On the geometry of stabilizer states

Abstract

Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by {\em stabilizer circuits} (consisting of CNOT, Hadamard and Phase gates) and can be represented compactly on conventional computers using $\Theta (n^2)$ bits, where $n$ is the number of qubits \cite{Gottes98}. As an additional application, the work in~\cite{Aaron, AaronGottes} suggests the use of superpositions of stabilizer states to represent arbitrary quantum states. To aid in such applications and improve our understanding of stabilizer states, we characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizer states, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizer states and short linear combinations of stabilizer states, develop an improved inner-product computation for stabilizer states via synthesis of compact canonical stabilizer circuits, propose an orthogonalization procedure for stabilizer states, and evaluate several of these algorithms empirically.