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arXiv:1207.4805 · New Journal of Physics

Symmetry protection of measurement-based quantum computation in ground states

Dominic V. Else, Stephen D. Bartlett, Andrew C. Doherty

Published Nov 9, 2012• 60 citations
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Abstract

The two-dimensional cluster state, a universal resource for measurement-based quantum computation, is also the gapped ground state of a short-ranged Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian. We prove that, provided the perturbation is sufficiently small and respects a certain symmetry, the perturbed ground state remains a universal resource. We do this by characterizing the operation of an adaptive measurement protocol throughout a suitable symmetry-protected quantum phase, relying on generic properties of the phase rather than any analytic control over the ground state.

DOI
10.1088/1367-2630/14/11/113016

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The two-dimensional cluster state, a universal resource for measurement-based quantum computation, is also the gapped ground state of a short-ranged Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian. We prove that, provided the perturbation is sufficiently small and respects a certain symmetry, the perturbed ground state remains a universal resource. We do this by characterizing the operation of an adaptive measurement protocol throughout a suitable symmetry-protected quantum phase, relying on generic properties of the phase rather than any analytic control over the ground state.
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.
A cluster state is a highly entangled quantum state that serves as a universal resource for measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the product state $\left\vert ++\cdots +\right\rangle $, and its parent Hamiltonian is the ZXZ model. This model exhibits a topological phase protected by the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry. By applying general quantum gates to the state $\left\vert ++\cdots +\right\rangle ,wesystematicallyobtainageneralshort−rangeentangledclusterstate.Ifweusethecontrolled−controlledZ(CCZ)gateinsteadoftheCZgate,weobtainhyper−clusterstatesfeaturingfive−bodyinteractionterms.ThesestatesenabletheimplementationoftheCZgatewithinmeasurement−basedquantumcomputation.Motivatedbythis,weproposeageneralizationtotheC, we systematically obtain a general short-range entangled cluster state. If we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain hyper-cluster states featuring five-body interaction terms. These states enable the implementation of the CZ gate within measurement-based quantum computation. Motivated by this, we propose a generalization to the C,wesystematicallyobtainageneralshort−rangeentangledclusterstate.Ifweusethecontrolled−controlledZ(CCZ)gateinsteadoftheCZgate,weobtainhyper−clusterstatesfeaturingfive−bodyinteractionterms.ThesestatesenabletheimplementationoftheCZgatewithinmeasurement−basedquantumcomputation.Motivatedbythis,weproposeageneralizationtotheC^{N}$Z gate, leading to a hyper-cluster model, in which (2N+1)(2N+1)(2N+1)-body entangled states are generated. This model retains the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry, which lies outside the Clifford group for N≥3N\geq 3N≥3. We demonstrate that an open chain in this model exhibits a 22N2^{2N}22N fold degenerate ground states protected by the symmetry, corresponding to the emergence of NNN effective free spins at each edge. These boundary degrees of freedom can serve as NNN qubits for measurement-based quantum computation. In particular, this framework makes it possible to implement the CN−1^{N-1}N−1Z gate in measurement-based quantum computation.
Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional (2D) plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms of symmetry fractionalization and that they are universal resource states for quantum computation. Our results of the nontrivial symmetry-protected topological order on arbitrary 2D lattices are based on an extension of the recent construction by Chen et al. [Phys. Rev. B 87, 155114 (2013)] on the square lattice.
We extend the hidden symmetry breaking picture, first proposed by Kennedy and Tasaki in the context of the Haldane phase, to a wider class of symmetry-protected topological (SPT) phases. We construct a generalization of the Kennedy-Tasaki transformation that transforms SPT phases into symmetry-breaking phases and relates long-range order in the latter to the more subtle “string order” in the former. In doing so we directly connect the form of the Kennedy-Tasaki transformation to the modern formulation of SPT order. We apply our generalized Kennedy-Tasaki transformation to solve the following problem in quantum information theory. We consider the 2-D cluster state, a simple “toy model” of a locally interacting system whose ground state is a universal resource for MBQC. We prove that, in the presence of a perturbation to the interaction Hamiltonian, the perturbed ground state remains a universal resource. We do this by using the generalized Kennedy-Tasaki transformation to prove that, if we employ the techniques of fault-tolerant quantum computation, the ground states of models in an appropriate SPT phases can serve as universal resources for MBQC provided that the symmetry-breaking is sufficiently strong in the symmetry-breaking phase obtained through the generalized Kennedy-Tasaki transformation.
Recent progress in the characterization of gapped quantum phases has also triggered the search for a universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought, in that nontrivial one-dimensional symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points. Here we provide two families of two-dimensional Z2{Z}_{2}Z2​ symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power appears to lose its universality at the boundary between the SPT and the symmetry-breaking phases.