スタビライザー形式と トポロジカル量子計算 藤井 啓祐 ψ pl eiθlpz ψ pl

Transcription

1 ψ p L e iθlp ψ p L

2 1990.-G. Wen Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces:topological order Phys. Rev. B A. Kitaev Surface (Torus) code: Topological quantum memory quantum information Kitaev model: exactly solvable model of topological ordered system condensed matter physics ariv: R. Raussendorf et al. Topologically protected fault-tolerant quantum computation Ann. Phys ; NJP 9 199; PRL

3 Monday QEC11 Schedule Tuesday 7:00-8:10 Continental breakfast 7:00-8:25 Continental breakfast :30-10:00.-G. On-site registration Wen 8:30-9:30 Raymond Laflamme 8:15-8:30 Opening statement (Lidar) 9:30-10:00 Jeongwan Haah Ground-state 8:30-9:30 Todd Brun degeneracy 10:00-10:30 Matthew of Reed the fractional quantum Hall states in the presence 9:30-10:30 Robert Raussendorf 10:30-10:50 Coffee break 10:30-10:50 Coffee break 10:50-11:30 Mike Biercuk of a 10:50-11:50 random Andrew Landahl potential 11:30-12:10 and Graeme on Smith high-genus Riemann surfaces:topological order 11:50-12:50 Daniel Lidar 12:10-12:30 Constantin Brif 12:50-2:20 Lunch 12:30-2:00 Lunch Phys. 2:20-3:00Rev. John Preskill B :00-2:40 Nir Davidson 3:00-3:40 Lorenza Viola 2:40-3:20 Dave Bacon 3:40-4:00 Gerardo Paz 3:20-3:40 Bryan Fong 4:00-4:20 Coffee break 3:40-4:00 Coffee break 4:20-5:00 David Cory 4:00-4:40 Hector Bombin 5:00-5:20 Jim Harrington 4:40-5:20 Sergey Bravyi :20-5:40 A. Andrew Kitaev Cross 5:20-5:40 Isaac Kim 5:40-6:00 Nicolas Delfosse 5:40-6:00 Prashant Kumar Surface 6:00-7:00 Put (Torus) posters up code: 6:00-7:00 Topological Poster session quantum memory quantum information Wednesday Thursday Kitaev 7:00-8:25 model: Continental breakfast exactly 7:00-8:25solvable Continental breakfast model of topological ordered system 8:30-9:30 Hideo Mabuchi 8:30-9:30 Emanuel Knill 9:30-10:10 Jason Twamley 9:30-10:10 Rainer Blatt 10:10-10:30 Cody Jones 10:10-10:30 Leonid Pryadko condensed matter physics 10:30-10:50 Coffee break 10:30-10:50 Coffee break ariv: :50-11:30 Dieter Suter 10:50-11:30 Sean Barrett 11:30-12:10 Markus Grassl 11:30-12:10 Eduardo Novais 12:10-12:30 Gonzalo Alvarez 12:10-12:30 Andrew Landahl 12:30-2:00 Lunch, dessert in lobby 12:30-2:00 Lunch 2:00-2:40 Ognyan Oreshkov 2:00-2:40 Viatechslav Dobrovitski 2:40-3:00 Liang Jiang 2:40-3:20 Kaveh Khodjasteh 3:00-3:20 Jason Dominy 3:20-3:40 Katherine Brown :20-3:40 R. Peter Raussendorf Brooks 3:40-4:00 et Coffee al. break 3:40-4:00 Coffee break 4:00-4:40 Martin Roetteler 4:00-4:40 Austin Fowler Topologically protected fault-tolerant Panel moderated by Mark 4:40-5:20 David Kribs 4:40-6:00 quantum computation Byrd 5:20-5:40 Cedric Beny Ann. Phys ; NJP 9 199; PRL :40-6:00 Prabha Mandayam 6:00-7:00 Poster session 6:00-7:00 Poster session 7:00-9:00 Banquet Friday 7:00-8:25 Continental breakfast 8:30-9:30 Daniel Gottesman 9:30-10:10 Mark Wilde 10:10-10:30 Simon Benjamin 10:30-10:50 Coffee break 10:50-11:30 Thaddeus Ladd 11:30-12:10 Guillaume Duclos-Cianci 12:10-12:30 Yuichiro Fujiwara 12:30-:12:40 Closing statement 12:40-2:00 box lunch, END All talks and meals will take place at the Davidson conference center Legend: Keynote Tutorial Best student paper prize talk

