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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 Gapped topological phase with

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Topological Quantum Error Correcting Codes

Kasper Duivenvoorden JARA IQI, RWTH Aachen

Topological Phases of Quantum Matter – 4 September 2014

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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

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Overview

  • Homological codes

A formalism using emphasizing topology

  • Fractal codes

Possibly thermally stable codes

  • Chamon code

Work in progress

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Example: Toric Code

Logicals: preserves the code space Ground states:

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Homological codes

Lattice  CW-complex Point  0-cells Line  1-cells Surface  2-cells ... ` k-cells Boundary map

= =

Co-Boundary map

= =

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Homological codes

Boundary map

=

Co-Boundary map

=

Properties 1: 2:

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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

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Homological codes

= =

Logicals

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Homological codes

= =

Logicals

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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

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Thermal Stability

Movement of anyons At T=0 Tunneling p = exp(-L) At T>0 Thermal excitation p = exp(-E/kT) No stability! E

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Thermal Stability

At T>0 Thermal excitation p = exp(-E/kT) Stability! E

Chesi, Loss, Bravyi, Terhal, 2010

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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

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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

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Overview

  • Homological codes

A formalism using emphasizing topology

  • Fractal codes

Possibly thermally stable codes

  • Chamon code

Work in progress

Yoshida, 2013

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Fractal Codes

Algebraic representation of Z-type operators

  • Generalize to higher dimensions:
  • More spins per site:

y z x

Yoshida, 2013

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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

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Fractal Codes

Hamiltonian Example: Z Z Z Logical X[g]  g(1+fy) = 0 modulo 2 Assumptions: Z y x

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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

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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

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Quantum Fractal Codes

Now in 3 dimensions: Terms commute since: Fractal logicals in x-y plane and in x-z plane

Yoshida, 2013

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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

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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:

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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

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Overview

  • Homological codes

A formalism using emphasizing topology

  • Fractal codes

Possibly thermally stable codes

  • Chamon code

Work in progress

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Chamon Code

Chamon, 2005

X ↔ Z

z z z z x x x x

Z X X Z

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Chamon Code

Nog

Bravyi, Leemhuis, Terhal, 2011

x-X-X-X-X-x

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Chamon Code

pfailure p pfailure p  Increased system size

Ben-Or, Aharonov, 1999

Noise (p) Encode Decode  pfailure Error threshold

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Chamon Code

Noise (p) Encode Decode  pfailure Error threshold Can be related to percolation

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Chamon Code

Bath (T) Time (t)  Encode Memory Time Decode  psucces

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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

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Fractal Codes

Logicals: Z Logicals: X Similarly:

Yoshida, 2013

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Fractal Codes

Logicals: Z Logicals: X Commutation relations Both fractal like!

Yoshida, 2013

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Error Correction

noise Communication

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Error Correction

noise Storage time space

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Error Correction

noise Solution: Build in Redundancy 1 11111 1 11001 Encoding Decoding

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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:

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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

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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

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Example: Toric Code

Ground states: Error:  Excitations

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Plaquette: Star:

Example: Toric Code

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Fractal Codes

Algebraic representation of Hamiltonian Example: toric code Plaquette: Star:

Yoshida, 2013

y x