Dynamical Decoupling and Quantum Error Correction Codes

Transcription

1 Dynamical Decoupling and Quantum Error Correction Codes Gerardo A. Paz-Silva and Daniel Lidar Center for Quantum Information Science & Technology University of Southern California GAPS and DAL paper in preparation 1

2 Dynamical Decoupling and Quantum Error Correction Codes (SXDD) Gerardo A. Paz-Silva and Daniel Lidar Center for Quantum Information Science & Technology University of Southern California GAPS and DAL paper in preparation 2

3 Motivation qmac H S 3

4 Motivation H B qmac H S 4

5 Motivation H B qmac H S 5

6 Motivation H B QEC + FT qmac H S 6

7 Motivation H B Dynamical Decoupling QEC + FT qmac H S 7

8 Motivation H B Dynamical Decoupling H SB QEC + FT qmac H S H SB H SB H SB 8

9 H = I H B + U τ min = e i(h τ min) η 0 = τ min [[n,k,d]] QEC code

10 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 [[n,k,d]] QEC code

11 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 [[n,k,d]] QEC code

12 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 DD DD DD DD DD Ng,Lidar,Preskill PRA 84, (2011) Enhanced fidelity of physical gates via appended DD sequences [[n,k,d]] QEC code

13 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 DD DD DD DD DD Ng,Lidar,Preskill PRA 84, (2011) Enhanced fidelity of physical gates via appended DD sequences Order of decoupling N cannot be arbitrarily large. [[n,k,d]] QEC code

14 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 DD DD DD DD DD Ng,Lidar,Preskill PRA 84, (2011) Enhanced fidelity of physical gates via appended DD sequences Order of decoupling N cannot be arbitrarily large. Unless has restricted locality Local-bath assumption [[n,k,d]] QEC code

15 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 DD DD DD DD DD [[n,k,d]] QEC code Ng,Lidar,Preskill PRA 84, (2011) Enhanced fidelity of physical gates via appended DD sequences Order of decoupling N cannot be arbitrarily large. Unless has restricted locality Local-bath assumption < FT

16 H = I H B + U τ min = e i(h τ min) η 0 = τ min U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 η DD < η 0 DD DD DD DD DD [[n,k,d]] QEC code n Ng,Lidar,Preskillhas qubit Pauli basis as restricted decoupling locality group Local-bath No local assumption bath assumption Length Enhanced of sequence fidelity of exponential physical gates in 2n via appended DD sequences Pulses Order look of decoupling like errors N to cannot the code be arbitrarily limits possible large. integration with other schemes Unless SB B SB has restricted locality Local-bath < FT assumption

17 H = I H B + U τ min = e i(h τ min) η 0 = τ min Desiderata for DD +QEC: U DD T = e i(h,eff T+,eff O T N+1 ) η DD (N) =,eff O T N+1 I. No extra locality assumptions II. Pulses in the code η DD < η 0 III. Shorter sequences than full decoupling approach. DD DD DD DD DD η DD < η 0 [[n,k,d]] QEC code n Ng,Lidar,Preskillhas qubit Pauli basis as restricted decoupling locality group Local-bath No local assumption bath assumption Length Enhanced of sequence fidelity of exponential physical gates in 2n via appended DD sequences Pulses Order look of decoupling like errors N to cannot the code be arbitrarily limits possible large. integration with other schemes Unless SB B SB has restricted locality Local-bath < FT assumption

18 The magic is in the decoupling group 18

19 The magic is in the decoupling group Too small No arbitrary order decoupling No general Hamiltonians Too large Overkill Shorter sequences are better 19

20 The magic is in the decoupling group Too small No arbitrary order decoupling No general Hamiltonians Too large Overkill Shorter sequences are better Mutually Orthogonal Operator (generator) Set = {Ω i } i=1,,k (Ω i ) 2 = I Ω i Ω j = 1 f(i,j) Ω j Ω i ; f(i, j) = {0,1} Ω i Ω j Ω k 20

21 The magic is in the decoupling group Too small No arbitrary order decoupling No general Hamiltonians Too large Overkill Shorter sequences are better Mutually Orthogonal Operator (generator) Set = {Ω i } i=1,,k (Ω i ) 2 = I Ω i Ω j = 1 f(i,j) Ω j Ω i ; f(i, j) = {0,1} Ω i Ω j Ω k Concatenated Dynamical Decoupling (CDD) [Khodjasteh and Lidar, Phys. Rev. Lett. 95, (2005)] Nested Uhrig Dynamical Decoupling (NUDD) [Wang and Liu, Phys. Rev. A 83, (2011)] Pulses <MOOS> (2 K ) N pulses Pulses MOOS (N + 1) K pulses 21

22 What we propose Stabilizer generators = {S i } i=1,,q MOOS = {S i } i=1,,q 22

23 What we propose Stabilizer generators = {S i } i=1,,q Logical operators (Pauli basis) = { X i (L), Z i (L) } i=1,,k MOOS = {S i } i=1,,q MOOS = {S i } i=1,,q { X i (L), Z i (L) } i=1,,k 23

