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: 7.73). 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: We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their PEPS representations and global symmetries. Following this, we explore the relationship between the generalized cluster states and the Kitaev quantum double models and sketch a protocol for universal adiabatic topological cluster state quantum computation that makes use of the generalized cluster states.
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    ABSTRACT: We give restrictions on the locality-preserving unitary automorphisms $U$, which are protected gates, for topologically ordered systems. For arbitrary anyon models, we show that such unitaries only generate a finite group, and hence do not provide universality. For abelian anyon models, we find that the logical action of $U$ is contained in a proper subgroup of the generalized Clifford group. In the case $D(\mathbb{Z}_2)$, which describes Kitaev's toric code, this represents a tightening of statement previously obtained within the stabilizer framework (PRL 110:170503). For non-abelian models, we find that such automorphisms are very limited: for example, there is no non-trivial gate for Fibonacci anyons. For Ising anyons, protected gates are elements of the Pauli group. These results are derived by relating such automorphisms to symmetries of the underlying anyon model: protected gates realize automorphisms of the Verlinde algebra. We additionally use the compatibility with basis changes to characterize the logical action.


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