Are random pure States useful for quantum computation?

We show the following: a randomly chosen pure state as a resource for measurement-based quantum computation is-with overwhelming probability-of no greater help to a polynomially bounded classical control computer, than a string of random bits. Thus, unlike the familiar "cluster states," the computing power of a classical control device is not increased from P to BQP (bounded-error, quantum polynomial time), but only to BPP (bounded-error, probabilistic polynomial time). The same holds if the task is to sample from a distribution rather than to perform a bounded-error computation.

Algorithmic complexity and entanglement of quantum states.

We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.