Improved Graph Formalism for Quantum Circuit Simulation

Abstract

Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a new way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulation and establish limitations on reducing the quadratic runtime of applying controlled-Pauli Z gates. We do so by creating a simpler formula for combining two Pauli-related stabilizer states into one. Third, to better understand the linear dependence of stabilizer states, we characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our novel controlled-Pauli Z algorithm, we improve runtime for inner product computation from O(n 3 ) to O(nd 2 ) where d is the maximum degree of the graph.

Complete flow-preserving rewrite rules for MBQC patterns with Pauli measurements

“…Based on a recent proposal by Hu and Khesin [17], we then show how to make this new pseudonormal form unique, yielding a canonical form for stabilizer state diagrams in the ZX-calculus 1 . In the process, we correct a flaw in the uniqueness proof of Hu and Khesin, and simplify the arguments by making use of formalisms and results from the literature about holant problems.…”

“…It will be useful to give a canonical choice of free variables, this is inspired by Hu and Khesin's normal form for stabilizer states [17], and will lead us to an analogous normal form for stabilizer diagrams. Definition 3.1.…”

“…Apart from the translation into our terminology, this differs from the normal form definition of Hu and Khesin [17] only by reversing the order: we ask for free variables to come first whereas they put them last. Our uniqueness proof, making use of the properties of the affine support of a stabilizer state is shorter and simpler than that in [17].…”

“…Apart from the translation into our terminology, this differs from the normal form definition of Hu and Khesin [17] only by reversing the order: we ask for free variables to come first whereas they put them last. Our uniqueness proof, making use of the properties of the affine support of a stabilizer state is shorter and simpler than that in [17]. Their uniqueness proof also contains a flaw: Claim III.8 of [17] (using very different terminology) effectively states that changing one of the free variables while keeping the condition that the overall bit string is in A has no effect on the values of any other variables, which is not true for dependent variables that depend on the given free variable.…”

“…To achieve completeness, we introduce a new unique normal form for stabilizer ZX-calculus diagrams, which is close to the MBQC form. This normal form is based on work by Hu and Khesin [17] using the stabilizer graph notation of Elliott, Eastin and Caves [16], like the original stabilizer ZX-calculus completeness result [2]. As the proof by Hu and Khesin is very difficult to follow and contains at least one incorrect claim, we give an alternative uniqueness proof using the language of affine spaces.…”