4 Monday QEC11 Schedule Tuesday 7:00-8:10 Continental breakfast 7:00-8:25 Continental breakfast :30-10:00.-G. On-site registration Wen 8:30-9:30 Raymond Laflamme 8:15-8:30 Opening statement (Lidar) 9:30-10:00 Jeongwan Haah Ground-state 8:30-9:30 Todd Brun degeneracy 10:00-10:30 Matthew of Reed the fractional quantum Hall states in the presence 9:30-10:30 Robert Raussendorf 10:30-10:50 Coffee break 10:30-10:50 Coffee break 10:50-11:30 Mike Biercuk of a 10:50-11:50 random Andrew Landahl potential 11:30-12:10 and Graeme on Smith high-genus Riemann surfaces:topological order 11:50-12:50 Daniel Lidar 12:10-12:30 Constantin Brif 12:50-2:20 Lunch 12:30-2:00 Lunch Phys. 2:20-3:00Rev. John Preskill B :00-2:40 Nir Davidson 3:00-3:40 Lorenza Viola 2:40-3:20 Dave Bacon 3:40-4:00 Gerardo Paz 3:20-3:40 Bryan Fong 4:00-4:20 Coffee break 3:40-4:00 Coffee break 4:20-5:00 David Cory 4:00-4:40 Hector Bombin 5:00-5:20 Jim Harrington 4:40-5:20 Sergey Bravyi :20-5:40 A. Andrew Kitaev Cross 5:20-5:40 Isaac Kim 5:40-6:00 Nicolas Delfosse 5:40-6:00 Prashant Kumar Surface 6:00-7:00 Put (Torus) posters up code: 6:00-7:00 Topological Poster session quantum memory quantum information Wednesday Thursday Kitaev 7:00-8:25 model: Continental breakfast exactly 7:00-8:25solvable Continental breakfast model of topological ordered system 8:30-9:30 Hideo Mabuchi 8:30-9:30 Emanuel Knill 9:30-10:10 Jason Twamley 9:30-10:10 Rainer Blatt 10:10-10:30 Cody Jones 10:10-10:30 Leonid Pryadko condensed matter physics 10:30-10:50 Coffee break 10:30-10:50 Coffee break ariv: :50-11:30 Dieter Suter 10:50-11:30 Sean Barrett 11:30-12:10 Markus Grassl 11:30-12:10 Eduardo Novais 12:10-12:30 Gonzalo Alvarez 12:10-12:30 Andrew Landahl 12:30-2:00 Lunch, dessert in lobby 12:30-2:00 Lunch 2:00-2:40 Ognyan Oreshkov 2:00-2:40 Viatechslav Dobrovitski 2:40-3:00 Liang Jiang 2:40-3:20 Kaveh Khodjasteh 3:00-3:20 Jason Dominy 3:20-3:40 Katherine Brown :20-3:40 R. Peter Raussendorf Brooks 3:40-4:00 et Coffee al. break 3:40-4:00 Coffee break 4:00-4:40 Martin Roetteler 4:00-4:40 Austin Fowler Topologically protected fault-tolerant Panel moderated by Mark 4:40-5:20 David Kribs 4:40-6:00 quantum computation Byrd 5:20-5:40 Cedric Beny Ann. Phys ; NJP 9 199; PRL :40-6:00 Prabha Mandayam 6:00-7:00 Poster session 6:00-7:00 Poster session 7:00-9:00 Banquet Friday 7:00-8:25 Continental breakfast 8:30-9:30 Daniel Gottesman 9:30-10:10 Mark Wilde 10:10-10:30 Simon Benjamin 10:30-10:50 Coffee break 10:50-11:30 Thaddeus Ladd 11:30-12:10 Guillaume Duclos-Cianci 12:10-12:30 Yuichiro Fujiwara 12:30-:12:40 Closing statement 12:40-2:00 box lunch, END All talks and meals will take place at the Davidson conference center Legend: Keynote Tutorial Best student paper prize talk

