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

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

3

Motivation

qMac 𝑯𝑺

4

Motivation

𝑯𝑩

qMac 𝑯𝑺

5

Motivation

𝑯𝑩

qMac 𝑯𝑺 𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

6

Motivation

𝑯𝑩

qMac 𝑯𝑺 QEC + FT 𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

7

Motivation

𝑯𝑩

qMac 𝑯𝑺

Dynamical Decoupling

QEC + FT 𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

8

Motivation

𝑯𝑩

qMac 𝑯𝑺

Dynamical Decoupling

QEC + FT 𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯𝑺𝑩

𝑯′𝑺𝑩

𝑯′𝑺𝑩

𝑯′𝑺𝑩

𝑯′𝑺𝑩

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

DD DD DD DD DD

Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

DD DD DD DD DD

Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

DD DD DD DD DD

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.

Unless 𝐻𝑆𝐵 has restricted locality ‘Local-bath assumption’

DD DD DD DD DD

< FT

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.

Unless 𝐻𝑆𝐵 has restricted locality ‘Local-bath assumption’

DD DD DD DD DD

Ng,Lidar,Preskillhas restricted locality ‘Local-bath assumption’ • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.

Unless 𝐻 𝑆𝐵 𝑆𝐵 𝐵 𝑆𝐵 has restricted locality ‘Local-bath assumption’

< FT

n –qubit Pauli basis as decoupling group No ‘local bath assumption’ • Length of sequence exponential in 2n • Pulses look like errors to the code limits possible integration with other schemes

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

DD DD DD DD DD

Ng,Lidar,Preskillhas restricted locality ‘Local-bath assumption’ • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.

Unless 𝐻 𝑆𝐵 𝑆𝐵 𝐵 𝑆𝐵 has restricted locality ‘Local-bath assumption’

< FT

n –qubit Pauli basis as decoupling group No ‘local bath assumption’ • Length of sequence exponential in 2n • Pulses look like errors to the code limits possible integration with other schemes

[[n,k,d]] QEC code

𝐻 = 𝐼⊗𝐻𝐵 + 𝐻𝑆𝐵 𝑈 τ𝑚𝑖𝑛 = 𝑒−𝑖(𝐻 τ𝑚𝑖𝑛)

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓 𝑇+𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ) 𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

𝜂𝐷𝐷 < 𝜂0

Desiderata for DD +QEC:

I. No extra locality assumptions

II. Pulses in the code

III. Shorter sequences than full decoupling approach.

𝜂𝐷𝐷 < 𝜂0

18

The magic is in the decoupling group

19

The magic is in the decoupling group

Too small No arbitrary order decoupling No general Hamiltonians Too large Overkill Shorter sequences are better

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 = Ω𝑖𝑖=1,…,𝐾

(Ω𝑖)2 = 𝐼

Ω𝑖 Ω𝑗 = −1𝑓(𝑖,𝑗)Ω𝑗 Ω𝑖; 𝑓(𝑖, 𝑗) = 0,1

Ω𝑖 Ω𝑗 ≠ Ω𝑘

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 = Ω𝑖𝑖=1,…,𝐾

(Ω𝑖)2 = 𝐼

Ω𝑖 Ω𝑗 = −1𝑓(𝑖,𝑗)Ω𝑗 Ω𝑖; 𝑓(𝑖, 𝑗) = 0,1

Ω𝑖 Ω𝑗 ≠ Ω𝑘

Concatenated Dynamical Decoupling (CDD) [Khodjasteh and Lidar, Phys. Rev. Lett. 95, 180501 (2005)]

Pulses <MOOS> (2𝐾)𝑁 pulses

Nested Uhrig Dynamical Decoupling (NUDD) [Wang and Liu, Phys. Rev. A 83, 022306 (2011)]

Pulses MOOS (𝑁 + 1)𝐾 pulses

22

What we propose…

• Stabilizer generators = S𝑖𝑖=1,…,𝑄

MOOS = S𝑖𝑖=1,…,𝑄

23

What we propose…

• Stabilizer generators = S𝑖𝑖=1,…,𝑄

MOOS = S𝑖𝑖=1,…,𝑄 MOOS = S𝑖𝑖=1,…,𝑄 ⋃ X𝑖(𝐿), Z𝑖

(𝐿)𝑖=1,…,𝑘

• Logical operators (Pauli basis) = X𝑖(𝐿), Z𝑖

(𝐿)𝑖=1,…,𝑘

24

What we propose…

• Stabilizer generators = S𝑖𝑖=1,…,𝑄

𝑈𝐷𝐷 𝑇 = 𝑒−𝑖(𝐻∅,𝑒𝑓𝑓𝑂 𝑇 +𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂(𝑇𝑁+1))

𝐻∅,𝑒𝑓𝑓 ∝ S𝑖𝑖=1,…,𝑄

Contains no physical or logical errors ! Only harmless terms ! Even if 𝐻𝑆𝐵 is a logical error!

