School US-Japan seminar 2013/4/4 @Nara Topological quantum computation -from topological order to fault-tolerant quantum computation- The Hakubi Center for Advanced Research, Kyoto University Graduate School of Informatics, Kyoto University Keisuke Fujii Outline (1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model (3) Thermal instability of topological order (4) Error correction on (Kitaevʼs toric code) surface code (5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing What is topological order? Topological order is.......... -a new kind of order in zero-temperature phase of matter. -cannot be described by Landauʼs symmetry breaking argument. -ground states are degenerated and it exhibits long-range quantum entanglement. -the degenerated ground states cannot be distinguished by local operations. -topologically ordered states are robust against local perturbations. -related to quantum spin liquids, fractional quantum Hall effect, fault-tolerant quantum computation. Landau’s symmetry breaking argument Ising model (e.g. two dimension): Pauli operators: H = J X X − i j 10 01ij 0 i 10 I = ,X = ,Y = − ,Z = . the 01Ising Hamiltonian10 is invariant spin flippingi 0 Z w.r.t. X-basis.0 1 − ComputationalMagnetization (local basis order (qubit): parameter) takes zero above the critical temperature (CT) and non-zero below the CT. 0 , 1 → Z 0 = 0 ,Z1 = 1 {| | } | | | −| + , → X + = + ,X = {| |−} | | |− −|− disordered N (para) Multi-qubit system 0 , 1 ⊗ : {| | } Ai = I1 ... Ii 1orderedAi Ii+1 ...IN − ⊗ ⊗magnetization (ferro)⊗ ⊗ ⊗ (A = X, Y, Z) temperature T Landau’s symmetry breaking argument Ising model (e.g. two dimension): H = J X X − i j ij The Ising Hamiltonian is invariant under spin flipping Z w.r.t. X-basis. The ground states: ++ + , (Z2 symmetry) {| ··· | −−···−} ... longitudinal magnetic field | ↑↑ ↑ gs h X 2Nh ++ + i | ··· | −−···− ... | ↓↓ ↓ →ground state degeneracy is lifted by longitudinal magnetic field. →ground state degeneracy is not robust against local perturbation. Topologically ordered states In topologically ordered system.... degenerated ground states: any local perturbation ΨGS1 Ψ | | GS2 well separated O(L) Only nonlocal operators (high weight operator A ⊗ ) can exchange the ground states. → The ground state degeneracy cannot be lifted by any local αL operations (energy shift is ~ e − ). e.g) Ψ =(0000 + 1111 )/√2 | 1 | | Ψ =(0011 + 1100 )/√2 | 2 | | two orthogonal state cannot be distinguished by measuring a single qubit in any basis → second order perturbation first lifts the degeneracy Topologically ordered states How can we describe topologically ordered states efficiently? → Theory of quantum error correction is a very useful tool to describe topologically ordered system. quantum error correction codes topologically ordered system code subspace ground state degeneracy correctability against errors robustness against local perturbation (k-error correction code) (robust up to (2k+1)-th order perturbation) stabilizer codes stabilizer Hamiltonian (D. Gottesman PhD thesis 97) locality and translation invariance Toric code (surface code) Kitaev model classical repetition code Ising model (non-topological-ordered) (can correct either X or Z errors) → thermal stability/ information capacity of discrete systems/ exotic topologically ordered state (fractal quantum liquid) Outline (1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model (3) Thermal instability of topological order (4) Error correction on (Kitaevʼs toric code) surface code (5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing Majorana fermions Ising model in one dimension (Ising chain): H = J X X − i j ij the Ising Hamiltonian is invariant spin flipping Z w.r.t. X-basis. The ground states: ++ + , (Z2 symmetry) {| ··· | −−···−} longitudinal magnetic field gs h X 2Nh ++ + i | ··· | −−···− a repetition code X i X i +1 cannot correct any X error, since any single bit-flip error X i changes the code space non-trivially. Majorana fermions Let us consider a mathematically equivalent but physically different system. N 1 − Ising chain: H = J X X Ising − i i+1 i=1 Jordan-Wigner transformation c2i 1 = Z1 ...Zi 1Xi ⇄ − − (spin fermion) c2i = Z1 ...Zi 1Yi − ci,cj = δijI,ci† = ci {(Majorana} fermion operator) N 1 − 2N spinless fermions: H = J ( i)c c Maj − − 2i 2i+1 2 superconductor, topological insulator, semiconducting heterostructure (see A. Kitaev and C. Laumann, arXiv:0904.2771 for review ) Majorana fermions N 1 − paired H = J ( i)c c ・・・ Maj − − 2i 2i+1 2 c1 c2 c3 c4 c2N 1 c2N − ground states: ( i)c c Ψ = Ψ for all i. − 2i 2i+1| | unpaired Majorana fermions at the edges of the chain → “zero-energy Majorana boundary mode” 0¯ , 1¯ {| | } ( i)c c 0¯ = 0¯ , ( i)c c 1¯ = 