Figure 1

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) for CZ (○) and $XY$ (×) two-qubit gates using single-qubit random rotations, and CZ gates using single-qubit HZ gates (◻) (8 qubits, 100 implementations, all computational basis states). When using random single-qubit gates, both indicators come within ${10}^{-4}$ much more quickly for $XY$ two-qubit gates than for CZ gates. Modifying the CZ-based algorithm to use single-qubit HZ gates boosts the convergence rate so much that it even outperforms the $XY$-based algorithm with random single-qubit rotations.Reuse & Permissions

Figure 2

Markov spectral gap, $\Delta$, versus local gate parameter, $c$, for various qubit topologies using two-qubit CZ (main panel) and $XY$ gates (inset). Dark lines: $n=6$; light lines: $n=8$. For CZ gates, the open chain topology (solid lines) has a larger gap than any of the other topologies for single-qubit HZ gates, $c=0$, but the smallest gap for single-qubit random gates, $c=1∕3$ (vertical dotted line). The HZ gate is the optimal single-qubit gate for the open chain topology. The AA topology (…) has the smallest or second smallest gap (depending on the number of qubits) for HZ gates but the largest for random gates. Random gates are close to optimal single-qubit rotations for AA topology. The closed-chain topology (–⋅) has the second largest gap for both random rotations and HZ gates. The optimal single-qubit rotation for this topology would be $c\approx 0.18$. The star topology (dashed line) has the second smallest gap for both random and HZ single-qubit rotations and the optimal single-qubit gate (for eight qubits) is also $c\approx 0.18$. For random single-qubit gates, going from 6 to 8 qubits decreases the size of the gap for all topologies. For single-qubit HZ gates, the gap increases when going from 6 to 8 qubits for the open- and closed-chain topologies but decreases for the star topology. Inset: $\Delta$ versus $c$ for two iterations (such that all couplings have been utilized) of PR algorithms using two-qubit $XY$ gates. The closed-chain topology (–⋅) has a significantly larger gap than the open chain (solid line). The optimal single-qubit gate for eight qubits is the random rotation, $c\approx 1∕3$, for the closed chain, but $c\approx 1∕2$ for the open chain. Note the smaller gap (hence slower convergence rate) of $n=8$ as compared to $n=6$.Reuse & Permissions

Figure 3

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) for different qubit topologies using CZ two-qubit gates and random single-qubit rotations (8 qubits, 100 implementations, all computational basis states). The qubits have been arranged in an open chain (○), a star formation (∗), a closed chain (◻), and in such a way that all the qubits are connected to each other (◇). The convergence rate as a function of topology agrees with Markov analysis, see Fig. 2.Reuse & Permissions

Figure 4

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of the average global entanglement from the random value) for different gate topologies using CZ gates and single-qubit HZ gates (eight qubits, 100 implementations, all computational basis states). The qubits have been arranged in an open chain (○), closed chain (◻), a star formation (∗), and in such a way that all the qubits are connected to each other (◇). The decay rates of the star and AA topologies are now slower than that of the open and closed chain, as expected from the Markov analysis, see Fig. 2.Reuse & Permissions

Figure 5

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) for different gate topologies using $XY$ two-qubit gates and random single-qubit rotations (eight qubits, 100 implementations, all computational basis states). The qubits have been arranged in an open chain (○) and closed chain (◻). As expected, the convergence rate of the closed chain is faster than that of the open chain.Reuse & Permissions

Figure 6

(Color online) Markov spectral gap, $\Delta$, versus local gate parameter, $c$, for an eight-qubit open-chain topology for different probabilities, $p$, that each CZ gate will be implemented. The gap increases with $p$ for all values of c. Inset: Gap (⋅) and convergence rate (×) for HZ gates as a function of $p$. The HZ gate $(c=0)$ is the optimal single-qubit gate for all $p$ until $p=0.98$.Reuse & Permissions

Figure 7

(Color online) Markov spectral gap, $\Delta$, versus local gate parameter, $c$, for an eight-qubit closed-chain topology with probability $p$ that each CZ gate will be implemented. The HZ gate is optimal until $p\simeq 0.85$. Above that, a hump grows towards greater $c$ with increasing $p$. The height of the hump (size of the gap) reaches a maximum at $p\simeq 0.89$, $c\simeq 4∕90$ (see upper inset). As $p$ continues to grow, the peak of the hump moves to higher values of $c$ but now $\Delta$ shrinks. Lower inset: Gap (⋅) and convergence rate (×) for HZ gates as a function of $p$. For $0.9<p<1$, the decreased gap size is due to an eigenvalue representing coupling between the two parity subspaces of the system (see text) and thus is not reliable to fully determine the actual convergence rate of the circuit.Reuse & Permissions