MOOS = S𝑖𝑖=1,…,𝑄 MOOS = S𝑖𝑖=1,…,𝑄 ⋃ X𝑖(𝐿), Z𝑖

(𝐿)𝑖=1,…,𝑘

• Logical operators (Pauli basis) = X𝑖(𝐿), Z𝑖

(𝐿)𝑖=1,…,𝑘

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)

𝑯𝑺𝑩 1 - 0

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.

𝑯𝑺𝑩 1 - 0 2 - 0

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 < 22𝑛𝑁

𝑁𝑈𝐷𝐷(Ω𝑖,𝑁) (𝑁 + 1)𝑛+𝑘−𝑔 < (𝑁 + 1)2𝑛

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 < 22𝑛𝑁

𝑁𝑈𝐷𝐷(Ω𝑖,𝑁) (𝑁 + 1)𝑛+𝑘−𝑔 < (𝑁 + 1)2𝑛

e.g. [[ 𝑛2, 1, 𝑛]] Bacon Shor code: Stabilizer generators: 2(n-1) Logical generators: 2

SXDD Full decoupling

D(MOOS) 2 𝑛 2 𝑛2

NUDD (𝑁 + 1)2 𝑛 (𝑁 + 1)2 𝑛2

CDD 22 𝑛 𝑁 22 𝑛2𝑁

𝑯𝑺𝑩 3 - 0

29

𝜼𝑫𝑫 < 𝜼𝟎 ?

Recall our (effective) noise rates:

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

Are an overestimation: bounds obtained without using the QEC code structure. (work in progress)

30

𝜼𝑫𝑫 < 𝜼𝟎 ?

Recall our (effective) noise rates:

𝜂0 = 𝐻𝑆𝐵 τ𝑚𝑖𝑛

𝜂𝐷𝐷(𝑁) = 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1

Are an overestimation: bounds obtained without using the QEC code structure. (work in progress)

How to compute 𝐻𝑆𝐵,𝑒𝑓𝑓 𝑂 𝑇𝑁+1 ?? NLP results (Eqs. 152-164)

Recursive relations for 𝐻𝑆𝐵,𝑒𝑓𝑓(𝑞) and 𝐻∅,𝑒𝑓𝑓(𝑞) at every degree of concatenation q.

𝐻𝑆𝐵,𝑒𝑓𝑓(𝑞) ≤ 𝑅𝑞(𝑞+3)/2 𝑐 𝐻𝑆𝐵 + 𝐻𝐵 τ0𝑞−1 𝐻𝑆𝐵

𝑇(𝑞) = 𝑅𝑞τ𝑚𝑖𝑛

where 𝑅 = 2𝐷(𝑀𝑂𝑂𝑆) and 𝑐 ~1

𝜼𝑫𝑫 < 𝜼𝟎

[[9,1,3]] – BS code: 𝜏𝑚𝑖𝑛 = 1 ;𝐷 𝑀𝑂𝑂𝑆 = 4 + 2

N=1 N=2 N=3

𝐼⊗𝐻𝐵 = 𝐽0

𝐻𝑆𝐵 = 𝐽𝑆𝐵

𝜼𝑫𝑫 < 𝜼𝟎

[[9,1,3]] – BS code: 𝜏𝑚𝑖𝑛 = 1 ;𝐷 𝑀𝑂𝑂𝑆 = 4 + 2

N=1 N=2 N=3

𝐼⊗𝐻𝐵 = 𝐽0

𝐻𝑆𝐵 = 𝐽𝑆𝐵 𝑯𝑺𝑩 4 - 0

𝜼𝑫𝑫 < 𝜼𝟎

[[9,1,3]] – BS code: 𝜏𝑚𝑖𝑛 = 1 ;𝐷 𝑀𝑂𝑂𝑆 = 4 + 2

N=1 N=2 N=3

𝐼⊗𝐻𝐵 = 𝐽0

𝐻𝑆𝐵 = 𝐽𝑆𝐵 𝑯𝑺𝑩 4 - 0 4 - 1

34

Beyond 𝐻𝑆 = 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, 012305(2011)]

(Concatenated) Dynamically corrected gates: based on CDD Eulerian cycle on the Caley graph of DD group

[Khodjasteh and Viola, PRL 102, 080501 (2009)]

[Khodjasteh, Lidar, Viola, PRL 104, 090501 (2010)]

𝑯𝑺𝑩 5 - 1

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

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)

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 ?