Topological Quantum Compiling

Transcription

1 Topologicl Quntum Compiling Work in collbortion with: Lyl Hormozi Georgios Zikos Steven H. Simon Michel Freedmn Nd Petrovic Florid Stte University Lucent Technologies Microsoft Project Q UCSB NEB, L. Hormozi, G. Zikos, S.H. Simon, Phys. Rev. Lett (25) S.H. Simon, NEB, M.Freedmn, N, Petrovic, L. Hormozi, Phys. Rev. Lett. 96, 753 (26). L. Hormozi, G. Zikos, NEB, nd S.H. Simon, In preprtion Support: US DOE

2 Fiboncci Anyons my ctully exist! ν = 2/5 Possibly Red-Rezyi k = 3 Prfermion stte. Red nd Rezyi, 99 Chrge e/5 qusiprticles with briding properties described by SU(2) 3 Chern- Simons Theory. Slingerlnd nd Bis, Non-Abelin content is tht of Fiboncci nyons. J.S. Xi et l., PRL (24).

3 Mybe, one dy, Fiboncci Anyons will be everywhere! Bosonic Red-Rezyi sttes (including k=3 t ν = 3/2) my be relizble in rotting Bose condenstes. Rezyi, Red, Cooper 5 Doubled Fiboncci string-nets my be found / relized. Levin nd Wen 4 Fendley nd Frdkin 5 Freedmn, Nyk, Shtengel, Wlker nd Wng 3 Think Golden!

4 τ τ = + τ

5 Topologicl Quntum Computtion Ψ f time Ψ f = M M MM Ψ i Ψ i When qusiprticles re present there is n exponentilly lrge Hilbert spce whose sttes cnnot be distinguished by locl mesurements. Qusiprticle world-lines forming brids crry out unitry trnsformtions on this Hilbert spce.

6 Topologicl Quntum Computtion Ψ f time Ψ f = M M MM Ψ i Ψ i Unitry trnsformtion depends only on the topology of the brid swept out by nyon world lines! Robust quntum computtion? (Kitev 97; Freedmn, Lrsen nd Wng )

7 Quntum Circuit U U Wht brid corresponds to this circuit?

8 Fiboncci Anyon Bsics A Fiboncci Anyon Fiboncci The lws of Fiboncci nyons:. Fiboncci nyons hve quntum ttribute I ll cll q-spin: q-spin = 2. A collection of Fiboncci nyons cn hve totl q-spin of either or :, Nottion: Ovls re lbeled by totl q-spin of enclosed prticles.

9 Fiboncci Anyon Bsics 3. The fusion rule for combining q-spin is: x = + This mens tht two Fiboncci nyons cn hve totl q-spin or, or be in ny quntum superposition of these two sttes. α + β Two dimensionl Hilbert spce Three Fiboncci nyons Three dimensionl Hilbert spce α + β + γ For N Fiboncci nyons Hilbert spce dimension is Fib(N-)

10 The F Mtrix Chnging fusion bses: c F b = c b c τ τ τ τ = c 5 F τ = b 2

11 The R Mtrix Exchnging Prticles: = /5 e i4π = i3π /5 e R = e i4π /5 e i3π /5 F nd R must stisfy certin consistency conditions (the pentgon nd hexgon equtions). For Fiboncci nyons these equtions uniquely determine F nd R.

12 Encoding Qubit (Freedmn, Lrsen, nd Wng, 2) Qubit Sttes Non-Computtionl Stte = = Stte of qubit is determined by q-spin of two leftmost prticles Trnsitions to this stte re lekge errors

13 Initilizing Qubit Pull two qusiprticle-qusihole pirs out of the vcuum.

14 Initilizing Qubit Pull two qusiprticle-qusihole pirs out of the vcuum. These three prticles hve totl q-spin

15 Initilizing Qubit Pull two qusiprticle-qusihole pirs out of the vcuum.

16 Mesuring Qubit Try to fuse the leftmost qusiprticle-qusihole pir.? α + β

17 Mesuring Qubit If they fuse bck into the vcuum the result of the mesurement is.

18 Mesuring Qubit If they cnnot fuse bck into the vcuum the result of the mesurement is.

19 c Briding Mtrices for 3 Fiboncci Anyons time c = c = c τ = 5 2 Ψi Ψf σ σ 2 σ σ 2 = M Ψf = M - Ψ i

20 Single Qubit Opertions Generl rule: Briding inside n ovl does not chnge the totl q-spin of the enclosed prticles. Importnt consequence: As long s we brid within qubit, there is no lekge error. Cn we do rbitrry single qubit rottions this wy?

