Anyons and topological quantum computing

Transcription

1 Anyons and topological quantum computing Statistical Physics PhD Course Quantum statistical physics and Field theory 05/10/2012

2 Plan of the seminar Why anyons? Anyons: definitions fusion rules, F and R matrices Summary of quantum computation and example models: Toric code Honeycomb model Universal topological quantum computation: compiling with Fibonacci anyons Conclusions

3 Plan of the seminar Why anyons? Anyons: definitions fusion rules, F and R matrices Summary of quantum computation and example models: Toric code Honeycomb model Universal topological quantum computation: compiling with Fibonacci anyons Conclusions

4 Plan of the seminar Why anyons? Anyons: definitions fusion rules, F and R matrices Summary of quantum computation and example models: Toric code Honeycomb model Universal topological quantum computation: compiling with Fibonacci anyons Conclusions

5 Why anyons and topology? For quantum computing we need: qbits and a way to operate. ξ = a 0 + b 1 U ξ Qbits any sistem which has two degenerate states; Gates done by applying unitary operators on states. Errors come from decoherence and inexact application of U. Error correction can in theory be done, but not in practice.

6 Why anyons and topology? Another approach: topology protects the qbits. For Topological Quantum Computation: qbit anyons (excitations of the system) unitary operator braiding of the anyons measurement fusion of the anyons

7 Why anyons and topology? Another approach: topology protects the qbits. For Topological Quantum Computation: qbit anyons (excitations of the system) unitary operator braiding of the anyons measurement fusion of the anyons

8 Anyons Take two identical particles and exchange them two times: in 3D it is equivalent to the identity ± phase on exchange; action of the permutation group (finite, abelian); Two types of particles: bosons and fermions. in 2D γ 1 and γ 2 involve a passing through nontrivial phase; action of the braid group (infinite, non abelian); Anyons.

9 Anyons: some properties Fusion: A new type of anyon is formed by putting together two anyons: φ a φ b = c N c ab φ c, N c ab = 0, 1 Abelian anyons: only one fusion channel is available. Connection with conformal field theories: This is the purely algebric part of an OPE. Example: Ising CFT σ σ = 1 + ɛ, ɛ σ = σ, ɛ ɛ = 1 Ising anyons

10 Anyons: some properties Fusion: A new type of anyon is formed by putting together two anyons: φ a φ b = c N c ab φ c, N c ab = 0, 1 Abelian anyons: only one fusion channel is available. Connection with conformal field theories: This is the purely algebric part of an OPE. Example: Ising CFT σ σ = 1 + ɛ, ɛ σ = σ, ɛ ɛ = 1 Ising anyons

11 Anyons: some properties Example: Tricritical Ising CFT Notation: even: φ 21 = ɛ, φ 31 = t, φ 41 = ɛ odd: φ 12 = σ, φ 22 = σ t t = 1 + t ɛ ɛ = 1 + t ɛ t = ɛ + ɛ ɛ σ = σ ɛ σ = σ + σ t σ = σ t σ = σ + σ Subalgebra of Fibonacci anyons Ising anyons σ σ = 1 + ɛ σ ɛ = σ ɛ ɛ = 1 ɛ ɛ = t ɛ t = ɛ ɛ σ = σ σ σ = ɛ + t σ σ = 1 + ɛ + t + ɛ

12 Anyons: some properties R matrices and F matrices: Non abelian anyons higher dim representations of B n : R matrix F matrix: relates the ways three anyons can fuse into a fourth. Consistency conditions: pentagon and hexagon equations

13 Anyons: some properties R matrices and F matrices: Non abelian anyons higher dim representations of B n : R matrix F matrix: relates the ways three anyons can fuse into a fourth. Consistency conditions: pentagon and hexagon equations

14 Anyons: some properties Action of braid group on an ordered basis is specified by: - R matrix - B matrix: B = F 1 RF Example: Fibonacci anyons: rich enough for universal computing Anyons: 1, σ Fusion rules: 1 1 = 1, σ 1 = σ, F matrices: Fσσa 1σb = δb σ δσ a F σσ σσ = R matrix: R = σ σ = 1 + σ ( φ 1 φ 1 2 φ 1 2 φ 1 ) ( e 4πı e 2πı 5 )

15 Anyons: some properties Action of braid group on an ordered basis is specified by: - R matrix - B matrix: B = F 1 RF Example: Fibonacci anyons: rich enough for universal computing Anyons: 1, σ Fusion rules: 1 1 = 1, σ 1 = σ, F matrices: Fσσa 1σb = δb σ δσ a F σσ σσ = R matrix: R = σ σ = 1 + σ ( φ 1 φ 1 2 φ 1 2 φ 1 ) ( e 4πı e 2πı 5 )

16 Quantum computation Steps of a quantum computation: - initialization with a state ψ of one or more qbits - unitary operation U ψ - measurement of the state Program unitary operator = building the quantum circuit. We can build any U with few 1- and 2-qbit gates. - CNOT = Hadamard = 1 2 ( π 8 gate = ( e ıπ ) )