5 Monday QEC11 Schedule Tuesday 7:00-8:10 Continental breakfast 7:00-8:25 Continental breakfast :30-10:00.-G. On-site registration Wen 8:30-9:30 Raymond Laflamme 8:15-8:30 Opening statement (Lidar) 9:30-10:00 Jeongwan Haah Ground-state 8:30-9:30 Todd Brun degeneracy 10:00-10:30 Matthew of Reed the fractional quantum Hall states in the presence 9:30-10:30 Robert Raussendorf 10:30-10:50 Coffee break 10:30-10:50 Coffee break 10:50-11:30 Mike Biercuk of a 10:50-11:50 random Andrew Landahl potential 11:30-12:10 and Graeme on Smith high-genus Riemann surfaces:topological order 11:50-12:50 Daniel Lidar 12:10-12:30 Constantin Brif 12:50-2:20 Lunch 12:30-2:00 Lunch Phys. 2:20-3:00Rev. John Preskill B :00-2:40 Nir Davidson 3:00-3:40 Lorenza Viola 2:40-3:20 Dave Bacon 3:40-4:00 Gerardo Paz 3:20-3:40 Bryan Fong 4:00-4:20 Coffee break 3:40-4:00 Coffee break 4:20-5:00 David Cory 4:00-4:40 Hector Bombin 5:00-5:20 Jim Harrington 4:40-5:20 Sergey Bravyi :20-5:40 A. Andrew Kitaev Cross 5:20-5:40 Isaac Kim 5:40-6:00 Nicolas Delfosse 5:40-6:00 Prashant Kumar Surface 6:00-7:00 Put (Torus) posters up code: 6:00-7:00 Topological Poster session quantum memory quantum information Wednesday Thursday Kitaev 7:00-8:25 model: Continental breakfast exactly 7:00-8:25solvable Continental breakfast model of topological ordered system 8:30-9:30 Hideo Mabuchi 8:30-9:30 Emanuel Knill 9:30-10:10 Jason Twamley 9:30-10:10 Rainer Blatt 10:10-10:30 Cody Jones 10:10-10:30 Leonid Pryadko condensed matter physics 10:30-10:50 Coffee break 10:30-10:50 Coffee break ariv: :50-11:30 Dieter Suter 10:50-11:30 Sean Barrett 11:30-12:10 Markus Grassl 11:30-12:10 Eduardo Novais 12:10-12:30 Gonzalo Alvarez 12:10-12:30 Andrew Landahl 12:30-2:00 Lunch, dessert in lobby 12:30-2:00 Lunch 2:00-2:40 Ognyan Oreshkov 2:00-2:40 Viatechslav Dobrovitski 2:40-3:00 Liang Jiang 2:40-3:20 Kaveh Khodjasteh 3:00-3:20 Jason Dominy 3:20-3:40 Katherine Brown :20-3:40 R. Peter Raussendorf Brooks 3:40-4:00 et Coffee al. break 3:40-4:00 Coffee break 4:00-4:40 Martin Roetteler 4:00-4:40 Austin Fowler Topologically protected fault-tolerant Panel moderated by Mark 4:40-5:20 David Kribs 4:40-6:00 quantum computation Byrd 5:20-5:40 Cedric Beny Ann. Phys ; NJP 9 199; PRL :40-6:00 Prabha Mandayam 6:00-7:00 Poster session 6:00-7:00 Poster session 7:00-9:00 Banquet Friday 7:00-8:25 Continental breakfast 8:30-9:30 Daniel Gottesman 9:30-10:10 Mark Wilde 10:10-10:30 Simon Benjamin 10:30-10:50 Coffee break 10:50-11:30 Thaddeus Ladd 11:30-12:10 Guillaume Duclos-Cianci 12:10-12:30 Yuichiro Fujiwara 12:30-:12:40 Closing statement 12:40-2:00 box lunch, END All talks and meals will take place at the Davidson conference center (condensed matter physics) Legend: Keynote Tutorial Best student paper prize talk

6

7

8 Qubit & Pauli matrices (1.1) qubit: ψ = α 0 + β 1 (1.2) Pauli matrices: = = Y = 0 i i 0 eg) 0 = 1 1 = 0 (bit-flip) Y = i 0 = 0 1 = 1 (phase-flip) Y 0 = i 1 Y 1 = i 0 (bit&phase-flip + global phase) note) 0, 1 are eigenstates of. ± ( 0 ± 1)/ 2 are eigenstates of.

