 
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 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
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 efficient algorithms
for computing the inner product between two stabilizer states, putting any nqubit stabilizer
circuit into a "canonical form" that requires at most O n2
/ 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.
