Computational power and correlation in a quantum computational tensor network

We investigate relationships between computational power and correlation in resource states for quantum computational tensor network, which is a general framework for measurement-based quantum computation. We find that if the size of resource states is finite, not all resource states allow correct projective measurements in the correlation space, which is related to nonvanishing two-point correlations in the resource states. On the other hand, for infinite-size resource states, we can always implement correct projective measurements if the resource state can simulate arbitrary single-qubit rotations, since such a resource state exhibits exponentially decaying two-point correlations. This implies that a many-body state whose two-point correlation cannot be upper bounded by an exponentially decaying function cannot simulate arbitrary single-qubit rotations.

Figure 1

(a) Matrix-product state. (b) By measuring from the first to $(k-1)$th qudits, a single-qubit rotation is simulated in the correlation space, $|L\rangle \to |\psi \rangle$. In the normalizable resource with a finite $l$, projective measurements are simulated correctly at every position which is more than $l$ qudits away from the right boundary.