9 Pauli group (1.3) n-qubit Pauli products: {±1, ±i} {I,,Y,} n which forms so-called Pauli group P n. eg) 2-qubit Pauli group: {II,I,IY,I,I,,Y,,Y I,Y,Y Y,Y,I,,Y,} {1, 1, i, i} (where AB A B)

10 Single-qubit Clifford gates (1.4) Clifford operations: = conjugation Pauli products Pauli products A UAU = B P P Hadamard (1.5) gate H: H = HH =, HH = eg) H 0 = +, H 1 = (1.6) Phase gate S: S = i SS = Y, SY S =, SS =

11 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t )U CNOT = c t, U CNOT (I c t )U CNOT = I c t, U CNOT ( c I t )U CNOT = c I t, U CNOT (I c t )U CNOT = c t (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

12 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

13 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

14 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

15 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

16 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

17 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

18 Two-qubit Clifford gates (1.7) CNOT gate: U CNOT = 00 c I t + 11 c t U CNOT ( c I t ) = ( c t )U CNOT, U CNOT (I c t ) = (I c t )U CNOT, U CNOT ( c I t ) = ( c I t )U CNOT, U CNOT (I c t ) = ( c t )U CNOT (1.8) C gate: U C = 00 c I t + 11 c t U C ( c I t )U C = c t, U C (I c t )U C = c t, U C ( c I t )U C = c I t, U C (I c t )U C = I c t

19 qubit, Pauli operator, Clifford gates

20 Stabilizer group Stabilizer group (1.9) S = {S i } eg) {I,I,II,} : = Pauli group no overlap [S i,s j ]=0 (1.10) Stabilizer eg) {II,,, YY} even overlap = generators S G = { S i } : stabilizer group stabilizer generator eg) eg) {I,I} {,} no overlap even overlap { S i } Stabilizer group { S i } eg) {,} = {II,,, YY}

21 Stabilizer state (1.12) Stabilizer state Ψ: S i Ψ = Ψ for all S i S. stabilizer operator +1 (stabilizer operators )

22 Stabilizer state (1.12) Stabilizer state Ψ: S i Ψ = Ψ for all S i S. stabilizer operator +1 (stabilizer operators ) n qubit 2 n stabilizer generator 2 S G stabilizer generator S G qubit n

23 Stabilizer state (1.12) Stabilizer state Ψ: S i Ψ = Ψ for all S i S. stabilizer operator +1 (stabilizer operators ) n qubit 2 n stabilizer generator 2 S G stabilizer generator S G qubit n ( )/ 2-1 ( 00 11)/ 2-1 ( )/ 2 ( 10 01)/ 2

24 Stabilizer state (1.12) Stabilizer state Ψ: S i Ψ = Ψ for all S i S. stabilizer operator +1 (stabilizer operators ) n qubit 2 n stabilizer generator 2 S G stabilizer generator S G qubit n +1-1 eg) S 2 =, Bell state ( )/ 2 +1 ( )/ 2 ( 00 11)/ 2-1 ( )/ 2 ( 10 01)/ 2

25 Stabilizer state (1.12) Stabilizer state Ψ: S i Ψ = Ψ for all S i S. stabilizer operator +1 (stabilizer operators ) n qubit 2 n stabilizer generator 2 S G stabilizer generator S G qubit n +1-1 eg) S 2 =, Bell state ( )/ 2 +1 ( )/ 2 ( 00 11)/ 2 eg) S 3 = I,I, -1 ( )/ 2 ( 10 01)/ 2 GH state ( )/ 2

