Overview

This repository contains a hybrid algorithm that integrates quantum simulations with classical machine learning for error correction in topological quantum codes. The approach leverages the mathematical structure of Lie algebras and topological anyons, combined with dynamic error correction strategies and machine learning techniques.

Mathematical Framework

1. Lie Algebra for Quantum Operations

We use the Lie algebra associated with the special unitary group $\text{SU}(2)$, defined by the Pauli matrices:

Pauli X ($σ_x$):

$$ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$

Pauli Y ($σ_y$):

$$ \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$

Pauli Z ($σ_z$):

$$ \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$

The commutator of two matrices $A$ and $B$ is given by:

$$ [A, B] = AB - BA $$

2. Monoid Structure

We define a monoid structure with an identity matrix $I$ and a matrix multiplication operation:

$$ M(A, B) = AB $$

For a sequence of matrices ${M_1, M_2, \ldots, M_n}$, the braiding operation is:

$$ \text{Braiding}(M_1, M_2, \ldots, M_n) = M_n \cdots M_2 M_1 $$

3. Topological Anyons

Topological anyons are described by their fusion and braiding rules. For anyons $A$ and $B$, fusion is governed by:

$$ A \times B = C $$

where $C$ is the resulting anyon. Braiding changes the quantum state by:

$$ \text{Braiding}(A, B) = B' $$

where $B'$ is the braided anyon.

4. Dynamic Error Correction

The dynamic error correction mechanism involves adjusting the correction matrix based on the fidelity $F$:

$$ F = \text{Tr}(\rho \sigma) $$

where $\rho$ is the actual state and $\sigma$ is the expected state. The correction matrix adapts as follows:

$$ \text{Correction}(F) = \begin{cases} I + \epsilon & \text{if } F < 0.1 \text{ (strong correction)} \\ I + 0.2 & \text{if } 0.1 \leq F < 0.5 \text{ (moderate correction)} \\ I + 0.05 & \text{if } 0.5 \leq F < 0.99 \text{ (mild correction)} \\ I & \text{if } F \geq 0.99 \end{cases} $$

5. Machine Learning for Error Detection

An MLPClassifier is trained on features extracted from noisy and ideal quantum states. The features are:

$$ \text{Features} = \text{Re}(\psi) \cup \text{Im}(\psi) $$

where $\psi$ is the quantum state vector.