No 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.

We present a quantum compilation algorithm that maps Clifford encoders, an equivalence class of quantum circuits that arise universally in quantum error correction, into a representation in the ZX calculus. In particular, we develop a canonical form in the ZX calculus and prove canonicity as well as efficient reducibility of any Clifford encoder into the canonical form. The diagrams produced by our compiler explicitly visualize information propagation and entanglement structure of the encoder, revealing properties that may be obscured in the circuit or stabilizer-tableau representation.

We study scenarios which arise when two spatially-separated observers, Alice and Bob, are try to identify a quantum state sampled from several possibilities. In particular, we examine their strategies for maximizing both the probability of guessing their state correctly as well as their information gain about it. It is known that there are scenarios where allowing Alice and Bob to use LOCC offers an improvement over the case where they must make their measurements simultaneously. Similarly, Alice and Bob can sometimes improve their outcomes if they have access to a Bell pair. We show how LOCC allows Alice and Bob to distinguish between two product states optimally and find that a LOCC is almost always more helpful than a Bell pair for distinguishing product states.

We extend past results on a family of formal power series K n,Λ , parameterized by n and Λ ⊆ [n], that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product K n,Λ K m,Ω of any two functions K n,Λ and K m,Ω from the family can be expressed as a linear combination of other functions from the family. In this paper, we show that this is indeed the case and that the span of the K n,Λ 's forms an algebra. We also provide techniques for examining similar families of functions and a formula for the product K n,Λ K m,Ω when n = 1.