1¯ ,c0¯ = 1¯ − 1 2N | | − 1 2N | −| 1| | Y1Z2 ...ZN 1YN (Z2 symmetry) X1 − If unpaired Majonara fermions are well (act on the ground subspace nontrivially) separated, this operator would not act. But! c 1 or c 2 N (odd weight fermionic operators) require coherent creation/ annihilation of a single fermion, which is prohibited by superselection rule. → X errors are naturally prohibited by the fermionic superselection rule. → Unpaired Majorana fermion is robust against any “physical” perturbation. Topological quantum computation Majorana fermions are non- Abelian, but do not allow universal quantum computation. Topologically protected gates + Magic state distillation [Bravyi-Kitaev PRA 71, 022316 (2005)] = Universal quantum computation pair creation of exchanging Majorana Majorana fermions fermions via T-junction Topological quantum computation Majorana fermions are non- Abelian, but do not allow universal quantum computation. Topologically protected gates What if your system has no + superselection rule, suchMagic asstate the distillation [Bravyi-Kitaev PRA 71, 022316 (2005)] fermionic parity preservation?= Universal quantum computation pair creation of exchanging Majorana Majorana fermions fermions via T-junction Kitaev’s honeycomb model A. Kitaev, Ann. Phys. 321, 2 (2006) Honeycomb model: H = J X X J Y Y J Z Z hc − x i j − y i j − z i j x-link y-link z-link x y x y z z z up y x y x dimerization z z z z left right y x y x y x Jx,Jy Jz z z z down y x y x J 2J 2 H = x y Y Y X X eff −16 J 3 left(p) right(p) up(p) down(p) z p | | Toric code Hamiltonian: local unitary H = J Z Z Z Z J X X X X transformation TC − l(f) r(f) d(f) u(f) − l(v) r(v) d(v) u(v) f v A. Kitaev, Ann. Phys. 303, 2 (2003) Kitaev’s toric code model Kitaevʼs toric code model is a representative example of topologically ordered system. face operator: Af = Zi Z i face f X ∈ Z Af Z X vertex operator: Bv = Xi Bv Z i vertex v X X ∈ Toric code Hamiltonian: H = J A J B − f − v v f Note that these operators are commutale: even number anti-commute × anti-commute crossover ＝ commute The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): A Ψ = Ψ ,BΨ = Ψ f | | v| | Kitaev toric code model Kitaevʼs toric code model is a representative example of topologically ordered system. ★Short note on the stabilizer formalism face operator: Af =n Zi •n-qubit Pauli group: 1, i I,X,Y,Z ⊗ Z {± ± } × { } i face f X ∈ Z•stabilizerAf Z group: = vertexSi , where operator: [Si,Sj] = 0 and Si = S† X S { } Bv = iXi Bv (commutative) (hermitian) Z i vertex v X X ∈ stabilizer generators: minimum independent set of stabilizer elements • Toric code Hamiltonian: H = J A J B − f − v v f •stabilizer state: S i Ψ = Ψ for all stabilizer generators S i |Note that| these operators are commutale: •example: X1X2,Z1Z2 ( 00 + 11 )/√2 even→ number| | anti-commute × anti-commute crossing ＝ commute dimension of the stabilizer subspace: 2^(# qubits - # generators) • The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): A Ψ = Ψ ,BΨ = Ψ f | | v| | Kitaev toric code model Kitaevʼs toric code model is a representative example of topologically ordered system. face operator: Af = Zi Z i face f X ∈ →stabilizer generators Z Af Z X vertex operator: Bv = Xi Bv Z i vertex v X X ∈ Toric code Hamiltonian: H = J A J B − f − v v →stabilizer Hamiltonian f Note that these operators are commutale: even number anti-commute × anti-commute crossing ＝ commute The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): →stabilizer subspace A Ψ = Ψ ,BΨ = Ψ f | | v| | Structure of grand states Degeneracy of the ground subspace: Z X [Torus] Z Af Z 2 X # qubits: (edges) on N×N torus = 2N Bv Z X # stabilizer generators: X 2 (faces + vertexes - 2) = 2N - 2 # dimension of ground subspace: 22=4 # logical qubits → 2 (two logical qubits) [General surface] (face)+(vertex)-(edge)=2-2g g = genus Euler characteristic # logical qubits → (edge)-[(face)+(vertex)-2]=2g Structure of ground states Degeneracy of the ground subspace: Z X [Torus] Z Af Z 2 X # qubits: (edges) in N×N torus = 2N Bv Z X # stabilizer generators: X 2 (faces + vertexes - 2) = 2N - 2 # dimension of ground subspace: 22=4 # logical qubits → 2 (two logical qubits) [General surface] How is the ground (face)+(vertex)-(edge)=2-2state degeneracy described?g g = genus Euler characteristic → Find a good quantum number! The operator that acts on the ground subspace #nontrivially, logical qubits “logical→ (edge)-[(face)+(vertex)-2]=2 operator”. g Non-trivial cycle: Logical operators L X(¯c1 ) L L The operators on non-trivial cycles Z ( c 1 ) ,X (¯ c 1 ) are commutable with all face and vertex operators, but L cannot given by a product of them. X(¯c1 ) L L L Z(c ),X(¯c ) =0 Z(c1 ) { 1 1 } → logical Pauli operators.