26 Clifford gates & GK theorem Clifford gates Pauli product Puali product stabilizer state stabilizer state S i ψ = ψ US i U U ψ = U ψ Gottesman-Knill Theorem S i (U ψ) =(U ψ) stabilizer group Pauli Clifford Pauli n-qubit stabilizer generator (2n n) -bit eg) (1, 1 0, 0) (0, 1 1, 0) i i 1 j (n+j) 1. universal quantum computation

27 Magic state distillation non-clifford gate OK by Solovay-Kitaev theorem magic state non-clifford gate ψ cos(π/8) 0 + i sin(π/8) 1 p e iπ/8 ψ magic state Clifford gate magic state distillation by Bravyi-Kitaev PRA (2005) noisy ancilla + Clifford gate = universal

28 (1.15) Stabilizer subspace: Stabilizer subspace stabilizer generator 2 n S G eg) S bit I,I S G qubit n stabilizer state eg) stabilizer subspace: { 00, 11} stabilizer subspace: { 000, 111}

29 (1.15) (1.16) Stabilizer Logical subspace: eg) Stabilizer subspace stabilizer generator 2 n S G S G qubit n stabilizer state eg) stabilizer subspace: { 00, 11} operators: S bit I,I stabilizer subspace: { 000, 111} L i L L i = L i L i [L A i,s i,l j ]=0 n S G i stabilizer i =1,,n S G eg) L 1 eg) =, L 1 = I S bit = I,I L 1 00 ± 11 L 1 00, 11 L 1 =, L 1 = II

30 (1.17) Logical basis: Stabilizer code S j, ( 1) i 1 L 1,, ( 1) i k L k stabilize logical computational basis i 1,,i k L ( k = n S G ) L i,l i logical basis logical Pauli operator L k i 1,,i k L =( 1) i k i 1,,i k L L j i 1,,i k L = i 1,,i j 1,,i k L ( L j (L j i 1,,i k L )= L j ( 1) i j i 1,,i k L =( 1) ij 1 (L j i 1,,i k L ) ) eg) I,I, L 1 = II 0 L = 000, 1 L = 111 (1.18) Stabilizer code: L 1 1 L = 1 L L 1 0 L = 1 L stabilizer group logical operator

31 qubit, Pauli operator, Clifford gates stabilizer group, state, subspace, logical operator, magic state

32 (1.19) Pauli error: Quantum error correction input bit-flip error: (1 p)ρ + pρ phase-flip error: (1 p)ρ + pρ S i stabilizer code Pauli operator {[A, S i ]=0 [A, S j ] = 0 stabilizer output depolarizing error: (1 p)ρ + p/3(ρ + Y ρy + ρ) Pauli error : (Pauli product) A stabilizer logical state A logical operator logical state stabilizer stabilizer subspace

33 Quantum error correction (1.20) ex α β 111 α β 111 α β 111 Syndrome subspace and error syndrome: s i S i I II II α β 111 α 000 β 111 α β 110 s i = ±1 S i H(s 1,,s n k ) (s 1,,s n k ) error syndrome (1.21) eg) S bit = I,I I I H(+1, +1) H(+1, 1) 000, , H( 1, +1) H( 1, 1) 001, , 101

34 Quantum error correction (1.22) Quantum error correction: (1) syndrome subspace (= error syndrome ) Stabilizer generator =syndrome measurement (2) error syndrome eg) bit-flip error 3 qubit error syndrome (+1,-1) II with probability ~p or I with probability ~p 2 most likely I +1-1 I , , 011 II 001, 110 H(+1, +1) H(+1, 1) H( 1, +1) II II H( 1, 1) 010, 101 p effective

35 (1.23) Indirect projective measurement: A ±1. + ψ Syndrome measurement A 00 I + 11 A projective measurement of A eg) S bit = I,I + + ( 0 m ψ + 1 m A ψ)/ 2 ++ I + A + ψ 2 A +1

36 Steane 7-qubit code 1 qubit error error. Stabilizer generators: S 1 = III S 2 = III S 3 = III S 4 = III S 5 = III S 6 = III Syndrome subspace 2 6 =64 error (1+7)(1+7)=64 t i=k 2 n 2 n 1 t i=0 (n,t)=(7,1),(23,3)