17 An example: the adder Quantum adder:

18 Topological Quantum computation: a model Let us consider a simple model: Toric code Lattice of 1 2 spins. H = J e A s J m s A s = B p = j star(s) j plaquette(p) p σj x σ z j A s, B p commute. B p

19 Topological Quantum computation: a model - on a torus, 4 times degenerate ground state - two kinds of excitations, with gap: two J e contributions at the extremes of l W (e) l = j l σz j two J m contributions at the extremes of l (dual lattice) W (m) l = j l σx j - W (e) and W (m) create no excitations for closed loops

20 Topological Quantum computation: a model Excitations in this model are (abelian) anyons: braiding e particles bosons (no phase) braiding m particles bosons (no phase) braiding couples ɛ = e m fermions ( 1 phase) Fusion rules: e e = 1 m m = 1 ɛ ɛ = 1 e m = ɛ e ɛ = m m ɛ = e

21 Topological Quantum computation The goal: universal topological quantum computation. - Degenerate ground states with quasiparticles at fixed points - A qbit is made from quasiparticles non local - Discrete operations: result depends only from topology of trajectory of braid Input Initialize the qbit by creating anyons in a specific way (e.g. 0 by pulling from vacuum) Operation Physically moving and braiding anyons Output Measurement of the state of the qbit

22 Topological Quantum computation The goal: universal topological quantum computation. - Degenerate ground states with quasiparticles at fixed points - A qbit is made from quasiparticles non local - Discrete operations: result depends only from topology of trajectory of braid Input Initialize the qbit by creating anyons in a specific way (e.g. 0 by pulling from vacuum) Operation Physically moving and braiding anyons Output Measurement of the state of the qbit

23 Topological Quantum computation: another model H = J x σj x σx k J y σ y j σy k J z x links y links z links σ z j σz k Honeycomb model: two phases - J x, J y, J z satisfy some conditions plus a perturbation: non abelian anyons - otherwise: abelian anyons, same as toric code

24 The honeycomb model: sketch of the solution Introduce four Majorana fermions for each lattice site c, b α H H = ı A jk c j c k 4 A jk = 2J αjk u jk, j,k having substituted σ α ıb α c. u jk = ıb α jk j b α jk k Note: [ H, u jk ] = 0 L = L {u} in the subspace L {u}, H {u} is free fermions. Introduce for each plaquette p the hexagon vortex operators W p = σ1σ x y 2 σz 3σ4σ x y 5 σz 6 j,k p [W p, H] = 0 L = L {w} Then L {w} = P L L {u}, P L projector on the physical space. u jk

25 The honeycomb model: sketch of the solution Introduce four Majorana fermions for each lattice site c, b α H H = ı A jk c j c k 4 A jk = 2J αjk u jk, j,k having substituted σ α ıb α c. u jk = ıb α jk j b α jk k Note: [ H, u jk ] = 0 L = L {u} in the subspace L {u}, H {u} is free fermions. Introduce for each plaquette p the hexagon vortex operators W p = σ1σ x y 2 σz 3σ4σ x y 5 σz 6 j,k p [W p, H] = 0 L = L {w} Then L {w} = P L L {u}, P L projector on the physical space. u jk

26 The honeycomb model: sketch of the solution Results: Minimum energy: state with w p = 1 p (no vortices) The spectrum is given by ɛ(q) = ± 2(J x e ıq n1 + J y e ıq n2 + J z ) J x, J y, J z satisfy triangle inequalities: gapless spectrum phase B For other values of J x, J y, J z : gapped spectrum phase A α

27 The honeycomb model In the gapless phase: Particle types: fermions (gapless), vortices (gapped). A gap appears if we apply a perturbation: Results: α=x,y,z j hx hy hz a gap ( J ) opens for the fermions 2 hexagon operators W p are no longer conserved vortices can hop between hexagons h α σ α j Anyon types: 1, ɛ (fermion), σ (vortex) Fusion rules: ɛ ɛ = 1 ɛ σ = σ σ σ = 1 + ɛ Ising anyons

28 Universal computation With Fibonacci anyons we can build a universal quantum computer: - Qbit encoding: one qbit four anyons 1234 (excitation); 0 fusion of 12 gives 1, 1 fusion of 12 gives σ Equivalently: three anyons (with one inadmissible state) - Measurement: fusing the couples of anyons observe vacuum or not;

29 Universal computation With Fibonacci anyons we can build a universal quantum computer: - Qbit encoding: one qbit four anyons 1234 (excitation); 0 fusion of 12 gives 1, 1 fusion of 12 gives σ Equivalently: three anyons (with one inadmissible state) - Measurement: fusing the couples of anyons observe vacuum or not;