21 Single Qubit Opertions re Rottions 2 π α 2 π 2 π The set of ll single qubit rottions lives in solid sphere of rdius 2π. ψ U α U α ψ 2 π U α = exp iα σ 2

22 2 π σ 2 σ π 2 π σ π σ -2

23 N =

24 N = 2

25 N = 3

26 N = 4

27 N = 5

28 N = 6

29 N = 7

30 N = 8

31 N = 9

32 N =

33 N =

34 Brute Force Serch = i i + error O( ε 3 ) For brute force serch: Brid Length lnε -2 ln ε Brid Length

35 Brute Force Serch = i i + O( 3 ) Brute force serching rpidly becomes infesible s brids get longer. Fortuntely, clever lgorithm due to Solovy nd Kitev llows for systemtic improvement of the brid given sufficiently dense covering of SU(2).

36 (Actul clcultion) Solovy-Kitev Construction i i + O( ε 4 ) Brid Length c lnε c 4,

37 Wht About Two Qubit Gtes? Qubit 2?? Qubit SU(5)+SU(8) Problems:. We re pulling qusiprticles out of qubits: Lekge error! dimensionl serch spce (s opposed to 3 for threeprticle brids). Strightforwrd brute force serch is problemtic.

38 Two Qubit Controlled Gtes b Control qubit Trget qubit Gol: Find brid in which some rottion is performed on the trget qubit only if the control qubit is in the stte. (b=)

39 Constructing Two Qubit Gtes by Weving Weve pir of nyons from the control qubit between nyons in the trget qubit. control pir b Importnt Rule: Briding q-spin object does not induce trnsitions. Trget qubit is only ffected if control qubit is in stte (b = )

40 Constructing Two Qubit Gtes by Weving Only nontrivil cse is when the control pir hs q-spin. control pir We ve reduced the problem to weving one nyon round three others. Still too hrd for brute force pproch!

41 OK, Try Weving Through Only Two Prticles We re bck to SU(2), so this is numericlly fesible. control pir Question: Cn we find weve which does not led to lekge errors?

42 A Trick: Effective Briding Actul Weving Effective Briding The effect of weving the blue nyon through the two green nyons hs pproximtely the sme effect s briding the two green nyons twice.

43 Controlled Knot Gte Effective briding is ll within the trget qubit No lekge! Not CNOT, but sufficient for universl quntum computtion.

44 Solovy-Kitev Improved Controlled Knot Gte

45 Another Trick: Injection Weving Step : Inject the control pir into the trget qubit. control pir

46 control pir Step 2: Weve the control pir inside the injected trget qubit. control pir i i

47 Step 3: Extrct the control pir from the trget using the inverse of the injection weve. control pir Putting it ll together we hve CNOT gte: Injection Rottion Extrction

48 Solovy-Kitev Improved CNOT

49 Universl Set of Fult Tolernt Gtes Single qubit rottions: ψ U φ U φ ψ Controlled NOT:

50 Quntum Circuit U U

51 Quntum Circuit U U

52 Brid

53 We know it is possible to crry out universl quntum computtion by moving only single qusiprticle. Cn we find n efficient CNOT construction in which only single prticle is woven through the other prticles?

54 Another Useful Brid: The F-brid F-Mtrix: τ τ τ τ = F-Brid: c c i τ τ τ τ

55 Single Prticle Weve Gte: Prt b

56 Single Prticle Weve Gte: Prt b F-Brid

57 Single Prticle Weve Gte: Prt c b c F-Brid

58 Single Prticle Weve Gte: Prt b b b Intermedite Stte b

59 Single Prticle Weve Gte: Prt 2 b b Phse b = b = b Phse Brid

60 Single Prticle Weve Gte: Prt 2 b b Phse b b

61 Single Prticle Weve Gte: Prt 3 b b Phse b b

62 Controlled-Phse Gte F-Brid Phse-Brid Inverse of F-Brid b b Intermedite stte b U = Finl result + O( -3 )

63 Solovy-Kitev-Improved Controlled-Phse Gte