State complexity and quantum computation

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.

15 pages, 9 figures. Annalen der Physik, Topical issue on Complex Quantum Systems, 2015

References in corpus (11)

• Matrix product states represent ground states faithfully
• Testing the limits of quantum mechanical superpositions
• Detecting Genuine Multipartite Entanglement with Two Local Measurements
• Valence Bond Solids for Quantum Computation
• Fast simulation of stabilizer circuits using a graph state representation
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Measures of macroscopicity for quantum spin systems
• Universal quantum computation with little entanglement
• Classical simulation versus universality in measurement based quantum computation
• Probability density function characterization of multipartite entanglement
• Interference versus success probability in quantum algorithms with imperfections

Cited by in corpus (7)

• The theory of variational hybrid quantum-classical algorithms
• Tripartite entanglement: Foundations and applications
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Quantifying Computational Advantage of Grover's Algorithm with the Trace Speed
• Classifying Entanglement by Algebraic Geometry
• Oracle problems as communication tasks and optimization of quantum algorithms
• Topologically driven no-superposing theorem with a tight error bound