Figure 1

A stabilizer state $\mid \psi ⟩$ represented in different ways: (a) as stabilizer tableau; i.e., the state is stabilized by the group of Pauli operators generated by the operators in the four rows. This representation needs space $O\left(N^2\right)$ for $N$ qubits. (b), (c) as LC equivalence to a graph state. (b) shows the graph, with the VOP’s given by their decomposition into the group generators $\{H,S\}$. (c) is the data structure that represents (b) in our algorithm. The VOP’s are now specified using numbers between 0 and 23 (which enumerate the $\mid {\mathcal{C}}_1\mid =24$ LC operators). Here, we need space $O\left(N\overline{d}\right)$, where $\overline{d}$ is the average vertex degree—i.e., the average length of the adjacency lists. Writing $G$ for the graph in (b), we can use the notation of Eq. (5) and write $\mid \psi ⟩=\mid G;H,I,HS,S⟩$.Reuse & Permissions

Figure 2

Pseudocode for controlled phase gate $\left(\Lambda Z\right)$ acting on vertices $a$ and $b$ (“cphase”) and for the two auxiliary routines “remove VOP” and “local complementation”.Reuse & Permissions

Figure 3

A simple example in PYTHON.Reuse & Permissions

Figure 4

Comparison of the performance of CHP and GRAPHSIM. A simulation of entanglement purification was used as sample application. The register has 1000 times the size of the states to hold an ensemble of 1000 states.Reuse & Permissions

Figure 5

Benchmark of GRAPHSIM for very large registers. Entanglement purification—specifically, the purification of ten-qubit cluster states with the protocol of Ref. 7—was used as sample problem. The register was filled up with cluster states to make a large ensemble, and two protocol steps were simulated. The average time per operation was obtained from the total run time [27].Reuse & Permissions