Randomized benchmarking in measurement-based quantum computing

Randomized benchmarking is routinely used as an efficient method for characterizing the performance of sets of elementary logic gates in small quantum devices. In the measurement-based model of quantum computation, logic gates are implemented via single-site measurements on a fixed universal resource state. Here we adapt the randomized benchmarking protocol for a single qubit to a linear cluster state computation, which provides partial, yet efficient characterization of the noise associated with the target gate set. Applying randomized benchmarking to measurement-based quantum computation exhibits an interesting interplay between the inherent randomness associated with logic gates in the measurement-based model and the random gate sequences used in benchmarking. We consider two different approaches: the first makes use of the standard single-qubit Clifford group, while the second uses recently introduced (non-Clifford) measurement-based 2-designs, which harness inherent randomness to implement gate sequences.

Figure 1

(a) Cluster state with wire graph (shown left) with an input state on the leftmost and green node. Measurement angles are labeled in the center of each node. (b) Measuring the input state in some basis in the $XY$ plane (i.e., a measurement in the eigenbasis of $X_{{\theta }_1}=Xcos{\theta }_1-Ysin{\theta }_1$) yields the output shown in the circuit on the right.

Figure 2

Here we show a small part of a noisy measurement-based quantum computation on a linear cluster state [33]. Perfect state preparation ($p$), entangling gate ($e$), and measurement ($m$) are all followed or preceded by some corresponding noise channel ${\mathcal{D}}_{p,e,m}$. Each measurement step will implement $M_{{\theta }_i,m_i}$, along with some effective single-qubit gate error ${\mathcal{D}}_{{\theta }_i,m_i}$. For simplicity, we have assumed that state preparation and measurement errors are the same for all cluster qubits, and therefore the effective noise channel per measurement step ${\mathcal{D}}_{\theta ,m}$ only depends on the $\theta$ and $m$. Note that such an error model is a generalization of those considered in Refs. [34, 35].

Figure 3

Each element of the 2-design (a) is implemented by making three measurements on the cluster wire [(b) and (c)]. We require a random sequence of Cliffords in each implementation. The measurement angles are all integer multiples of $\frac{\pi }{2}$ ($n,n^{\prime },n^{\prime \prime }\in \{0,1,2,3\})$. The noise operator per 2-design element ${\mathcal{D}}_{\text{seq(3)}}$ describes the noise added after three measurements.

Figure 4

Each element of the unitary 2-design is implemented by making five fixed measurements on the cluster wire. The noise operator per 2-design element ${\mathcal{D}}_{\text{seq(5)}}$ describes the noise added after five measurements.