Symmetry protection of measurement-based quantum computation in ground states

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

doi:10.1088/1367-2630/14/11/113016

Cited by 61 publications

(90 citation statements)

References 65 publications

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“…(II A 1)). As we discussed before, according to the classification of SPT phases in 1-dimensional spin systems, there is only one non-trivial SPT phase protected by Z 2 × Z 2 symmetry, and it has been shown that the cluster states is in this non-trivial phase 11,12 . (Note that in the absence of the symmetry, the cluster state can be transformed to a product state via a low-depth circuit, and therefore it does not have topological order).…”

Section: Example: Perturbed Cluster Hamiltonianmentioning

confidence: 93%

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Symmetry-protected topological entanglement

Marvian

2017

Phys. Rev. B

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We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT-entanglement, is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPTentanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.

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Section: Example: Perturbed Cluster Hamiltonianmentioning

confidence: 93%

“…Recently SPT order has attracted a considerable amount of attention in quantum information community (See e.g. [11][12][13][14][15][16] ). It is often claimed that (symmetry-protected) topological order of the system is fundamentally related to its entanglement structure.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-protected topological entanglement

Marvian

2017

Phys. Rev. B

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“…The Haldane phase is useful for quantum operations (for instance, as a perfect quantum wire) [17,18] and can be destroyed only by crossing a phase transition. The finite energy gap in topological spin-1 systems makes them a potential candidate for long-lived, robust quantum memories [19], and schemes using symmetry-protected spin-1 phases for measurement-based quantum computation have also been proposed [20,21].…”

Section: Introductionmentioning

confidence: 99%

Realization of a Quantum Integer-Spin Chain with Controllable Interactions

et al. 2015

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The physics of interacting integer-spin chains has been a topic of intense theoretical interest, particularly in the context of symmetry-protected topological phases. However, there has not been a controllable model system to study this physics experimentally. We demonstrate how spin-dependent forces on trapped ions can be used to engineer an effective system of interacting spin-1 particles. Our system evolves coherently under an applied spin-1 XY Hamiltonian with tunable, long-range couplings, and all three quantum levels at each site participate in the dynamics. We observe the time evolution of the system and verify its coherence by entangling a pair of effective three-level particles ("qutrits") with 86% fidelity. By adiabatically ramping a global field, we produce ground states of the XY model, and we demonstrate an instance where the ground state cannot be created without breaking the same symmetries that protect the topological Haldane phase. This experimental platform enables future studies of symmetry-protected order in spin-1 systems and their use in quantum applications.

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“…Studies of SPT phases have provided better understanding of various quantum phases of matter, including topological insulators, the topological gauge theory and gauge/gravitational anomalies. However, studies of SPT phases have yet to find interesting applications in quantum information science except for few instances [18][19][20]. ⊗d symmetry.…”

Section: Introductionmentioning

confidence: 99%

Topological color code and symmetry-protected topological phases

Yoshida

2015

Phys. Rev. B

133

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We study (d − 1)-dimensional excitations in the d-dimensional color code that are created by transversal application of the R d phase operators on connected subregions of qubits. We find that such excitations are the superpositions of electric charges and can be characterized by the fixed-point wave functions of (d − 1)-dimensional bosonic symmetry-protected topological (SPT) phases with (Z 2 ) ⊗d symmetry. While these SPT excitations are localized on (d − 1)-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the nontriviality of emerging SPT wave functions. Moreover, these SPT excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal R 3 operator, exchanges a magnetic flux and a composite of a magnetic flux and the looplike SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and the looplike SPT excitations exhibit nontrivial three-loop braiding statistics in three dimensions as a result of the fact that the R 3 phase operator belongs to the third level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.

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Symmetry protection of measurement-based quantum computation in ground states

Cited by 61 publications

References 65 publications

Symmetry-protected topological entanglement

Symmetry-protected topological entanglement

Realization of a Quantum Integer-Spin Chain with Controllable Interactions

Topological color code and symmetry-protected topological phases

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