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Topological Quantum Error Correcting Codes
Kasper Duivenvoorden
JARA IQI, RWTH Aachen
Topological Phases of Quantum Matter – 4 September 2014
Topology Physical errors are local Store information globally
Definition: locally ground states are the same
Gapped topological phase with ground space degeneracy:
Use ground space manifold of a topological phase to encode quantum information
A
B
Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
Example: Toric Code
Logicals: preserves the code space
Ground states:
Homological codes Lattice CW-complex Point 0-cells Line 1-cells Surface 2-cells ... ` k-cells
Boundary map
=
=
Co-Boundary map
=
=
Homological codes Boundary map
=
Co-Boundary map
=
Properties
1: 2:
Homological codes
Step 1: CW-complex
Step 2: Spins on every k-cells Example: k=1
Step 3: Hamiltonian = z
z
z
z
= x
x
x
x
Homological codes =
=
Logicals
Homological codes =
=
Logicals
Toric Codes D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
Dimensionality of logicals
Z X 1 1 1 2 2 2
Thermal Stability
Movement of anyons
At T=0 Tunneling p = exp(-L)
At T>0 Thermal excitation p = exp(-E/kT) No stability!
E
Thermal Stability
At T>0 Thermal excitation p = exp(-E/kT) Stability!
E
Chesi, Loss, Bravyi, Terhal, 2010
Toric Codes D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
Dim. of logicals Stable Memory
Z X 1 1 No 1 2 2 2 Quantum
Toric Codes
Drawbacks - Constant information - We only have 3 dimensions
Fractal codes
D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
Yoshida, 2013
Fractal Codes Algebraic representation of Z-type operators
• Generalize to higher dimensions: • More spins per site:
y z x
Yoshida, 2013
Fractal Codes
Commutation relations
Interpretation: shift X[g] by 0 , Z[f] and X[g] anticommute shift X[g] by 2 , Z[f] and X[g] anticommute
Yoshida, 2013
Fractal Codes Hamiltonian
Example:
Z Z Z
Logical X[g] g(1+fy) = 0 modulo 2 Assumptions:
Z
y x
Fractal Codes Logical: X[g]
X X X X X X X X X X X X X X
Z Z Z Z
X
Hamiltonian terms:
Set of Logicals: fractal like point like
y x
X X X X X X X X X 1 2 3 2 1
Fractal Codes Excitations
Z
Z Z Z
X X X X X
X X X X X X X
Compare with Toric code: Possible energy barrier
Yoshida, 2013
Quantum Fractal Codes Now in 3 dimensions:
Terms commute since:
Fractal logicals in x-y plane and in x-z plane
Yoshida, 2013
First spin propagation in y direction Second spin propagation in z direction
Quantum Fractal Codes Excitations:
Example:
X
X X X
X X X
Yoshida, 2013
X
X X X
X X X
Quantum Fractal Codes
X
X y
z
first
second cancellation
Excitations can propagate freely in the z-y direction Due to algebraic dependence of g and f
X X X
X
X
X second
first
Yoshida, 2013
Excitations:
Quantum Fractal Codes
X
X y
z
first
second cancellation X X X
X
X
X
No algebraic dependence (at least) logarithmic energy barrier
E ≈ Log(L) p ≈ exp(-E/kT) Polynomial rate
Optimal system size
Memory time
System size
Bravyi, Haah, 2011
Bravyi, Haah, 2013
Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
Chamon Code
Chamon, 2005
X ↔ Z
z
z
z
z x
x
x
x
Z
X X
Z
Chamon Code
Nog
Bravyi, Leemhuis, Terhal, 2011
x-X-X-X-X-x
Chamon Code
pfailure
p
pfailure
p
Increased
system size
Ben-Or, Aharonov, 1999
Noise (p) Encode Decode pfailure
Error threshold
Chamon Code
Noise (p) Encode Decode pfailure
Error threshold
Can be related to percolation
Chamon Code
Bath (T) Time (t)
Encode
Memory Time
Decode psucces
Future Work
• Homological codes Determine a more general condition for thermal
stability in terms of Hamiltonian properties
• Fractal codes Understand relation between fractal codes and
topological order
• Chamon code Consider better (but computational more
demanding) decoders
Conclustion: Existence of a quantum memory in 3D is still open
Fractal Codes
Logicals: Z
Logicals: X
Similarly:
Yoshida, 2013
Fractal Codes Logicals: Z
Logicals: X
Commutation relations
Both fractal like!
Yoshida, 2013
Error Correction
noise
Communication
Error Correction
no
ise
Storage
time
space
Error Correction
noise
Solution: Build in Redundancy
1 11111 1 11001
Encoding Decoding
Stabilizer Codes
Trade off Information k Stability d (number of qubits) (weight of a logical)
Logicals / Symmetries:
Gottesman, PRA 1996
• Allow for fault tollerant computations: errors do not accumalate when correcting • Overhead independent of computational time
Memory time
Gottesman, PRA 1998 Gottesman, 1310.2984
Stabilizers:
Ground states:
Error:
Exitated states:
Relation to topological order
Topological order at T > 0 ↔ Stable quantum memory at T > 0
Mazac, Hamma, 2012 Caselnovo, Chamon , 2007/2008
Topological Entropy
2D 3D 4D
Adiabatic Evolution
Hastings, 2011
No quantum memory in 2D
Homological codes
What can we learn k: stored information, related to genus d: distance, related to systole n: number of qubits, related to volume Intuitively... (sys)2 ≤ volume ... or better genus x (sys)2 ≤ volume
In general genus / log2 (genus) x (sys)2 ≤ volume
k / log2 (k) x (d)2 ≤ n
Gromov, 1992 Delfosse, 1301.6588
Example: Toric Code
Ground states:
Error:
Excitations
Plaquette:
Star:
Example: Toric Code
Fractal Codes Algebraic representation of Hamiltonian Example: toric code
Plaquette:
Star:
Yoshida, 2013
y x
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