Topological Quantum Error Correcting Codes Kasper Duivenvoorden - PowerPoint PPT Presentation

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 ground space degeneracy: Definition: locally ground states are the same B A Use ground space manifold of a topological phase to encode quantum information

Overview • Homological codes A formalism using emphasizing topology • Fractal codes Possibly thermally stable codes • Chamon code Work in progress

Example: Toric Code Ground states: Logicals: preserves the code space

Homological codes Lattice  CW-complex Point  0-cells  1-cells Line 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 Dimensionality of logicals Z X 2D 1 1 3D 1 2 4D 2 2

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

Thermal Stability E At T>0 Thermal excitation p = exp(-E/kT) Stability! Chesi, Loss, Bravyi, Terhal, 2010

Toric Codes D – dimensional lattice, qubits on k cells  k and D-k dimensional logicals Dim. of logicals Stable Memory Z X 2D 1 1 No 3D 1 2 4D 2 2 Quantum

Toric Codes D – dimensional lattice, qubits on k cells  k and D-k dimensional logicals 2D Drawbacks - Constant information - We only have 3 dimensions 3D  Fractal codes 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: y z x • More spins per site: 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: y Z Z Z Z x Logical X[g]  g(1+fy) = 0 modulo 2 Assumptions:

Fractal Codes Logical: X[g] Hamiltonian terms: Z Z Z Z X X X X X X X X 1 2 3 2 1 X X X X X X X X X X X X X X X y X x Set of Logicals: fractal like point like

Fractal Codes Z Z Z Z Excitations 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

Quantum Fractal Codes Excitations: Example: X X X First spin propagation in y direction X X X X Second spin propagation in z direction Yoshida, 2013

Quantum Fractal Codes Excitations: y second cancellation X X X X X second X X X X z X X X X X X first first Excitations can propagate freely in the z-y direction Due to algebraic dependence of g and f Yoshida, 2013

Quantum Fractal Codes y No algebraic dependence second cancellation X X X X  (at least) logarithmic energy barrier X Bravyi, Haah, 2011 X z X E ≈ Log(L) p ≈ exp( -E/kT) X first Memory time  Polynomial rate  Optimal system size Bravyi, Haah, 2013 System size

Overview • Homological codes A formalism using emphasizing topology • Fractal codes Possibly thermally stable codes • Chamon code Work in progress

Chamon Code X ↔ Z Z z x X X z z x x  z x Z Chamon, 2005

Chamon Code x-X-X-X-X-x Nog Bravyi, Leemhuis, Terhal, 2011

Chamon Code Error threshold Decode  p failure Encode Noise (p) p failure p failure  Increased system size p p Ben-Or, Aharonov, 1999

Chamon Code Error threshold Decode  p failure Encode Noise (p) Can be related to percolation

Chamon Code Memory Time Decode  p succes Encode Bath (T) Time (t) 

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 Communication noise

Error Correction Storage time noise space

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

Gottesman, PRA 1996 Stabilizer Codes Stabilizers: Ground states: Error: Exitated states: • Allow for fault tollerant computations: errors do not accumalate when correcting • Overhead independent of computational time Logicals / Symmetries: Gottesman, PRA 1998 Gottesman, 1310.2984 Trade off Memory time Information k Stability d (number of qubits) (weight of a logical)

Relation to topological order Topological order at T > 0 ↔ Stable quantum memory at T > 0 Topological Entropy Adiabatic Evolution 2D  No quantum memory in 2D 3D 4D Mazac, Hamma, 2012 Caselnovo, Chamon , 2007/2008 Hastings, 2011

Homological codes What can we learn k: stored information, related to genus d: distance, related to systole n: number of qubits, related to volume (sys) 2 Intuitively... ≤ volume genus x (sys) 2 ... or better ≤ volume Gromov, 1992 In general genus / log 2 (genus) x (sys) 2 ≤ volume Delfosse, 1301.6588 k / log 2 (k) x (d) 2 ≤ n

Example: Toric Code Ground states: Error:  Excitations

Example: Toric Code Plaquette: Star:

Fractal Codes Algebraic representation of Hamiltonian Example: toric code Plaquette: Star: y x Yoshida, 2013