Quantum-information processing with noisy cluster states

We provide an analysis of basic quantum-information processing protocols under the effect of intrinsic nonidealities in cluster states. These nonidealities are based on the introduction of randomness in the entangling steps that create the cluster state and are motivated by the unavoidable imperfections faced in creating entanglement using condensed-matter systems. Aided by the use of an alternative and very efficient method to construct cluster-state configurations, which relies on the concatenation of fundamental cluster structures, we address quantum-state transfer and various fundamental gate simulations through noisy cluster states. We find that a winning strategy to limit the effects of noise is the management of small clusters processed via just a few measurements. Our study also reinforces recent ideas related to the optical implementation of a one-way quantum computer.

Figure 1

(Color online) (a) The ${\mathrm{CBB}}_1$ layout. (b) The operation simulated when qubit 1 is measured in the $B_1\left(\alpha \right)$ basis and $s_1^{xy\left(\alpha \right)}=0$ is obtained.

Figure 2

(Color online) The $C_{\pi }\mathrm{PHASE}$ gate simulated by ${\mathrm{CBB}}_2$ on two logical qubits $\mid Q_1⟩$ and $\mid Q_2⟩$.

Figure 3

(Color online) (a) The ${\mathrm{CBB}}_3$ configuration. (b) The equivalent quantum circuit corresponding to the operation simulated by ${\mathrm{CBB}}_3$ with $\alpha =\pi ∕2$ and $s_2^{xy(\pi ∕2)}=0$.

Figure 4

(Color online) The concatenation of two ${\mathrm{CBB}}_1$’s (Fig. 1) with two ${\mathrm{CBB}}_2$’s (Fig. 2).

Figure 5

(Color online) The concatenation of two ${\mathrm{CBB}}_1$’s and one ${\mathrm{CBB}}_2$ with measurements in the $B_1\left(0\right)$ and $B_3\left(0\right)$ bases.

Figure 6

(Color online) (a) The configuration obtained by concatenating two ${\mathrm{CBB}}_1$’s with a ${\mathrm{CBB}}_3$. The single-qubit measurements are in the $B_{1\to 3}(\pi ∕2)$ bases. (b) The corresponding equivalent quantum circuit. The boxed part is equivalent to a $C_{\pi }\mathrm{PHASE}$.

Figure 7

(Color online) Information flow through an $N$-qubit linear cluster.

Figure 8

(Color online) (a) Fidelity of information transfer with $a$ averaged over the single-qubit Bloch sphere and all phases ${\theta }_j=\theta (\forall j)$. From top to bottom curve, we show $N=3$ (◆), $N=5$ (★), $N=7$ (∎), and $N=9$ (▴). The horizontal line (•) represents the classical threshold $2∕3$. (b): Fidelity of information transfer with $a$ averaged over the Bloch sphere and ${\theta }_j$ averaged over Gaussian distributions, centered on ${\theta }_j=0$, with standard deviation $\sigma =0.5$ (★, dotted line) and $\sigma =1$ (◆, solid line). Again, the classical threshold is shown for comparison (∎).

Figure 9

(Color online) (a) The layout for a single-qubit rotation where the input logical state encoded on qubit 1 is rotated by $U_R$ and transferred to qubit 5 after the measurements shown. (b): A modified configuration with two redundant qubits (qubits 3 and 6) which are removed via ${\sigma }_x$ measurements.

Figure 10

(Color online) (a) The fidelity of one-qubit rotations decomposed using the Euler angles $(\zeta ,\nu ,\xi )$, when our model of noise is considered. The fidelity is plotted against a common standard deviation $\sigma$ on the unwanted phases for various Euler angles. We have considered $\zeta =\pi ∕4,\nu =\xi =0$ (◆, solid line), $\zeta =\nu =\pi ∕2,\xi =0$ (★, dashed line), $\zeta =0,\nu =\xi =\pi ∕4$ (∎, dot-dashed line), and $\xi =\zeta =0,\nu =\pi$ (▴, dotted line). (b) The difference $F_{R5}-F_{R7}$ between the gate fidelity of the configurations in Figs. 9a, 9b.

Figure 11

(Color online) (a) Squashed-I configuration for a CNOT simulation. The input (output) control and target logical qubit are qubits 1 (7) and 9 (15), respectively, with qubit 8 as a bridging qubit. (b) The equivalent quantum circuit as a concatenation of CBB’s and ECBB’s.

Figure 12

(Color online) (a) Fidelity for a squashed-I CNOT plotted against the unwanted phase $\theta$ and the input control-state coefficient $a$. In this plot $a=c$. (b) Gate fidelities for different simulations of a CNOT. From top to bottom curve, we show the fidelity of the four-qubit CNOT of Fig. 5, the helix configuration, the squashed-I, and the squashed-I with an additional bridging qubit.

Figure 13

(Color online) (a) Helix layout and measurement pattern for a CNOT simulated through a $C_{\pi }\mathrm{PHASE}$ gate (within the dashed square) and two Hadamard gates involving qubits 6, 9 and 8, 10. (b) Equivalent quantum circuit drawn by exploiting the concatenation of eight ${\mathrm{CBB}}_1$’s and one box-shaped ECBB involving qubits 1,2,3,4 (see Fig. 4).