Improved simulation of stabilizer circuits

TLDR

The Gottesman–Knill theorem establishes that stabilizer circuits, composed of CNOT, Hadamard, and phase gates, can be efficiently simulated on a classical computer. This work extends the theorem by improving its efficiency and exploring its computational limits. The authors replace Gaussian elimination with a faster algorithm that doubles the bit representation, provide efficient inner‑product and canonical‑form procedures reducing gate counts to O(n²/ log n), and generalize the simulation to mixed states, limited non‑stabilizer gates, and tensor‑product initial states with few measurements. The resulting CHP program handles thousands of qubits, and the authors prove that stabilizer‑circuit simulation is ParityL‑complete, suggesting these circuits are unlikely to be classically universal.

Abstract

The Gottesman-Knill theorem says that a stabilizer circuit -- that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates -- 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-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 ParityL, 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/log n) gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.