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