Quantum pseudorandomness from cluster-state quantum computation

We show how to efficiently generate pseudorandom states suitable for quantum information processing via cluster-state quantum computation. By reformulating pseudorandom algorithms in the cluster-state picture, we identify a strategy for optimizing pseudorandom circuits by properly choosing single-qubit rotations. A Markov chain analysis provides the tool for analyzing convergence rates to the Haar measure and finding the optimal single-qubit gate distribution. Our results may be viewed as an alternative construction of approximate unitary $2$-designs.

Figure 1

Schematics of cluster-state PR architectures. Pairs of qubits subjected to CZ gates are connected by solid lines and each qubit is identified by the angle in the $x\text{−}y$ plane that defines its measurement basis. Dashed lines represent additional CZ gates for the enhanced version of the algorithm.

Figure 2

(Color online) Distance of the (normalized) distribution of squared moduli state-vector components from $P_{PT}\left(y\right)$ for random states as a function of run time. $◻$: Standard (6 rows and connections every third column), vs $○$: Enhanced (connections at every column) PR patterns. The run time equals the number of columns in the cluster state. Inset: Difference of global entanglement, $Q$, from the expected random-state value, $Q_R$, vs run time. For both test functions, the enhanced version of the algorithm converges to the Haar average with a rate about $6$ times faster than the standard rate.

Figure 3

(Color online) Gap, $\Delta \left(c\right)$, between ${\lambda }_1=1$ and the largest non-unit eigenvalue of $M^{\prime }$ vs $c$, for $n=6$ (solid line) and $n=10$ (dashed line). The gap at $c=0$ [$HZ\left(\alpha \right)$ gates] is significantly larger than the gap at $c=1∕3$ (arbitrary random single-qubit gates), identifying the optimal gate set. Inset: Gap for $c=1∕3$ (circles) and $c=0$ (squares) vs $n$.

Figure 4

(Color online) Total variation distance, $TV(\ell )=1∕2{\sum }_{\nu }\mid c_{\nu }^2(\ell )-c_{\nu }^2(\infty )\mid$, as a function of iteration for a PR circuit with random single-qubit gates (circles) and $HZ\left(\alpha \right)$ gates (squares). Data are shown for $n=6$ (solid lines) and $n=12$ (dashed lines). Inset: Difference of global Meyer-Wallach entanglement from the expected random-state value as a function of iteration for a PR map on $n=6$ qubits.