A streamlined demonstration that stabilizer circuits simulation reduces to Boolean linear algebra
Vsevolod I. Yashin
TL;DR
This work reframes stabilizer-circuit simulation by extending the stabilizer tableau formalism to describe Clifford channels via modified Choi-state tableaux. It shows that composing channels reduces to a Boolean linear-algebra problem (finding a basis in the intersection of vector spaces), and that a circuit can be viewed as a diagram of tableaux whose contraction implements the overall channel, with Gaussian elimination driving the simulation. The approach unifies stabilizer-state and stabilizer-channel perspectives, provides a diagrammatic view of simulation, and highlights avenues for tensor-network style contraction techniques and efficient contraction strategies. The practical impact lies in offering a pedagogical and potentially performance-oriented framework for stabilizer simulation and error-propagation studies, with natural extensions to qudits and other stabilizer theories.
Abstract
Gottesman-Knill theorem states that computations on stabilizer circuits can be simulated on a classical computer, conventional simulation algorithms extensively use linear algebra over bit strings. For instance, given a non-adaptive stabilizer circuit, the problem of computing the probability of a given outcome (strong simulation) is known to be log-space reducible to solving the system of linear equations over Boolean variables, which is commonly done by Gaussian elimination. This note aims to make the connection between stabilizer circuits and Boolean linear algebra even more explicit. To do this, we extend the stabilizer tableau formalism to include stabilizer tableau descriptions of arbitrary stabilizer operations (Clifford channels). Finding the tableau corresponding to the composition of two channels becomes a linear algebra problem. Any stabilizer circuit rewrites to a diagram with stabilizer tableaux on vertices, contracting an edge means to take the composition of channels, to compute the result of the circuit means to fully contract the diagram. Thus, simulating stabilizer circuits reduces to a sequence of Gaussian eliminations. This approach gives a new perspective on explaining the work of stabilizer tableau methods (reproducing the asymptotics) and creates opportunity for exploring various tensor-contraction techniques in stabilizer simulation.