Figure 1

Twelve graph-state bases on the loop graph $L_9$ for the ((9, 12, 3)) code $\mathbb{D}$. Each graph represents a graph state that is the unique common eigenstate of the vertex stabilizers $\{{\mathcal{G}}_a\}$ with eigenvalue $+1$ if $a$ is uncolored and $-1$ if the vertex is blue colored.Reuse & Permissions

Figure 2

The encoding-decoding circuit for the ((9, 12, 3)) code in the case that the outcomes of the measurements of ${\widehat{\alpha }}_1$, ${\widehat{\alpha }}_2$, and ${\widehat{\alpha }}_3$ are all positive. A boxed $H$ represents the Hadmard gate; a pair of connected black dots represents the controlled-phase gate; a pair of connected black (source) and white (target) dots represents the controlled-not gate; a triple of connected red (or gray) dots represents the controlled-controlled-PHASE gate; $M_X$ represents a measurement of the observable $\mathcal{X}$; ${\mathcal{R}}_M$ is the recovery operation dependent on the outcomes of the measurements as specified in Table . Between the encoding and decoding boxes arbitrary single-qubit errors may occur.Reuse & Permissions

Figure 3

The decoding circuit box $M_{{\alpha }_1{\alpha }_2{\alpha }_3}$ for the syndrome (${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$). Here a triple of connected white dot and 2 red (or gray) dots represents the Toffli gate with the white dot being the target.Reuse & Permissions