• DOI:10.1103/PhysRevA.70.052328
• Corpus ID: 5289248

Improved Simulation of Stabilizer Circuits

@article{Aaronson2004ImprovedSO,
  title={Improved Simulation of Stabilizer Circuits},
  author={Scott Aaronson and Daniel Gottesman},
  journal={ArXiv},
  year={2004},
  volume={quant-ph/0406196}
}

• S. Aaronson, D. Gottesman
• Published 25 June 2004
• Computer Science
• ArXiv

The Gottesman-Knill theorem says that a stabilizer circuit\char22{}that is, a quantum circuit consisting solely of controlled-NOT (CNOT), Hadamard, and phase gates\char22{}can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor of 2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely available program called CHP (CNOT-Hadamard-phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class $\ensuremath{\bigoplus}\mathsf{L}$, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any $n$-qubit stabilizer circuit into a ``canonical form'' that requires at most $O({n}^{2}∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)$ gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.