37 qubit, Pauli operator, Clifford gates stabilizer group, state, subspace, logical operator, code syndrome measurement, indirect measurement

38 Topological quantum computation Abelian non-abelian Topological quantum computation with Abelian anyon on the surface which consists of qubits. surface (torus) code(memory): Kitaev, Annals Phys. 303, 2 (2003) Topological quantum computation with non-abelian anyon. quantum memory: ν=5/2 fractional quantum Hall state (thought to be non-abelian anyon) topological quantum computation: Raussendorf-Harrington-Goyal, Annals Phys. 321, 2242 (2006) Raussendorf-Harrington-Goyal, NJP 9, 199 (2007) Raussendorf-Harrington, PRL 98, (2007) what we want to study universal quantum computation: Fibonacci anyon Lecture Notes for Physics 219: Quantum Computation by Preskill Sarma-Freedman-Nayak Physics Today 32 July 2006 Sarma-Freedman-Nayak Rev. Mod. Phys. 80, 1083 (2008).

39 (2.1) Surface (torus) code Stabilizer of the surface code: by A. Kitaev ʼ97 (ariv: ) The face and vertex operators are defined for all faces and vertexes. A F = i F i B V = j V j note) AF and BV are commutable, since each face and vertex share 0 or 2 qubits. Thus {AF, BV} is a stabilizer group. qubit A F B V (2.2) Surface code state Ψ : A F Ψ = Ψ and B V Ψ = Ψ for all F and V. 25

40 (2.3) Surface (torus) code by A. Kitaev ʼ97 (ariv: ) the number of the logical qubits encoded on the torus: Recall (1.15). # of qubits (edges) in N N torus: 2N 2 # of stab. generators (faces & vertexes): 2N 2-2 note) -2 comes from the fact that F and B V = I 2N 2, and one face and one vertex operator V are not independent. A F = I 2N 2 qubit A F B V (2.4) # of logical qubits: 2 In general... (face)+(vertex)-(edge)=2-2g Euler characteristic where g is the genus of the surface. # of logical qubits (edge)-[(face)+(vertex)-2]=2g 26

41 Trivial loop operator (2.5) Trivial loop operators act on the code space trivially: = there is no hole or defect inside the loop Since the loop operator is a product of, it is also a stabilizer operator. Thus its multiplication does not change the code state. (2.6) Non-trivial loop operators commute with all stabilizer operators, but they are not products of stabilizer operators. Thus non-trivial loop operators are logical operators (recall (1.16)), which represent logical qubits. x z x z z z x x x L (1) L (1) L (2) L (2)

42 How to correct error (2.7) Error syndromes: Incorrect error syndromes are found at boundaries of an error chain, since Pauli operators on the boundaries anti-commute with stabilizers (recall (1.19)). error (2.8) Error correction: Infer the most-likely locations of errors conditioned on the error syndrome (recall (1.22)) minimum-weight-perfect matching algorithm errors

43 How to correct error (2.9) Successful error correction: Estimated error is exactly as same as the actual error estimated error chain -1 actual error chain (2.10) note) Since + = trivial loop operator, the recovery operation successfully corrects the error. logical error: If errors are too dense, recovery operation results in a nontrivial loop operator, which changes the code state. The critical error probability, below which topological error correction succeeds, is ~0.11, which is very close to the quantum Gilbert-Varshamov bound in the limit of zero asymptotic rate. actual error estimated error

44 Threshold value p=3% p=10% p=15% Random-bond Ising model Nishimori line Surface code

45 Kitaev model Hamiltonian: H = J F A F + V B V qubit A F translationally invariant B V ground state subspace = stabilizer subspace topological order = ground state degeneracy cannot be distinguished by local operator (cannot be explained by Landauʼs sponteneous symmetry breaking) anyonic excitation String-net condensate by.-g. Wen Ψ vac = I + BV 2 V 00 0

46 qubit, Pauli operator, Clifford gates stabilizer group, state, subspace, logical operator, code syndrome measurement, indirect measurement surface code, trivial & non-trivial loop operator

47 How to increase logical qubits (2.11) Changing the topology of the surface: recall (2.4): # of logical qubits is 2g But, this approach is somewhat complicated. Is there any way to increase logical degree of freedom systematically?

