Symmetry-protected phases for measurement-based quantum computation

Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia.
Physical Review Letters (Impact Factor: 7.94). 01/2012; 108(24). DOI: 10.1103/PhysRevLett.108.240505
Source: arXiv

ABSTRACT Ground states of spin lattices can serve as a resource for measurement-based
quantum computation. Ideally, the ability to perform quantum gates via
measurements on such states would be insensitive to small variations in the
Hamiltonian. Here, we describe a class of symmetry-protected topological orders
in one-dimensional systems, any one of which ensures the perfect operation of
the identity gate. As a result, measurement-based quantum gates can be a robust
property of an entire phase in a quantum spin lattice, when protected by an
appropriate symmetry.

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    ABSTRACT: The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond states on the hexagonal and other trivalent Archimedean lattices were shown to be universal resource states for measurement-based quantum computation (MBQC). It is still unclear whether AKLT states of higher spin magnitude can also support universal MBQC. We demonstrate that several 2D AKLT states involving mixture of spin-2 and other lower-spin entities are also universal for MBQC. This includes a spin-2 spin-3/2 mixture and two other spin-2 spin-1 mixtures. In addition, we examine the universality for the spin-2 AKLT state on the Kagome lattice and provide evidence and argument that, however, it is likely not universal.
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    ABSTRACT: We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite abelian group. This generalization takes the form of a non-local unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterise the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterised by hidden symmetry-breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1) and SU(k).
    Physical review. B, Condensed matter 04/2013; 88(8). · 3.77 Impact Factor
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    ABSTRACT: Universal quantum computation can be achieved by simply performing single-spin measurements on a highly entangled resource state, such as cluster states. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states has recently been explored; for example, the spin-1 AKLT chain can be used to simulate single-qubit gate operations on a single qubit, and the spin-3/2 two-dimensional AKLT state on the honeycomb lattice can be used as a universal resource. However, it is unclear whether such universality is a coincidence for the specific state or a shared feature in all two-dimensional AKLT states. Here we consider the family of spin-3/2 AKLT states on various trivalent Archimedean lattices and show that in addition to the honeycomb lattice, the spin-3/2 AKLT states on the square octagon (4,82) and the “cross” (4,6,12) lattices are also universal resource, whereas the AKLT state on the “star” (3,122) lattice is likely not due to geometric frustration.
    Physical Review A 12/2013; · 3.04 Impact Factor


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