Estimation of coherent error sources from stabilizer measurements

In the context of measurement-based quantum computation a way of maintaining the coherence of a graph state is to measure its stabilizer operators. Aside from performing quantum error correction, it is possible to exploit the information gained from these measurements to characterize and then counteract a coherent source of errors; that is, to determine all the parameters of an error channel that applies a fixed—but unknown—unitary operation to the physical qubits. Such a channel is generated, e.g., by local stray fields that act on the qubits. We study the case in which each qubit of a given graph state may see a different error channel and we focus on channels given by a rotation on the Bloch sphere around either the $\widehat{\mathit{x}}$, the $\widehat{\mathit{y}}$, or the $\widehat{\mathit{z}}$ axis, for which analytical results can be given in a compact form. The possibility of reconstructing the channels at all qubits depends nontrivially on the topology of the graph state. We prove via perturbation methods that the reconstruction process is robust and supplement the analytic results with numerical evidence.

Figure 1

Summary of fully reconstructible fields. Each cell in the arrays corresponds to a different square lattice. In each array the entry in position $(m_1,m_2)$ is filled if the adjacency matrix of a square-lattice graph having $m_1\times m_2$ vertices is singular. In the case of cylindrical geometry, the open boundary condition is in the direction having $m_1$ rows and the closed boundary condition is in the direction having $m_2$ columns.

Figure 2

Summary of the effect on the reconstruction errors ${\mathcal{E}}_a$ (see text) on all vertices $a\in V$ of an open-ended chain with ten vertices. The reconstruction errors are induced by the depolarizing channel (with parameter $q$), by the misalignment of the axis (with average misalignment given by $\epsilon$), and by the finiteness of the number of measurements of the correlators (given by $M$); we plot the results for the $X$-, $Y$-, and $Z$-field cases. In each plot, we fix two of the parameters among $(q,\epsilon ,M)$ and vary the third one; the reference values in all plots are $q=0.01$, $\epsilon =0.01$, $M={10}^4$. The plots of the $M$ dependence of ${\mathcal{E}}_a$ are in bilogarithmic scale. Each curve in each plot represents the reconstruction error ${\mathcal{E}}_a$ for a specific vertex $a$ among the ten in the graph state, as indicated by the labels next to the corresponding curves. Each data point is obtained extracting an ensemble of ${10}^4$ field configurations, computing the reconstruction for each of these, and then averaging the reconstruction errors ${\mathcal{E}}_a$ over the ensemble.

Figure 3

Example of the reconstruction of stray fields through the data collected from the stabilizer measurements associated with graph states with four different (randomly selected) logical Pauli bases. The graph state here is a linear chain with three qubits, as represented in panel (a); in these figures graphical objects sharing the same color refer to the same qubit. The three vectors in panel (b) point in the direction ${\widehat{\mathit{n}}}_a$ of the stray field at each vertex $a$ and have lengths proportional to ${\beta }_a=cos{\lambda }_a$, where ${\lambda }_a=\gamma \mathrm{\Delta }t\left|{\mathit{B}}_a\right|$ is proportional to the strength of the field. We assume, without loss of generality, that the $\widehat{\mathit{z}}$ component of all vectors is positive, since in this representation opposite vectors are associated with the same physical stray field. We simulate the result of ${10}^4$ stabilizer measurements, for each of the three qubits in the graph state, for each of the four selected logical Pauli bases, giving, in total, $3\times 4\times {10}^4$ measurements of correlator operators. We then estimate from these $cos{\lambda }_a$ and ${\widehat{\mathit{n}}}_a$ through the numerical minimization of cost function $C\left({\{{\beta }_a,{\widehat{\mathit{n}}}_a\}}_{a\in V}\right)$, given by Eq. (58). We have repeated this procedure ${10}^3$ times; we display the resulting ${10}^3$ estimations of the parameters with colored points in the hemisphere in panel (b). In panel (c) the histograms show the number of points against the squared distance from the correct solution, i.e., the squared distance of the points from the tip of the corresponding arrow in panel (b).