Figure 1

A resource state for quantum computation on a single qubit together with the input state. Circles are qubits and the qubit $\left({\mathrm{in}}_1\right)$ carries the input state. The figure shows the state before the measurements have been performed. Solid lines indicate CZ gates. The resource qubits are in the ∣+⟩ state before the CZ gate is applied. They are labeled $r_{11},r_{12},\dots ,r_{1l}$, where $l$ signifies the length of the circuit and the first and second index is the row and column in the graph, respectively.Reuse & Permissions

Figure 2

The initial weighted graph state resource used in our random circuit generation scheme prior to any measurements, together with the column of qubits carrying the input state. Circles represent qubits and the qubits $({\mathrm{in}}_1,\dots ,{\mathrm{in}}_N)$ carry the input state. Thin horizontal lines indicate CZ gates, while thicker vertical lines are $\varphi$–gates, defined in Eq. (2). The resource qubits are in the ∣+⟩ state before the CZ gates and $\varphi$–gates are applied. Each lie on a vertex $a=(j,k)$ in a graph $G$ and take the labeling $r_{jk}$. $j$ is the row index that runs from 1 to $N$, the number of qubits in the input state. $k$ is the column index that runs from 1 to $l$, the length of the circuit.Reuse & Permissions

Figure 3

(Color online) The average entanglement, $\mathbb{E}\mathbf{\left(}S\left({\rho }_A\right)\mathbf{\right)}$, (blue diamonds), of our randomized circuits compared with the average expected entanglement given by Page’s conjecture (dashed line) for $N=8$ and various $N_A$. The solid red line shows the same quantities for stabilizer gates chosen at random. Our final entanglement values are calculated by first specifying an initial input product state of $N$ qubits, where each qubit is chosen at random according to the parametrization $(\alpha \mid 0⟩+\beta \mid 1⟩)∕\sqrt{{\alpha }^2+{\beta }^2}$. This state is evolved for ${10}^4$ iterations, and averages are then taken over ${10}^5$ separate realizations. In each case, the fractional difference is less than ${10}^{-4}$ from that expected from true random circuits (see inset).Reuse & Permissions

Figure 4

(Color online) Comparisons of the entanglement distributions between the weighted graph states with random initial product states (see caption of Fig. 3) and states chosen at random with respect to the Haar measure, for $N=8$ and $N_A=1,2,3,4$ (plots from top to bottom). In both cases, we calculate ${10}^5$ samples and group into 500 evenly spaced bins. The distributions are then rescaled such that $\int \rho \left(S_A\right)=1$. Black dashed lines represent the weighted graph state entanglements, and blue solid lines represent the Haar distribution. For comparison, we also include the distributions for stabilizer states chosen at random (solid lines capped with circles), which are given in Eq. (8).Reuse & Permissions

Figure 5

(Color online) The scaling with $N$ of the time taken for the average entanglement to agree with that associated with the Haar measure to a fixed accuracy $\epsilon <0.0001$ (red squares), $\epsilon <0.001$ (blue circles), and $\epsilon <0.01$ (green triangles). For $N=2,3,4$, averages are over ${10}^6$ realizations, for $N=5,6$, averages are over ${10}^5$ realizations, and for $N=7,8$, averages are over ${10}^4$ realizations. In all cases, $\varphi =5\pi ∕8$ and the initial states are randomly chosen product states (see the caption of Fig. 3).Reuse & Permissions