Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Williamson, Dominic J.; Bultinck, Nick; Mariën, Michael; Sahinoǧlu, Mehmet; Haegeman, Jutho; Verstraete, Frank

doi:10.1103/physrevb.94.205150

Cited by 59 publications

(94 citation statements)

References 75 publications

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“…where the first term vanishes due to the gauge condition (44). From Lemma 4.2(ii) we have E t F ≤ 1, and repeating the same arguments as before, we get the bound…”

Section: The Proofsupporting

confidence: 60%

“…where we have used the fixed-point equations (15) and the gauge condition (44). The proof for j > j is analogous.…”

Section: Bounds On Transfer Operators Associated With the Excitation mentioning

confidence: 99%

“…Since we assume that p = 0, the gauge condition (44) states that |E B(p) = 0, hence the first term vanishes and it follows that…”

Section: The Proofmentioning

confidence: 99%

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Quantum error-detection at low energies

Gschwendtner

Koenig

Şahinoğlu

et al. 2019

J. High Energ. Phys.

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Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance Ω(n 1−ν ) for any ν ∈ (0, 1) and Ω(log n) encoded qubits. This shows that gapped systems contain -within isolated energy bands -error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains -within its low-energy eigenspace -an error-detecting code with the same parameter scaling. All these codes detect arbitrary d-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic errordetecting features.

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“…where the first term vanishes due to the gauge condition (44). From Lemma 4.2(ii) we have E t F ≤ 1, and repeating the same arguments as before, we get the bound…”

Section: The Proofsupporting

confidence: 60%

“…where we have used the fixed-point equations (15) and the gauge condition (44). The proof for j > j is analogous.…”

Section: Bounds On Transfer Operators Associated With the Excitation mentioning

confidence: 99%

See 1 more Smart Citation

Quantum error-detection at low energies

Gschwendtner

Koenig

Şahinoğlu

et al. 2019

J. High Energ. Phys.

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show abstract

“…and then apply the unitary U φ , which can always be expressed as a matrix product unitary (MPU) due to its local nature (cf. [46][47][48]). Exploiting this fact U φ can be written as the MPU, we describe graphically, for the case of a square lattice, ,…”

Section: Phases Of Symmetry-protected Topological Ordermentioning

confidence: 99%

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Daniel

Alexander

Miyake

2020

Quantum

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What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.

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“…This implies certain (generalized) injectivity conditions on the relevant wavefunction ansatz, which we implicitly assume from hereon (See Refs. [42,48] for details).…”

Section: Projective Representations and 1d Spt Phasesmentioning

confidence: 99%

Interacting symmetry-protected topological phases out of equilibrium

McGinley

Cooper

2019

Phys. Rev. Research

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We show that the dynamics of generic isolated quantum systems admits a topological classification which captures universal features of many-body wavefunctions far from equilibrium. Specifically, we consider two short-ranged entangled wavefunctions to be topologically equivalent if they can be interconverted via finite-time unitary evolution governed by a symmetry-respecting Hamiltonian. By analogy to the classification of symmetry-protected topological phases, we consider the projective action of the symmetry group on a time-evolved wavefunction to explicitly calculate this nonequilibrium topological classification, focussing on systems of strongly interacting bosons in a variety of symmetry classes. The physical consequences of the non-equilibrium classification are described. In particular, we show that the characteristic zero-frequency spectroscopic peaks associated with topologically protected edge modes will be broadened by external noise only when the system is trivial in the non-equilibrium classification. arXiv:1908.06875v1 [cond-mat.str-el]

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Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Cited by 59 publications

References 75 publications

Quantum error-detection at low energies

Quantum error-detection at low energies

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Interacting symmetry-protected topological phases out of equilibrium

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