48 How to increase logical qubits (2.12) Injecting the defects on the surface: Consider a plane surface instead of torus. A stabilizer A is removed from the stabilizer gp. We call this defect. injection of a logical qubit A L (1) L (1) # of qubits: 2N(N-1) # of stabilizers: (N-1) 2 +N 2-1 stabilizers on the plane surface defines a state (not subspace). vacuum Ψ vac The operator A is commutable with all stabilizers, thus it is a logical operator, say L (1). We can also find L (1).

49 Primal defect pair (2.13) Pairing the defects as logical qubit: Boundary might be far away from the defects. If we inject a lot of defects, the logical operators might be complicated... We introduce a logical by creating a defect pair. A1 and A2 are removed from stabilizer group, but A1A2 is still an element of stabilizer group. L (1) L (1) The and are both commutable with all stabilizers. L (1) L (1) A 1 A 2 L (1) L (1) The and anti-commute, since they share one qubit. The defect pair represents a logical qubit.

50 Defect pair creation (state preparation) (2.14) How to create the defect pair: Measure a qubit in the basis. Aa Ab Aa and Ab are removed from the stabilizer group, since they are anti-commute with. But, AaAb is still a stabilizer.

51 Defect pair creation (2.15) How to move the defect: Measure the neighboring qubit in the basis. (state preparation) Measure the Ab, and it is revived as a stabilizer. Aa Ab A c separation Aa A c repeating this procedure Ac is removed from the stabilizer group, but AaAbAc is still a stabilizer. Since the state is the eigenstate of, which is the logical operator, + L =( 0 L + 1 L )/ 2 is now injected. time logical operator + p L primal defect pair creation primal defect pair

52 Defect pair creation (state preparation) (2.16) How to prepare logical eigenstate: The defect pair created in (2.14) & (2.15) is the logical eigenstate. How is the eigenstate prepared? L (1) L (1) The logical operator is the removed face operator. Thus the eigenstate of, that is the stabilizer state before the defect injection. equivalent to or Ψ vac = 0 p L 0 p L primal defect pair logical operator

53 Dual defect pair creation (2.17) We can also consider the dual defect pair: L (state preparation) Defect pair creation (with basis measurement), separation can be done similarly to the primal defect (2.14,15). L Similarly to (2.16), Ψ vac = + d L. dual defect pair creation 0 d L time dual defect pair + d L primal defect pair

54 Defect pair annihilation (2.18) (Logical qubit measurement) Primal logical measurement: Since logical operator is the tensor product of on the chain which connects the primal defect pair, physical Pauli measurements tell us logical measurement outcome. p L L (1) primal defect pair annihilation (2.19) Primal logical measurement: Stabilizer measurement of the removed face operator gives logical measurement outcome. p L L (1)

55 (2.20) Diagram Diagram for primal & dual defect creation: dual defect pair creation time 0 p L logical operator primal defect pair time 0 d L dual defect pair (2.21) + p L primal defect pair creation primal defect pair + d L Diagram for primal & dual logical measurements: p L d L primal defect pair primal defect pair annihilation p L d L primal defect pair annihilation

56 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding. input state

57 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding.

58 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding.

59 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding.

60 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding. multiplication of loop operator does not change the code state

61 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding.

62 CNOT gate by braiding (2.22) Braiding the primal defect around the dual defect: Let us first consider the time evolution of logical operators under the braiding. contraction does not change the logical operator (2.23) L p Id is transformed to L p Ld by the braiding.

63 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding. input state

64 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding.

65 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding.

66 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding.

67 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding. multiplication of loop operator does not change the code state

68 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding.

69 CNOT gate by braiding (2.24) Braiding the primal defect around the dual defect: Next we consider the time evolution of logical operators under the braiding. output state contraction does not change the logical operator (2.25) I p L d is transformed to L p Ld by the braiding.

70 CNOT gate by braiding L p Id L p Ld I p L d L p Ld (2.26) CNOT gate by braiding: The transformation rule is equivalent to that of CNOT gate (1.7),which indicates that the braiding operation acts as the CNOT gate for primal (control) and dual (target) qubits.

