Quantum Computing from Graphs

Abstract

While stabilizer tableaus have proven exceptionally useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or coding algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures. Specifically, the graphs take a semi-bipartite form wherein input nodes map to output nodes, such that output nodes may connect to each other but input nodes may not. Intuitively, the graph’s input-output edges represent information propagation of the encoding circuit, while output-output edges represent the code’s entanglement structure. We prove that this graph representation is in bijection with tableaus and give an efficient compilation algorithm that transforms tableaus into graphs. We then show that this map is efficiently invertible, which gives a new universal recipe for code construction by way of finding graphs with sufficiently nice properties.

The graph representation gives insight into both code construction and algorithms. To the former, we argue that graphs provide a flexible platform for building codes particularly at small non-asymptotic scales. We construct as examples several constant-size codes and several infinite families codes. We also leverage graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. To the latter, we show that key coding algorithms, distance approximation, weight reduction, and decoding, are unified as instances of a single optimization game on a graph. Moreover, key code properties such as distance, weight, and encoding circuit depth, are all controlled by the graph degree. We give efficient algorithms for producing simple encoding circuits whose depths scale as twice the degree and for implementing logical diagonal and certain Clifford gates with non-constant but reduced depth. Finally, we construct a simple efficient decoding algorithm and prove a performance guarantee for a certain classes of graphs. These results give evidence that graphs are generically useful for the study of quantum computing and its practical implementations.