Unifying Graph Measures and Stabilizer Decompositions for the Classical Simulation of Quantum Circuits

TL;DR

A unified framework is presented that bridges these two approaches based on stabilizer decompositions and tensor network contraction, placing them under a common formalism, and the refined complexity measures are introduced, always at least as efficient as their standard equivalent.

Abstract

Various algorithms have been developed to simulate quantum circuits on classical hardware. Among the most prominent are approaches based on \emph{stabilizer decompositions} and \emph{tensor network contraction}. In this work, we present a unified framework that bridges these two approaches, placing them under a common formalism. Using this, we present two new algorithms to simulate an $n$-qubit circuit $C$: one that runs in $\tilde{O}(T^{\mathsf{tw}(C)})$ time and the other in $\tilde{O}(T^{γ\cdot \mathsf{tw}(C)})$ time, where $\mathsf{tw}(C)$ and $\mathsf{rw}(C)$ refer to the the tree-width and rank-width, respectively, of a tensor network associated to $C$, $T$ is the number of non-Clifford gates in $C$, and $γ\approx 3.42$. The proposed algorithms are simple, only require a linear amount of memory, are trivially parallelizable, and interact nicely with ZX-diagram simplification routines. Furthermore, we introduce the refined complexity measures \emph{focused tree-width} and \emph{focused rank-width}, which are always at least as efficient as their standard equivalent; these can be directly applied within our simulation algorithms, allowing for a more precise upper bound on the run time.

Figures (4)

• Figure 1: An example of constructing a rank decomposition and tree of cuts for a ZX-diagram. The diagram we use in this example consists of eight nodes with arbitrary phases, connected as a clique with four edges removed. Left: This graph has rank-width one, and mixed rank-width two. A rank decomposition of the graph is given by the black nodes and edges that connect the green Z-spiders of the ZX-diagram. Each edge is labelled with its cut-rank. Center: Here we show a potential decomposition of the graph, first into the two green sub-diagrams obtained by performing a single bipartite decomposition cut along the center. Each of the sub-diagrams is then reduced to only one vertex by cutting the three vertices outlined in red and blue, respectively. Right: The same decomposition of the graph organized as a tree of cuts. Each interior node represents one decomposition operation, and the leaves represent the vertices of the graph. Leaves connected to interior nodes by dashed edges are the vertices to be cut. Each interior node is labelled with the total number of terms represented by its subtree. This example decomposes $8$ vertices into $64$ terms, resulting in $\alpha = 0.75$.
• Figure 2: The mixed rank-width and computed $\alpha$ values to decompose an Erdős-Rényi random graph with a varying number of vertices $n$ and probability of each edge $p$. We evaluated graphs with $n = 16$ to $64$ and $p = 0.1$ to $1$. We do not evaluate $p < 0.1$ to ensure that the graphs are not comprised of many small disconnected components. We see that when $p$ is small or large, the mixed rank-width and $\alpha$ decrease, while for intermediate $p$ the graphs almost always have maximal mixed rank-width. This is to be expected as rank-width is maximal with high probability for any constant $p$ as $n \to \infty$rwRandom. Note that the maximum $\alpha$ is not at $p = \frac{1}{2}$ because at lower densities vertex deletions are more prevalent, while at higher densities bipartite decompositions are required.
• Figure 3: The mixed rank-width and computed $\alpha$ values to decompose a graph-like ZX-diagram obtained from simplifying a uniformly random Clifford+$R_Z$ circuit. We evaluated circuits with between $16$ and $40$ qubits, and between $100$ and $1000$ gates, with a fixed $20\%$ fraction of non-Clifford gates. We observe that the computed $\alpha$ is essentially independent of both qubit count and non-Clifford vertex count. The mean $\alpha \approx 0.653$ is marked in a dashed line. Moreover, we see that the mixed rank-width appears to grow linearly with non-Clifford vertex count, but does not saturate the upper bound. This suggests that the graphs obtained from this process do not look like Erdős-Rényi graphs.
• Figure 4: The mixed rank-width and computed $\alpha$ values to decompose a graph-like ZX-diagram obtained from simplifying a circuit composed of Pauli gadgets pauliGadget with weight at most half the number of qubits. We sampled circuits with between $8$ and $32$ qubits and scaled the number of Pauli gadgets proportional to the number of qubits, by factors between one and four. We observe that, except for small graphs, the mixed rank-width upper bound is almost always saturated; the points that appear to exceed the upper bound have some remaining vertices that are Clifford, which are not taken into account in the bound. As the non-Clifford vertex count grows, the $\alpha$ value trends towards $1$. This indicates that these graphs have similar characteristics to Erdős-Rényi graphs with intermediate values of $p$, and are essentially the hardest case for our methods. We observe a rough trend of $\alpha \approx 1 - \frac{4}{n}$, which is consistent with the theoretical value obtained by cutting every vertex except a constant number.

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Definition 5
• Lemma 2: cygan2015parameterized, Lemma 7.19
• Lemma 3
• proof