Figure 8

(Color online) Markov spectral gap, $\Delta$, versus local gate parameter, $c$, for an eight-qubit star topology with probability $p$ that a given CZ gate will be implemented. The gap increases with $p$ and the HZ gate $(c=0)$ is the optimal single-qubit gate for $p\gtrsim 0.7$. Inset: Gap (⋅) and convergence rate (×) for HZ gates as a function of $p$.Reuse & Permissions

Figure 9

(Color online) Markov spectral gap, $\Delta$, versus local gate parameter, $c$, for an eight-qubit AA topology with probability $p$ that a given CZ gate will be implemented. The largest gap appears as a singularity at $p=0.5$ and $c=0$. For $p<0.5$, the gap increases with decreasing $c$, the HZ gate $(c=0)$ being optimal. For $p>0.5$, the largest possible gap decreases and the optimal single-qubit gate approaches random, $c=1∕3$, with increasing $p$. In addition, the size of the gap as a function of $c$ exhibits a sharp change of behavior after reaching its maximum. Inset: Gap (⋅) and convergence rate (×) for HZ gates as a function of $p$. The singularity at $p=0.5$ is easily seen, as is the difference in the behavior of the gap size as a function of $p$ above and below this singularity.Reuse & Permissions

Figure 10

$1-\Delta$, versus number of qubits for the AA topology using probabilistically applied CZ gates. Dark curves: $c=0.01$ and $p=0.25$ (○), 0.35 (×), and 0.45 (◻). Light curves: $c=0.1$ using the same values of $p$. In both cases, the gap approaches $1-c$ for increasing $n$, according to Eq. (6). Upper inset: $1-\Delta$ versus number of qubits for the AA topology when $c=0$ and $p=0.2$ (dotted line), 0.4 (dashed line), and 0.5 (solid line). The gap approaches one at an exponential rate for all $p$, with the fastest convergence at $p=0.5$, according to Eqs. (7, 9). Lower inset: Gap versus number of qubits for deterministic AA topology, $p=1$, for single-qubit HZ (○) and random (×) gates. The gap grows with increased $n$ until saturating at about 0.3596 for the HZ gates and 0.4444 for the random gates.Reuse & Permissions

Figure 11

Spectral gap, $\Delta$, of a PR circuit using an AA topology as a function of the number of qubits, $n$, with two-qubit gates applied with probability $p=\frac{2}{n-1}$ such that at each iteration the number of gates performed is on average $n$. The gap asymptotically approaches a constant value of $\approx 0.705$ (dashed line). The upper inset shows the inverse gap, $1∕\Delta$, as a function of $n$ for $p=\frac{2}{n(n-1)}$ so that on average one two-qubit gate is performed per iteration. The gap scales as $\propto 1∕n$. The lower inset shows $(1-\Delta )$ for 50 qubits as a function of $p$. The gap is largest, and hence convergence quickest, at $p=0.5$. The approximate formula for the gap when $p<0.5$ is given by $\Delta =1-1∕2^{np}$ and is shown as a dashed line.Reuse & Permissions

Figure 12

$l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) versus number of iterations for open-chain topologies (eight qubits, 100 implementations) in which each qubit undergoes equivalent random rotations (○) or every other qubit undergoes equivalent random rotations (×). Both random single-qubit gates (dark) and HZ gates (light) were used. These are compared to circuits in which each qubit undergoes a different rotation (◻). Interestingly, using different rotations on all odd and all even qubits does not improve the convergence rate beyond the one achievable by using only one collective rotation for all qubits.Reuse & Permissions

Figure 13

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) versus number of iterations of a PR algorithm for eight qubits (500 samples) and initial states $a=0$ (○), $a\simeq 0.02337$, corresponding to a state with random entanglement (△),0.1 (×), 0.2 (◻), 0.3 (+), and 1 (◇). All states converge to random at the same rate. Note that the $a=0$ case has a four or five iteration advantage over the $a=1$ initial state. Lower inset: Entanglement of the initial states versus $a$. The random entanglement value is $\simeq 0.9883$.Reuse & Permissions

Figure 14

(Color online) $l_2$ distance between PR distribution of squared moduli of components in the computational basis and the PT distribution (inset: distance of average global entanglement from random value) versus number of iterations for different cluster state topologies (eight qubits, 500 implementations): two-dimensional lattice (△), cylinder (◻), star topology measuring the central qubit first (×), star topology measuring the central qubit last (○), and AA (◇). The cylindrical lattice converges most quickly. The GHZ-type topologies (star and AA) converge more slowly and are, in this way, similar to the AA topologies of circuit-model PR algorithms using two-qubit CZ and one-qubit HZ gates. The open and closed chain exhibit cutoff behavior in the entanglement convergence [23].Reuse & Permissions