Correction Codes Gerardo A. Paz-Silva and Daniel Lidar Center for - PowerPoint PPT Presentation

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

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

Motivation qMac 𝑰 𝑻 3

Motivation 𝑰 𝑪 qMac 𝑰 𝑻 4

Motivation 𝑰 𝑪 𝑰 𝑻𝑪 𝑰 𝑻𝑪 qMac 𝑰 𝑻 𝑰 𝑻𝑪 𝑰 𝑻𝑪 5

Motivation 𝑰 𝑪 𝑰 𝑻𝑪 𝑰 𝑻𝑪 QEC + FT qMac 𝑰 𝑻 𝑰 𝑻𝑪 𝑰 𝑻𝑪 6

Motivation 𝑰 𝑪 Dynamical Decoupling 𝑰 𝑻𝑪 𝑰 𝑻𝑪 QEC + FT qMac 𝑰 𝑻 𝑰 𝑻𝑪 𝑰 𝑻𝑪 7

Motivation 𝑰 𝑪 Dynamical Decoupling 𝑰 𝑻𝑪 𝑰 𝑻𝑪 𝑰′ 𝑻𝑪 QEC + FT 𝑰′ 𝑻𝑪 qMac 𝑰 𝑻 𝑰 𝑻𝑪 𝑰′ 𝑻𝑪 𝑰′ 𝑻𝑪 𝑰 𝑻𝑪 8

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 [[n,k,d]] QEC code 

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = [[n,k,d]] QEC code 

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code 

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code DD DD DD DD DD Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences 

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code 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. 

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code 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.  Unless 𝐼 𝑇𝐶 has restricted locality  ‘ Local-bath assumption ’

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code 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.  Unless 𝐼 𝑇𝐶 has restricted locality  ‘ Local-bath assumption ’ < FT

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = 𝜃 𝐸𝐸 < 𝜃 0 [[n,k,d]] QEC code DD DD DD DD DD n – qubit Pauli basis as decoupling group  No ‘local bath assumption’ Ng,Lidar,Preskil lhas restricted locality ‘ Local-bath assumption ’ • • Length of sequence exponential in 2n Enhanced fidelity of physical gates via appended DD sequences • Pulses look like errors to the code  limits possible integration with other schemes • Order of decoupling N cannot be arbitrarily large.  Unless 𝐼 𝑇𝐶 𝑇𝐶 𝐶 𝑇𝐶 has restricted locality  ‘ Local-bath < FT assumption ’

𝐼 = 𝐽⊗𝐼 𝐶 + 𝐼 𝑇𝐶  𝑉 τ 𝑛𝑗𝑜 = 𝑓 −𝑗(𝐼 τ 𝑛𝑗𝑜 ) 𝜃 0 = 𝐼 𝑇𝐶 τ 𝑛𝑗𝑜 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑈+𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 ) Desiderata for DD +QEC: 𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃 𝑈 𝑂+1 𝜃 𝐸𝐸 (𝑂) = I. No extra locality assumptions 𝜃 𝐸𝐸 < 𝜃 0 II. Pulses in the code III. Shorter sequences than full decoupling approach. [[n,k,d]] QEC code DD DD DD DD DD 𝜃 𝐸𝐸 < 𝜃 0 n – qubit Pauli basis as decoupling group  No ‘local bath assumption’ Ng,Lidar,Preskil lhas restricted locality ‘ Local-bath assumption ’ • • Length of sequence exponential in 2n Enhanced fidelity of physical gates via appended DD sequences • Pulses look like errors to the code  limits possible integration with other schemes • Order of decoupling N cannot be arbitrarily large.  Unless 𝐼 𝑇𝐶 𝑇𝐶 𝐶 𝑇𝐶 has restricted locality  ‘ Local-bath < FT assumption ’

The magic is in the decoupling group 18

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

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 𝑗 } 𝑗=1,…,𝐿 • ( Ω 𝑗 ) 2 = 𝐽 Ω 𝑗 Ω 𝑘 = −1 𝑔(𝑗,𝑘) Ω 𝑘 Ω 𝑗 ; 𝑔(𝑗, 𝑘) = {0,1} Ω 𝑗 Ω 𝑘 ≠ Ω 𝑙 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 𝑗 } 𝑗=1,…,𝐿 • ( Ω 𝑗 ) 2 = 𝐽 Ω 𝑗 Ω 𝑘 = −1 𝑔(𝑗,𝑘) Ω 𝑘 Ω 𝑗 ; 𝑔(𝑗, 𝑘) = {0,1} Ω 𝑗 Ω 𝑘 ≠ Ω 𝑙 Pulses  <MOOS> Concatenated Dynamical Decoupling (CDD) (2 𝐿 ) 𝑂 pulses [Khodjasteh and Lidar, Phys. Rev. Lett. 95, 180501 (2005)] Pulses  MOOS Nested Uhrig Dynamical Decoupling (NUDD) (𝑂 + 1) 𝐿 pulses [Wang and Liu, Phys. Rev. A 83, 022306 (2011)] 21

What we propose… Stabilizer generators = {S 𝑗 } 𝑗=1,…,𝑅 • MOOS = {S 𝑗 } 𝑗=1,…,𝑅 22

What we propose… Stabilizer generators = {S 𝑗 } 𝑗=1,…,𝑅 • (𝑀) , Z 𝑗 (𝑀) } 𝑗=1,…,𝑙 Logical operators (Pauli basis) = { X 𝑗 • (𝑀) , Z 𝑗 (𝑀) } 𝑗=1,…,𝑙 MOOS = {S 𝑗 } 𝑗=1,…,𝑅 MOOS = {S 𝑗 } 𝑗=1,…,𝑅 ⋃ { X 𝑗 23

What we propose… Stabilizer generators = {S 𝑗 } 𝑗=1,…,𝑅 • (𝑀) , Z 𝑗 (𝑀) } 𝑗=1,…,𝑙 Logical operators (Pauli basis) = { X 𝑗 • (𝑀) , Z 𝑗 (𝑀) } 𝑗=1,…,𝑙 MOOS = {S 𝑗 } 𝑗=1,…,𝑅 MOOS = {S 𝑗 } 𝑗=1,…,𝑅 ⋃ { X 𝑗 𝑉 𝐸𝐸 𝑈 = 𝑓 −𝑗(𝐼 ∅,𝑓𝑔𝑔 𝑃 𝑈 +𝐼 𝑇𝐶,𝑓𝑔𝑔 𝑃(𝑈 𝑂+1 )) 𝐼 ∅,𝑓𝑔𝑔 ∝ {S 𝑗 } 𝑗=1,…,𝑅 Contains no physical or logical errors ! Only harmless terms ! Even if 𝐼 𝑇𝐶 is a logical error! 24

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) 𝑰 𝑻𝑪 1 - 0 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) 𝑰 𝑻𝑪 1 - 0 2 - 0  DD Pulses are bitwise / transversal in the code Pulses do not look like errors to the code  Allows interaction with other protection schemes. 26

What else do we gain ? • Shorter sequences than full decoupling approach: For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators 𝐷𝐸𝐸 (<Ω 𝑗 >,𝑂)  2 𝑜+𝑙−𝑕 𝑂 < 2 2𝑜𝑂 𝑂𝑉𝐸𝐸 ({Ω 𝑗 ,𝑂})  (𝑂 + 1) 𝑜+𝑙−𝑕 < (𝑂 + 1) 2𝑜 27