Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes
@inproceedings{Aasen2022AdiabaticPO, title={Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes}, author={David Aasen and Zhenghan Wang and Matthew B. Hastings}, year={2022} }
The recent “honeycomb code” is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and third homotopy groups of this space in two spatial dimensions. A single cycle of such a path can implement some automorphism of the…
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References
SHOWING 1-10 OF 27 REFERENCES
On adiabatic cycles of quantum spin systems
- Physics, Mathematics
- 2021
Motivated by the Ω-spectrum proposal of unique gapped ground states by Kitaev [1], we study adiabatic cycles in gapped quantum spin systems from various perspectives. We give a few exactly solvable…
String-net condensation: A physical mechanism for topological phases
- Physics
- 2005
We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,''…
Interacting invariants for Floquet phases of fermions in two dimensions
- PhysicsPhysical Review B
- 2019
We construct a many-body quantized invariant that sharply distinguishes among two dimensional non-equilibrium driven phases of interacting fermions. This is an interacting generalization of a…
Periodic table for topological insulators and superconductors
- Physics, Mathematics
- 2009
Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a…
The Group Structure of Quantum Cellular Automata
- MathematicsCommunications in Mathematical Physics
- 2022
We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of…
Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
- Physics
- 2021
This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems,…
Higher-dimensional generalizations of Berry curvature
- Mathematics
- 2020
A family of finite-dimensional quantum systems with a non-degenerate ground state gives rise to a closed 2-form on the parameter space: the curvature of the Berry connection. Its cohomology class is…
Long-range entanglement from measuring symmetry-protected topological phases
- Physics
- 2021
A fundamental distinction between many-body quantum states are those with shortand longrange entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the…
Topological order with a twist: Ising anyons from an Abelian model.
- MathematicsPhysical review letters
- 2010
The realization of topological defects that, under monodromy, transform anyons according to a symmetry are studied in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons.