Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes - INSPIRE

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Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes

David Aasen
(
- Station Q, UCSB and
- Santa Barbara, KITP
)
,
Zhenghan Wang
(
- Station Q, UCSB and
- UC, Santa Barbara (main)
)
,
Matthew B. Hastings
(
- Station Q, UCSB and
- Unlisted, US, WA
)

Mar 21, 2022

36 pages

Published in:

Phys.Rev.B 106 (2022) 8, 085122

Published: Aug 15, 2022

e-Print:

2203.11137 [quant-ph]

DOI:

10.1103/PhysRevB.106.085122 (publication)

View in:

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Abstract: (APS)

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 topological order of that Hamiltonian. We construct such paths for arbitrary automorphisms of two-dimensional doubled topological order. Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.

Note:

36 pages, 10 figures; v2: additional references and minor revisions

phase: topological
symmetry: topological
dimension: 2
Hamiltonian
adiabatic
homotopy

References(33)

Figures(10)

[1]

Dynamically Generated Logical Qubits

Matthew B. Hastings
(
- Station Q, UCSB and
- Microsoft, Redmond
)
,
Jeongwan Haah
(
- Microsoft, Redmond
)

- Quantum 5 (2021) 564
•
e-Print:
- 2107.02194
•
DOI:
- 10.22331/q-2021-10-19-564

[2]

A Fault-Tolerant Honeycomb Memory

- Quantum 5 (2021) 605,
- Quantum 5 (2021) 605
•
e-Print:
- 2108.10457
•
DOI:
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[3]

Benchmarking the Planar Honeycomb Code

Craig Gidney
(
- Unlisted, US, CA
)
,
Michael Newman
(
- Unlisted, US, CA
)
,
Matt McEwen
(
- Geodynamics, Santa Barbara and
- UC, Santa Barbara
)

- Quantum 6 (2022) 813,
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•
e-Print:
- 2202.11845
•
DOI:
- 10.22331/q-2022-09-21-813

[4]

Performance of Planar Floquet Codes with Majorana-Based Qubits

Adam Paetznick
(
- Station Q, UCSB
)
,
Christina Knapp
(
- Station Q, UCSB
)
,
Nicolas Delfosse
(
- Station Q, UCSB
)
,
Bela Bauer
(
- Station Q, UCSB
)
,
Jeongwan Haah
(
- Station Q, UCSB
)

et al.

- PRX Quantum 4 (2023) 1, 010310
•
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•
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Boundaries for the Honeycomb Code

Jeongwan Haah
(
- Microsoft, Redmond
)
,
Matthew B. Hastings
(
- Microsoft, Redmond and
- Station Q, UCSB
)

- Quantum 6 (2022) 693,
- Quantum 6 (2022) 693
•
e-Print:
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•
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[6]

Talk at Simons Center for Geometry and Physics

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Talk at Simons Center for Geometry and Physics

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Talk at Simons Center for Geometry and Physics

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These states do not support topological order. Rather, each such state has an inverse, some other short-range entangled state, such that their tensor product is related to a product state by a local quantum circuit (possibly with tails)

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(
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Lev Spodyneiko
(
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)
,
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)
,
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(
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)
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)
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Ken Shiozaki
(
- Kyoto U., Yukawa Inst., Kyoto
)

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•
e-Print:
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•
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Lukasz Fidkowski
(
- SUNY, Stony Brook and
- Santa Barbara, KITP
)
,
Hoi Chun Po
(
- UC, Berkeley and
- Harvard U.
)
,
Andrew C. Potter
(
- Texas U.
)
,
Ashvin Vishwanath
(
- UC, Berkeley and
- Harvard U.
)

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•
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Alexei Kitaev
(
- Caltech, Pasadena (main)
)

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•
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•
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[20]

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Michael Kamfor
(
- Dortmund U.
)
,
Sebastien Dusuel
(
- Lycee Saint-Louis
)
,
Julien Vidal
(
- Paris U., VI-VII
)
,
Kai Phillip Schmidt
(
- Dortmund U.
)

- J.Stat.Mech. 1008 (2010) P08010
•
e-Print:
- 1005.5103
•
DOI:
- 10.1088/1742-5468/2010/08/P08010

[21]

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- Phys.Rev.B 91 (2015) 13, 134419
•
e-Print:
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•
DOI:
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[22]

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H. Bombin
(
- Perimeter Inst. Theor. Phys.
)

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•
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•
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[23]

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Michael Freedman
(
- Station Q, UCSB and
- UC, Santa Barbara, Dept. Math.
)
,
Jeongwan Haah
(
- Microsoft, Redmond
)
,
Matthew B. Hastings
(
- Station Q, UCSB
)

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•
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[24]

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(
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)
,
Xiao-Gang Wen
(
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)

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