State complexity and quantum computation

TL;DR

This paper explores the relationship between the complexity of quantum states, measured by tree size, and their usefulness in quantum computation, demonstrating that superpolynomial complexity is crucial for measurement-based quantum computing.

Contribution

It reviews tree size complexity, identifies states with superpolynomial tree size, and links this complexity to the power of quantum computational models, providing new bounds and verification methods.

Findings

Superpolynomial tree size states include Shor's and cluster states.

Superpolynomial tree size is necessary for measurement-based quantum computation.

Efficient verification of complex states is possible.

Abstract

The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. We discuss this link using the tree size of a multiqubit state, a complexity measure that has two noticeable (and, so far, unique) features: it is in principle computable, and non-trivial lower bounds can be obtained, hence identifying truly complex states. In this paper, we first review the definition of tree size, together with known results on the most complex three and four qubits states. Moving to the multiqubit case, we revisit a mathematical theorem for proving a lower bound on tree size that scales superpolynomially in the number of qubits. Next, states with superpolynomial tree size, the Immanant states, the Deutsch-Jozsa states, the Shor's states and the subgroup states, are described. We show that the universal resource state for measurement based quantum computation, the 2D-cluster state, has superpolynomial tree size. Moreover, we show how the complexity of subgroup states and the 2D cluster state can be verified efficiently. The question of how tree size is related to the speed up achieved in quantum computation is also addressed. We show that superpolynomial tree size of the resource state is essential for measurement based quantum computation. The necessary role of large tree size in the circuit model of quantum computation is still a conjecture; and we prove a weaker version of the conjecture.