Efficient Inner-product Algorithm for Stabilizer States

Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits aredescribed via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Suchstates are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase only) and can be represented compactly onconventional computers using Ω(n) bits, where n is the number of qubits [10]. Although techniques for the efficient simulationof stabilizer circuits have been studied extensively [1], [9], [10], techniques for efficient manipulation of stabilizer states are notcurrently available. To this end, we leverage the theoretical insights from [1] and [16] to design new algorithms for: (i) obtainingcanonical generators for stabilizer states, (ii) obtaining canonical stabilizer circuits, and (iii) computing the inner product betweenstabilizer states. Our inner-product algorithm takes O(n) time in general, but observes quadratic behavior for many practicalinstances relevant to QECC (e.g., GHZ states). We prove that each n-qubit stabilizer state has exactly 4(2−1) nearest-neighborstabilizer states, and verify this claim experimentally using our algorithms. We design techniques for representing arbitrary quantumstates using stabilizer frames and generalize our algorithms to compute the inner product between two such frames.