Computational Power and Correlation in Quantum Computational Tensor Network

TL;DR

This paper explores how the correlation properties of resource states in quantum tensor networks influence their computational capabilities, revealing that exponential decay of correlations is essential for universal single-qubit operations.

Contribution

It establishes a link between correlation decay and computational universality in quantum tensor networks, especially distinguishing finite and infinite resource states.

Findings

Finite resource states with non-vanishing correlations cannot perform all projective measurements.

Infinite resource states with exponential correlation decay can implement arbitrary single-qubit rotations.

States with non-exponentially decaying correlations cannot simulate universal quantum computation.

Abstract

We investigate relations 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 non-vanishing 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 upperbounded by an exponentially-decaying function cannot simulate arbitrary single-qubit rotations.