The paper "Improved Simulation of Stabilizer Circuits" by Scott Aaronson and Daniel Gottesman improves the Gottesman-Knill theorem, which states that stabilizer circuits can be efficiently simulated on a classical computer. The improvements include: 1. **Efficiency**: By removing the need for Gaussian elimination, the simulation algorithm becomes much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. The improved algorithm is implemented in a freely available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits. 2. **Complexity**: The problem of simulating stabilizer circuits is shown to be complete for the classical complexity class $\oplus L$, indicating that stabilizer circuits are not even universal for classical computation. 3. **Canonical Form**: The authors provide efficient algorithms for computing the inner product between two stabilizer states and put any $n$-qubit stabilizer circuit into a "canonical form" that requires at most $O(n^2 / \log n)$ gates. 4. **Extensions**: The simulation algorithm is extended to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states with a limited number of measurements. The paper also discusses the implementation and experiments of the CHP program, demonstrating its practicality for simulating large stabilizer circuits. Additionally, it explores the complexity of simulating stabilizer circuits, proving that the problem is in $\oplus L$. Finally, the authors present a canonical form theorem for stabilizer circuits, showing that any stabilizer circuit can be transformed into a sequence of Hadamard, CNOT, and phase gates, with an equivalent circuit requiring only $O(n^2 / \log n)$ gates.The paper "Improved Simulation of Stabilizer Circuits" by Scott Aaronson and Daniel Gottesman improves the Gottesman-Knill theorem, which states that stabilizer circuits can be efficiently simulated on a classical computer. The improvements include: 1. **Efficiency**: By removing the need for Gaussian elimination, the simulation algorithm becomes much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. The improved algorithm is implemented in a freely available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits. 2. **Complexity**: The problem of simulating stabilizer circuits is shown to be complete for the classical complexity class $\oplus L$, indicating that stabilizer circuits are not even universal for classical computation. 3. **Canonical Form**: The authors provide efficient algorithms for computing the inner product between two stabilizer states and put any $n$-qubit stabilizer circuit into a "canonical form" that requires at most $O(n^2 / \log n)$ gates. 4. **Extensions**: The simulation algorithm is extended to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states with a limited number of measurements. The paper also discusses the implementation and experiments of the CHP program, demonstrating its practicality for simulating large stabilizer circuits. Additionally, it explores the complexity of simulating stabilizer circuits, proving that the problem is in $\oplus L$. Finally, the authors present a canonical form theorem for stabilizer circuits, showing that any stabilizer circuit can be transformed into a sequence of Hadamard, CNOT, and phase gates, with an equivalent circuit requiring only $O(n^2 / \log n)$ gates.