Decoherence-based exploration of $d$-dimensional one-way quantum computation: Information transfer and basic gates

We study the effects of amplitude and phase damping decoherence in $d$-dimensional one-way quantum computation. We focus our attention on low dimensions and elementary unidimensional cluster state resources. Our investigation shows how information transfer and entangling gate simulations are affected for $d⩾2$. To understand motivations for extending the one-way model to higher dimensions, we describe how basic qudit cluster states deteriorate under environmental noise of experimental interest. In order to protect quantum information from the environment, we consider encoding logical qubits into qudits and compare entangled pairs of linear qubit-cluster states to single qudit clusters of equal length and total dimension. A significant reduction in the performance of cluster state resources for $d>2$ is found when Markovian-type decoherence models are present.

Figure 1

(Color online) The linear qudit clusters studied in this work. Each physical qudit is affected by its local environment (jagged surroundings) by AD and PD decoherence. (a) Linear qudit clusters. (b) Entangled pair of linear qubit clusters.

Figure 2

(Color online) Fidelities of decoherence-affected linear qudit-clusters. In (a) and (b) the dashed (solid) lines correspond to $n=2$ $(n=5)$ with dimension $d=2\to 4$ from top to bottom in each line style. In (c) and (d), we compare $n=3$ qudit cluster states (solid lines) with three-qudit GHZ states (dashed lines) for $d=2\to 4$ from top to bottom in each line-style. We consider AD (a, c) and PD channels (b, d).

Figure 3

(Color online) Bipartite entanglement decay in two-qudit cluster states when decoherence affects the individual qudits. The concurrence of qubit cluster states affected by PD and AD (top solid and dashed lines, respectively) is compared to the normalized quasiconcurrence for $d=3$ (a) and $d=4$ qudit-cluster states (b). PD and AD correspond to the lower solid and dashed lines in each panel, respectively.

Figure 4

(Color online) Fidelity decay for arbitrary single qudits with $d=2$, 3, 4, and $2\otimes 2$ when affected by AD (a) and PD (b). In both panels the upper solid, lower solid, dashed-dotted, and dashed lines correspond to $d=2$, $2\otimes 2$, 3, and 4, respectively.

Figure 5

(Color online) Average fidelities of AD-affected qudits propagated along qudit cluster states of lengths $n=2\to 5$. In (a–c) the solid, dashed, dashed-dotted, and dotted lines correspond to $n=2$, 3, 4, and 5 length clusters, respectively. (a) AD for $d=2$. (b) AD for $d=3$. (c) AD for $d=4$ compared to $2\otimes 2$ (top lines at each $n$). (d) Even and odd length average fidelity as ${\Gamma }_{\mathrm{A}}t\to \infty$. The solid (dashed) line corresponds to odd (even) lengths.

Figure 6

(Color online) Average fidelities of PD-affected qudits propagated along qudit cluster states of lengths $n=2\to 5$. The solid, dashed, dashed-dotted, and dotted lines correspond to $n=2$, 3, 4, and 5, respectively. The top, middle, and bottom four lines correspond to dimensions $d=2$, 3, 4.

Figure 7

(Color online) Average fidelities of PD-affected qudits propagated along qudit cluster states of lengths $n=2$ and $n=5$ corresponding to solid and dotted lines, respectively. Comparison between $d=4$ and $d=2\otimes 2$. The top lines always correspond to an entangled pair for each $n$.

Figure 8

(Color online) Average fidelities of qubits encoded into decoherence-affected single qudits. The upper solid line corresponds to $d=2$ and $G$ encodings for $d=3$ and 4 in both (a) and (b). In (a), we consider AD. The dashed-dotted lines correspond to an $L$-encoded qubit for $d=2\otimes 2$, 3, and 4 (from top to bottom). The dotted lines correspond to $M$ encoding for $d=2\otimes 2$, $T$ encoding for $d=3$ and 4 from top to bottom, respectively. In (b), we consider PD. The $G$ and $T$ encodings for $d=3$ and 4 match the $d=2$ case. The dashed-dotted line is for an $O∕M$-encoded qubit in $d=2\otimes 2$. The dotted lines correspond to $E$ encoding for $d=3$ and 4 (from top to bottom). The lower solid line is for $L$ encoding in $d=2\otimes 2$.

Figure 9

(Color online) Average fidelities of AD-affected encoded qubits propagated along qudit cluster states of length $n=2$. The solid line corresponds to $d=2$, the dashed-dotted lines represent $L$ encoding for $d=2\otimes 2$, 3, and 4 (from top to bottom), and finally, dashed lines represent $G$ encoding for $d=3$ and 4, and $M∕O$ encoding for $d=2\otimes 2$ (from top to bottom).

Figure 10

(Color online) Average fidelities of AD-affected encoded qubits propagated along qudit cluster states of length $n=3$. The solid line represents $d=2$, dash-dotted lines represent $L$ encoding for $d=2\otimes 2$, 3 and 4 (from top to bottom), dashed lines represent $G$ encoding for $d=3$ and 4 (from top to bottom) and dotted lines represent $T$ encoding for $d=3$ and 4, and $M$ encoding for $d=2\otimes 2$ (from top to bottom).

Figure 11

(Color online) Average fidelities of PD-affected encoded qubits propagated along qudit cluster states of length $n=2$. The top solid line corresponds to $d=2$, the dashed lines to $T$ and $G$ encodings for $d=3$ and 4 (from top to bottom), the dashed-dotted lines to $L$ encoding for $d=3$ and 4 (from top to bottom). For $d=2\otimes 2$, the middle (bottom) solid line corresponds to $L$ $(O∕M)$ encoding.

Figure 12

(Color online) Average fidelities of PD-affected encoded qubits propagated along qudit cluster states of length $n=3$. The top solid line corresponds to $d=2$, the dashed lines to $G$ encoding for $d=3$ and $4$ (from top to bottom), the dashed-dotted lines to $L$ encoding for $d=3$ and 4 (from top to bottom), and the dotted lines to $T$ encoding for $d=3$ and $4$ (from top to bottom). For $d=2\otimes 2$, the middle (bottom) solid line corresponds to $L∕M$ $\left(E\right)$ encoding.

Figure 13

(Color online) Entanglement in decohered ${\mathrm{BBB}}_3$-produced cluster states. In all graphs, the solid (dashed) lines correspond to PD (AD). We show $c\left(\varrho \right)$ for $d=2$ (top two curves), ${\tilde{c}}_{qp}\left(\varrho \right)$ for $d=4$ (bottom two curves), and $c\left(\varrho \right)$ for two qubits encoded in the lowest levels of $d=4$ logical qudits and propagated through ${\mathrm{BBB}}_3$ (middle two curves).

Figure 14

(a), (c), (e) show the layouts of ${\mathrm{BBB}}_1$, ${\mathrm{BBB}}_2$ and ${\mathrm{BBB}}_3$. (b) The operation simulated on logical qudit $\mid Q_1⟩$, when physical qudit 1 is measured in the $B_1\left(\left\{\alpha \right\}\right)$ basis and $s_1=0$ is obtained. (d) The $S_{12}$ gate simulated by ${\mathrm{BBB}}_2$ on two logical qudits $\mid Q_1⟩$ and $\mid Q_2⟩$. (f) The quantum circuit corresponding to the operation ${\mathrm{BBB}}_3$ with $\left\{\alpha \right\}$ satisfying the conditions in the text, when $s_2=0$. Here, $C-E=F_2{S}_{12}{F}_2$.