 
Summary: Improved Simulation of Stabilizer Circuits
Scott Aaronson #
University of California, Berkeley
Daniel Gottesman +
Perimeter Institute
The GottesmanKnill theorem says that a stabilizer circuitthat is, a quantum circuit con
sisting solely of CNOT, Hadamard, and phase gatescan be simulated e#ciently 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
factor2 increase in the number of bits needed to represent a state. We have implemented the
improved algorithm in a freelyavailable program called CHP (CNOTHadamardPhase), which
can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer
circuits is complete for the classical complexity class #L, which means that stabilizer circuits
are probably not even universal for classical computation. Third, we give e#cient algorithms
for computing the inner product between two stabilizer states, putting any nqubit stabilizer
circuit into a ``canonical form'' that requires at most O # n 2 / log n # gates, and other useful tasks.
Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits con
taining a limited number of nonstabilizer gates, and circuits acting on general tensorproduct
initial states but containing only a limited number of measurements.
PACS numbers: 03.67.Lx, 03.67.Pp, 02.70.c
