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arXiv:0807.4797 · Physical Review A

Transitions in the computational power of thermal states for measurement-based quantum computation

S. D. Barrett, Stephen D. Bartlett, Andrew C. Doherty, David H. Jennings, Terry Rudolph

Published Dec 18, 2009• 22 citations
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Abstract

We show that the usefulness of the thermal state of a specific spin-lattice model for measurement-based quantum computing exhibits a transition between two distinct ``phases''---one in which every state is a universal resource for quantum computation, and another in which any local measurement sequence can be simulated efficiently on a classical computer. Remarkably, this transition in computational power does not coincide with any phase transition, classical, or quantum in the underlying spin-lattice model.

DOI
10.1103/physreva.80.062328

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We show that the usefulness of the thermal state of a specific spin-lattice model for measurement-based quantum computing exhibits a transition between two distinct ``phases''---one in which every state is a universal resource for quantum computation, and another in which any local measurement sequence can be simulated efficiently on a classical computer. Remarkably, this transition in computational power does not coincide with any phase transition, classical, or quantum in the underlying spin-lattice model.
We construct two spin models on lattices (both two and three dimensional) to study the capability of quantum computational power as a function of temperature and the system parameter. There exists a finite region in the phase diagram such that the thermal equilibrium states are capable of providing a universal fault-tolerant resource for measurement-based quantum computation. Moreover, in such a region the thermal resource states on the three-dimensional lattices can enable topological protection for quantum computation. The two models behave similarly in terms of quantum computational power. However, they have different properties in terms of the usual phase transitions. The first model has a first-order phase transition only at zero temperature whereas there is no transition at all in the second model. Interestingly, the transition in the quantum computational power does not coincide with the phase transition in the first model.
We study how heralded qubit losses during the preparation of a two-dimensional cluster state, a universal resource state for one-way quantum computation, affect its computational power. Above the percolation threshold we present a polynomial-time algorithm that concentrates a universal cluster state, using resources that scale optimally in the size of the original lattice. On the other hand, below the percolation threshold, we show that single qubit measurements on the faulty lattice can be efficiently simulated classically. We observe a phase transition at the threshold when the amount of entanglement in the faulty lattice directly relevant to the computational power changes exponentially.
In this paper, we build a framework allowing for a systematic investigation of the fundamental issue: ' Which quantum states serve as universal resources for measurement-based (one-way) quantum computation? ' We start our study by re-examining what is exactly meant by 'universality' in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, we find that the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs—we refer to this property as CQ-universality. We then show that a systematic study of CQ-universality in one-way quantum computation is possible by identifying entanglement features that are required to be present in every universal resource. In particular, we find that a large class of entanglement measures must reach its supremum on every universal resource. These insights are used to identify several families of states as being not universal, such as one-dimensional (1D) cluster states, Greenberger–Horne–Zeilinger (GHZ) states, W states, and ground states of non-critical 1D spin systems. Our criteria are strengthened by considering the efficiency of a quantum computation, and we find that entanglement measures must obey a certain scaling law with the system size for all efficient universal resources. This again leads to examples of non-universal resources, such as, e.g. ground states of critical 1D spin systems. On the other hand, we provide several examples of efficient universal resources, namely graph states corresponding to hexagonal, triangular and Kagome lattices. Finally, we consider the more general notion of encoded CQ-universality, where quantum outputs are allowed to be produced in an encoded form. Again we provide entanglement-based criteria for encoded universality. Moreover, we present a general procedure to construct encoded universal resources.
Universal quantum computation can be realised by measuring individual particles in a specially entangled state of many particles, called a universal resource state. This model of quantum computation, called measurement-based quantum computation (MBQC), provides a framework for studying the intrinsic computational power of physical systems. In this thesis I will investigate how universal resource states may arise naturally as ground states of interacting spin systems. In particular, I will describe new 'phases' of quantum matter, which are characterised by having universal resource states as ground states. This direction of research allows us to draw on techniques from both many-body quantum physics and quantum information theory.
Measurement-based quantum computation utilizes an initial entangled resource state and proceeds with subsequent single-qubit measurements. It is implicitly assumed that the interactions between qubits can be switched off so that the dynamics of the measured qubits do not affect the computation. By proposing a model spin Hamiltonian, we demonstrate that measurement-based quantum computation can be achieved on a thermal state with always-on interactions. Moreover, computational errors induced by thermal fluctuations can be corrected and thus the computation can be executed fault tolerantly if the temperature is below a threshold value.
Recently, it was shown that the non-local correlations needed for measurement-based quantum computation (MBQC) can be revealed in the ground state of the Affleck–Kennedy–Lieb–Tasaki (AKLT) model involving nearest-neighbour spin-3/2 interactions on a honeycomb lattice. This state is not singular but resides in the disordered phase of the ground states of a large family of Hamiltonians characterized by short-range-correlated valence bond solid states. By applying local filtering and adaptive single-particle measurements, we show that most states in the disordered phase can be reduced to a graph of correlated qubits that is a scalable resource for MBQC. At the transition between the disordered and Néel ordered phases, we find a transition from universal to non-universal states as witnessed by the scaling of percolation in the reduced graph state.