30 Universal computation: universal gates - Unitary operations: implement 1 and 2 qbit gates - braid matrices: R and B = Fσσ σσ R (Fσσ σσ) 1 - on one qbit: restrict on weaves (only one anyon moves) e 4πı 5 0 φ 1 e πı 5 φ 1 2 e 3πı 5 σ 1 = 0 e 2πı 5, σ 2 = φ 1 2 e 3πı e 2πı 5 5 φ 1 e 3πı 5 A generic weave is U({n i }) = σ nm 1 σnm σ n2 2 σn1 1, with n i = ±2, ±4 or 0 for the extremes. Search for the weave which approximate best U: brute force search or icosahedral group hashing. best accuracy best time

31 Universal computation: universal gates - Unitary operations: implement 1 and 2 qbit gates - braid matrices: R and B = Fσσ σσ R (Fσσ σσ) 1 - on one qbit: restrict on weaves (only one anyon moves) e 4πı 5 0 φ 1 e πı 5 φ 1 2 e 3πı 5 σ 1 = 0 e 2πı 5, σ 2 = φ 1 2 e 3πı e 2πı 5 5 φ 1 e 3πı 5 A generic weave is U({n i }) = σ nm 1 σnm σ n2 2 σn1 1, with n i = ±2, ±4 or 0 for the extremes. Search for the weave which approximate best U: brute force search or icosahedral group hashing. best accuracy best time

32 Icosahedral group hashing Largest finite subgroup of SU(2) we use it to do iteratively better approximations. Icosahedral group: I = {g 0 = 1, g 1,..., g 59 }. 1 brute force search (length L) for Ĩ(L) = { g 0, g 1,..., g 59 } 2 construct 1 = g i1... g in g in+1, with g n+1 = [g i1... g in ] 1 ; for the approximate rotations, S(L, n) = { g i1... g in g in+1 } {ik } is a mesh of fine rotations around 1 3 search for approximation of U with minimum distance: - initial approximation: search for best U 0 in Ĩ(L 0 ) m - k th iteration: search for best correction V S(L k, n) so that U k = U k 1 V has minimum distance from U; - iterate with bigger L k. Note: L k are chosen so that error of 1 error of U k.

33 Universal computation: universal gates - for two qbits: controlled gates one{ control qbit. 1 vacuum 1 action of control pair: σ one anyon Let s build a controlled gate: 1 injection of the control pair of the control qbit in the controlled qbit; (found by searching for σ 2U({n i })σ 2 = 1) 2 one qbit operation. Example: CNOT σ x; 3 inverse injection of the control pair.

34 Universal computation: universal gates - for two qbits: controlled gates one{ control qbit. 1 vacuum 1 action of control pair: σ one anyon Let s build a controlled gate: 1 injection of the control pair of the control qbit in the controlled qbit; (found by searching for σ 2U({n i })σ 2 = 1) 2 one qbit operation. Example: CNOT σ x; 3 inverse injection of the control pair.

35 Universal computation: universal gates - for two qbits: controlled gates one{ control qbit. 1 vacuum 1 action of control pair: σ one anyon Let s build a controlled gate: 1 injection of the control pair of the control qbit in the controlled qbit; (found by searching for σ 2U({n i })σ 2 = 1) 2 one qbit operation. Example: CNOT σ x; 3 inverse injection of the control pair.

36 Universal computation: universal gates - for two qbits: controlled gates one{ control qbit. 1 vacuum 1 action of control pair: σ one anyon Let s build a controlled gate: 1 injection of the control pair of the control qbit in the controlled qbit; (found by searching for σ 2U({n i })σ 2 = 1) 2 one qbit operation. Example: CNOT σ x; 3 inverse injection of the control pair.

37 Universal computation: universal gates It can also be done with one anyon instead of a couple: Swap of one anyon: action of F matrix. U({n i })σ 2 = F Summing up: for a controlled gate with control pair: Injection + U + Ejection with control anyon: F + U + F 1 All ingredients to build any circuit up to arbitrary precision.

38 Conclusions In 2D: anyons, characterised by fusion rules, braidings; It is possible to do topologically protected quantum computation; For some types of anyons: universal quantum computation; Proposed models: lattice surface codes; It is possible to efficiently compile a quantum circuit into braidings.

39 References: Thank you for your attention 1 Preskill, (lecture notes) 2 Nielsen, Chuang Quantum computation and quantum information, Cambridge University Press 3 Brennen, Pachos, Why should anyone care about computing with anyons?, Proc. R. Soc. A , Das Sarma, Freedman, Nayak, Simon, Stern, Non Abelian anyons and topological quantum computation, ArXiv v1 5 Trebst, Troyer, Wang, Ludwig, A short introduction to Fibonacci anyon models, Progress of Theoretical Physics 176, Kitaev, Laumann, Topological phases and quantum computation, ArXiv v1 7 Kitaev, Anyons in an exactly solved model and beyond, ArXiv cond-mat/ v3 8 Hormozi, Zikos, Bonesteel, Simon, Topological quantum compiling, Phys. Rev. B, 75, (2007) 9 Burrello, Mussardo, Wan, Topological quantum gate construction by iterative pseudogroup hashing, New J. Phys. 13, (2011)