g$ surface; the torus has $g = 1$. The two logical qubits correspond to the two independent noncontractible cycles, with logical operators $\bar Z_i$ ($Z$-strings around direct cycles) and $\bar X_i$ ($X$-strings around dual cycles) satisfying $\{\bar X_i, \bar Z_i\} = 0$.

Distance. A logical operator is a noncontractible loop; the shortest such loop wraps one direction of the torus in a straight line of $L$ edges. Hence

Putting it together, the $L \times L$ toric code is

A concrete instance

For $L = 4$: $n = 2 \cdot 16 = 32$ qubits, $k = 2$ logical qubits, $d = 4$. The code is $[[32, 2, 4]]$ and corrects $\lfloor (4-1)/2 \rfloor = 1$ arbitrary error (even distance, so a weight-2 error can be ambiguous — another reason odd distances are preferred in practice).

Try it

Implement toricParams(L) returning $\{n, k, d\}$ for the $L \times L$ toric code from the derivation above, then return the sum $n + k + d$ for $L = 4$. The grader checks the combined value.

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