24 What we propose Stabilizer generators = {S i } i=1,,q Logical operators (Pauli basis) = { X i (L), Z i (L) } i=1,,k MOOS = {S i } i=1,,q MOOS = {S i } i=1,,q { X i (L), Z i (L) } i=1,,k U DD T = e i(h,effo T +,eff O(T N+1 )) H,eff {S i } i=1,,q Contains no physical or logical errors! Only harmless terms! Even if is a logical error! 24

25 What do we gain? No extra locality assumptions: The DD group is powerful enough. CDD: NO higher order Magnus term is UNDECOUPLABLE and HARMFUL The next level of concatenation can deal with it NUDD: (See proof in W.-J. Kuo, GAPS, G. Quiroz, D. Lidar in preparation)

26 What do we gain? No extra locality assumptions: The DD group is powerful enough. CDD: NO higher order Magnus term is UNDECOUPLABLE and HARMFUL The next level of concatenation can deal with it NUDD: (See proof in W.-J. Kuo, GAPS, G. Quiroz, D. Lidar in preparation) DD Pulses are bitwise / transversal in the code Pulses do not look like errors to the code Allows interaction with other protection schemes. 26

27 What else do we gain? Shorter sequences than full decoupling approach: For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators CDD (<Ωi >,N) 2 n+k g N < 2 2nN NUDD ({Ωi,N}) (N + 1) n+k g < (N + 1) 2n 27

28 What else do we gain? Shorter sequences than full decoupling approach: For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators CDD (<Ωi >,N) 2 n+k g N < 2 2nN NUDD ({Ωi,N}) (N + 1) n+k g < (N + 1) 2n e.g. [[ n 2, 1, n]] Bacon Shor code: Stabilizer generators: 2(n-1) Logical generators: 2 SXDD Full decoupling D(MOOS) 2 n 2 n 2 NUDD (N + 1) 2 n (N + 1) 2 n2 CDD 2 2 n N 2 2 n2 N

29 η DD < η 0? Recall our (effective) noise rates: η 0 = τ min η DD (N) =,eff O T N+1 Are an overestimation: bounds obtained without using the QEC code structure. (work in progress) 29

30 η DD < η 0? Recall our (effective) noise rates: η 0 = τ min η DD (N) =,eff O T N+1 Are an overestimation: bounds obtained without using the QEC code structure. (work in progress) How to compute,eff O T N+1?? NLP results (Eqs ) Recursive relations for,eff (q) and H,eff (q) at every degree of concatenation q. where R = 2 D(MOOS) and c ~1,eff (q) R q(q+3)/2 c + H B τ 0 q 1 T(q) = R q τ min 30

31 η DD < η 0 N=1 N=2 N=3 I H B = J 0 = J SB [[9,1,3]] BS code: τ min = 1 ; D MOOS = 4 + 2

32 η DD < η 0 N=1 N=2 N=3 I H B = J = J SB [[9,1,3]] BS code: τ min = 1 ; D MOOS = 4 + 2

33 η DD < η 0 N=1 N=2 N=3 I H B = J = J SB [[9,1,3]] BS code: τ min = 1 ; D MOOS = 4 + 2

34 Beyond H S = 0 DD-based methods for fidelity enhanced gates can be directly ported: Dynamically protected gates: works for both CDD and NUDD Append SXDD sequence to a gate. [NLP, PRA 84, (2011)] (Concatenated) Dynamically corrected gates: based on CDD Eulerian cycle on the Caley graph of DD group [Khodjasteh and Viola, PRL 102, (2009)] [Khodjasteh, Lidar, Viola, PRL 104, (2010)]

35 Conclusions We have shown how to integrate dynamical decoupling and quantum error correction codes in a natural way. No extra locality conditions Pulses in the code. Shorter sequences than full decoupling approach. Improve effective error rates AND deal where Hamiltonians QEC fails. The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance friendly 35

36 Conclusions We have shown how to integrate dynamical decoupling and quantum error correction codes in a natural way. No extra locality conditions Pulses in the code. Shorter sequences than full decoupling approach. Improve effective error rates AND deal where Hamiltonians QEC fails. The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance friendly What we would like to do now: Detailed calculation of the effective error rate considering correctable errors, etc. in a DD + QEC scenario (at least for one encoded qubit) 36

37 Conclusions We have shown how to integrate dynamical decoupling and quantum error correction codes in a natural way. No extra locality conditions Pulses in the code. Shorter sequences than full decoupling approach. Improve effective error rates AND deal where Hamiltonians QEC fails. The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance friendly What we would like to do now: Detailed calculation of the effective error rate considering correctable errors, etc. in a DD + QEC scenario (at least for one encoded qubit) THANKS! QUESTIONS? 37