71 CNOT gate by braiding (2.27) Diagram for the CNOT gate: L p Id L p Ld circuit diagram topological diagram I p L d L p Ld

72 Why Abelian anyon supports non-abelian operations? (2.28) Braiding & anyon: = braiding = 2 swap Thus braiding operations of boson or fermion result in trivial operations. Since braiding of the defect change the code state, they are anyons. The defect is Abelian anyon, since the control qubit of the CNOT is always the primal qubit.

73 Why Abelian anyon supports non-abelian operations? (2.29) How to support non-abelian operations by using Abelian anyons: control in 0 d L control out d = + p L target in target out p = This can be understood that Abelian anyons support non-abelian operations by changing the topology of surface!

74 Universal topological quantum computation (2.30) Non-Clifford gates: So far, we have Pauli basis preparation, CNOT gate, Pauli basis measurements, which allow us an arbitrary Clifford operations. (2.31) However, quantum computation with Clifford operations can be efficiently simulated by classical computer. Thus non-clifford gates are necessary. State injection: e iθ e iθ e iθ Ψ vac = (cos θi + i sin θl p ) 0p L = cos θ 0 p L + i sin θ 1p L e iθ Ψ vac = cos θ + d L + i sin θ d L = e iθ ( 0 d L + e i2θ 1 d L) (2.32) One-bit teleportation for non-clifford gate ψ p L p ψ p L cos θ 0 p L + i sin θ 1p L e iθlp ψ p L e iθlp ψ p L

75 Fault-tolerant QC in 2D Recall that + ψ A 00 I + 11 A syndrome qubit A F B V + syndrome qubit measurement-based quantum computation +

76 Fault-tolerant QC in 3D = H H syndrome qubit cluster state time + H syndrome qubit H H H H + H H H 3D cluster state

77 Fault-tolerant quantum computation with high noise threshold ~5%: KF & K. Yamamoto, Phys. Rev. A 82, (R) (2010) [ KF & K. Yamamoto, Phys. Rev. A 81, (2010)] Fault-tolerant quantum computation with probabilistic two-qubit gate: KF & Y. Tokunaga, Phys. Rev. Lett (2010) Topological quantum computation on the thermal state of spin-3/2 system: KF & T. Morimae, ariv: , to be appeared in PRA Rapid Com. [ KF & T. Morimae, ariv: ] Topological blind quantum computation: T. Morimae & KF, ariv:

78 qubit, Pauli operator, Clifford gates stabilizer group, state, subspace, logical operator, code syndrome measurement, indirect measurement surface code, trivial & non-trivial loop operator braiding, diagram, 2D, 3D

79 qubit, Pauli operator, Clifford gates ψ p L stabilizer group, state, subspace, logical operator, code syndrome measurement, indirect measurement e iθlp ψ p L surface code, trivial & non-trivial loop operator braiding, diagram, 2D, 3D

80 Gottesman-Knill s theorem Theorem If the input state is product states of eigenstates of Pauli operators, all unitary operations are Clifford gates, and the output state is measured in Pauli basis, then such quantum computation can be efficiently simulated by classical computer. Input state is a stabilizer state. Each stabilizer generator of the n-qubit system can be represented by 2n-bit. (1.14) eg) (1, 1 0, 0) (0, 1 1, 0) i th is i th 1, j th is (n+j)th 1. Since Clifford gates map a stabilizer state to another stabilizer state, they can be expressed as a linear map of 2 2n. Thus stabilizer generators of the output state can be easily calculated by using classical computer. Furthermore, the probability distribution of the output state under the Pauli basis measurement is easily calculated by using classical computer. n I +( 1) ν i A i p(ν 1,, ν i ) = Ψ Ψ 2 i=1 1 n = Ψ 2 n [( 1) ν i A i ] ζ i Ψ where = Ψ := (ζ 1,,ζ n ) 1 2 n (ζ 1,, ζ n ) i=1 n [( 1) ν i A i ] ζ i Ψ i=1 n i=1 ±A ζ i i, ±iaζ i i S A i = I,,Y, ( ν i =0, 1 measurement outcome) ζ i =0, 1 ( )