Adaptive strategies for graph-state growth in the presence of monitored errors

Graph states (or cluster states) are the entanglement resource that enables one-way quantum computing. They can be grown by projective measurements on the component qubits. Such measurements typically carry a significant failure probability. Moreover, they may generate imperfect entanglement. Here we describe strategies to adapt growth operations in order to cancel incurred errors. Nascent states that initially deviate from the ideal graph states evolve toward the desired high fidelity resource without impractical overheads. Our analysis extends the diagrammatic language of graph states to include characteristics such as tilted vertices, weighted edges, and partial fusion, which arise from experimental imperfections. The strategies we present are relevant to parity projection schemes such as optical path erasure with distributed matter qubits.

Figure 1

(Color online) Graph generalizations (left-hand side), the corresponding proper graphs (right-hand side), and the relevant addition rules. (a) Tilted vertices are hollow circles labeled $\alpha$, denoting a qubit in a state $\cos \left(\alpha \right)\mid 0⟩+\sin \left(\alpha \right)\mid 1⟩$ prior to applying two qubit operations; (b) weighted graph edges, illustrated as a solid edge between two qubits $A$ and $B$, and labeled $\theta$. In operator notation a weighted edge is $U_{AB}\left(\theta \right)=\cos \left(\theta \right)\mathbb{1}+i\sin \left(\theta \right)Z_A{Z}_B$; (c) partial fusions, illustrated as a dashed line between two qubits $A$ and $B$ labeled $\theta$. In operator notation a partial fusion is $P_{AB}\left(\theta \right)=\cos \left(\theta \right)\mathbb{1}+\sin \left(\theta \right)Z_A{Z}_B$.

Figure 2

(Color online) (a) A distributed quantum computer; the qubits are stored in matter systems which are coupled to optical modes. Pairs of optical paths are routed toward a beam splitter with two detectors (a PM device). (b) Emission rates from two mismatched cavities; detection events within the shaded region correspond to high-fidelity entanglement. (c) Fusing tilted GHZ states by a projective measurement. Upon success, this will result in a larger tilted GHZ state. This can be probabilistically purified into a proper GHZ state. (d) Fusing two cherries generates a tilted central vertex with a single cherry (which is used for realignment). If realignment fails the state can be used for merging [Fig. 3a] or bridging [Fig. 4a].

Figure 3

(Color online) Connecting two subgraphs by merging one qubit from each subgraph. The qubits are labeled by $3$ and $4$, and a successful merger corresponds to $P_{34}(\pm \pi ∕4)$. (a) An attempt to merge two qubits attached to a tilted vertex with success outcome $P_{34}(\pm \pi ∕4)$ or failure outcome $P_{34}[\mp R({\alpha }_1\left)\right]$. After a failure, projective measurements must be repeated until two more cherries are fused. A realignment attempt can then be made on the resulting tilted vertex by measuring its only cherry. If realignment is successful then implement (c), but if failed then implement (b). (b) A probabilistic procedure that on success generates $P_{34}(\pm \pi )$ and on failure generates $P_{34}[\mp R({\alpha }_2\left)\right]$. (c) A deterministic procedure that projects with $P_{34}(\pm \pi ∕4)$.

Figure 4

(Color online) Connecting two subgraphs by bridging one qubit from each subgraph. The qubits are labeled by $3$ and $4$, and a successful bridge corresponds to $U_{34}(\pm \pi ∕4)$. (a) An attempt to bridge two qubits attached to a tilted vertex with success outcome $U_{34}(\pm \pi ∕4)$ or failure outcome $U_{34}[\mp R({\alpha }_1\left)\right]$. After a failure, projective measurements must be repeated until two more cherries are fused. A realignment attempt can then be made on the resulting tilted vertex by measuring its only cherry. If realignment is successful then implement (c), but if it fails then implement (b). (b) A probabilistic procedure that on success generates $U_{34}(\pm \pi ∕4-{\theta }_1)$ and on failure generates $U_{34}\left({\lambda }_f\right)$, where ${\beta }_{\pm }$ and ${\lambda }_f$ are, respectively, defined by (3, 5). (c) A deterministic procedure that projects with $U_{34}(\pm \pi ∕4-{\theta }_1)$.