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 푯푺 3 Motivation 푯푩 qMac 푯푺 4 Motivation 푯푩 푯푺푩 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 5 Motivation 푯푩 푯푺푩 QEC + FT 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 6 Motivation 푯푩 Dynamical Decoupling 푯푺푩 QEC + FT 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 7 Motivation 푯푩 Dynamical Decoupling 푯′푺푩 푯푺푩 QEC + FT 푯푺푩 푯′푺푩 qMac 푯푺 푯푺푩 푯′푺푩 푯′푺푩 푯푺푩 8 −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[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 −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[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. −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[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 퐻푆퐵 has restricted locality ‘Local-bath assumption’ −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[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 퐻푆퐵 has restricted locality ‘Local-bath assumption’ < FT −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[n,k,d]] QEC code DD DD DD DD DD n Ng,Lidar,Preskil–qubit Pauli basislhas as restricted decoupling locality group ‘Local No- ‘localbath assumption bath assumption’’ • • LengthEnhanced of sequence fidelity of exponential physical gates in 2n via appended DD sequences • • PulsesOrder look of decoupling like errors Nto cannotthe code be arbitrarily limits possible large. integration with other schemes Unless 퐻 푆퐵 푆퐵 퐵 푆퐵 has restricted locality ‘Local-bath < FT assumption’ −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 Desiderata−푖(퐻∅,푒푓푓 for 푇DD+퐻 +QEC:푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 I. No extra locality assumptions II. Pulses in the code 휂퐷퐷 < 휂0 III. Shorter sequences than full decoupling approach [[n,k,d]] QEC. code DD DD DD DD DD 휂퐷퐷 < 휂0 n Ng,Lidar,Preskil–qubit Pauli basislhas as restricted decoupling locality group ‘Local No- ‘localbath assumption bath assumption’’ • • LengthEnhanced of sequence fidelity of exponential physical gates in 2n via appended DD sequences • • PulsesOrder look of decoupling like errors Nto cannotthe code be arbitrarily limits possible large. integration with other schemes Unless 퐻 푆퐵 푆퐵 퐵 푆퐵 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 = {Ω푖}푖=1,…,퐾 2 (Ω푖) = 퐼 푓(푖,푗) Ω푖 Ω푗 = −1 Ω푗 Ω푖; 푓(푖, 푗) = {0,1} Ω푖 Ω푗 ≠ Ω푘 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 = {Ω푖}푖=1,…,퐾 2 (Ω푖) = 퐼 푓(푖,푗) Ω푖 Ω푗 = −1 Ω푗 Ω푖; 푓(푖, 푗) = {0,1} Ω푖 Ω푗 ≠ Ω푘 Concatenated Dynamical Decoupling (CDD) Pulses <MOOS> 퐾 푁 [Khodjasteh and Lidar, Phys. Rev. Lett. 95, 180501 (2005)] (2 ) pulses Nested Uhrig Dynamical Decoupling (NUDD) Pulses MOOS 퐾 [Wang and Liu, Phys. Rev. A 83, 022306 (2011)] (푁 + 1) pulses 21 What we propose… • Stabilizer generators = {S푖}푖=1,…,푄 MOOS = {S푖}푖=1,…,푄 22 What we propose… • Stabilizer generators = {S푖}푖=1,…,푄 (퐿) (퐿) • Logical operators (Pauli basis) = { X푖 , Z푖 }푖=1,…,푘 (퐿) (퐿) MOOSMOOS == {{SS푖푖}}푖푖==11,…,…,푄,푄 ⋃{ X푖 , Z푖 }푖=1,…,푘 23 What we propose… • Stabilizer generators = {S푖}푖=1,…,푄 (퐿) (퐿) • Logical operators (Pauli basis) = { X푖 , Z푖 }푖=1,…,푘 (퐿) (퐿) MOOSMOOS == {{SS푖푖}}푖푖==11,…,…,푄,푄 ⋃{ X푖 , Z푖 }푖=1,…,푘 푁+1 −푖(퐻∅,푒푓푓푂 푇 +퐻푆퐵,푒푓푓 푂(푇 )) 푈퐷퐷 푇 = 푒 퐻∅,푒푓푓 ∝ {S푖}푖=1,…,푄 Contains no physical or logical errors ! Only harmless terms ! Even if 퐻푆퐵 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) 1 - 0 푯푺푩 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) 21 - 0 푯푺푩 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 푛+푘−푔 푁 2푛푁 퐶퐷퐷(<Ω푖>,푁) 2 < 2 푛+푘−푔 2푛 푁푈퐷퐷({Ω푖,푁}) (푁 + 1) < (푁 + 1) 27 What else do we gain ? • Shorter sequences than full decoupling approach: For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators 푛+푘−푔 푁 2푛푁 퐶퐷퐷(<Ω푖>,푁) 2 < 2 푛+푘−푔 2푛 푁푈퐷퐷({Ω푖,푁}) (푁 + 1) < (푁 + 1) e.g. [[ 푛2, 1, 푛]] Bacon Shor code: Stabilizer generators: 2(n-1) Logical generators: 2 SXDD Full decoupling 2 D(MOOS) 2 푛 2 푛 2 NUDD (푁 + 1)2 푛 (푁 + 1)2 푛 2 CDD 22 푛 푁 22 푛 푁 3 - 0 푯 푺푩 28 휼푫푫 < 휼ퟎ ? Recall our (effective) noise rates: 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 Are an overestimation: bounds obtained without using the QEC code structure. (work in progress) 29 휼푫푫 < 휼ퟎ ? Recall our (effective) noise rates: 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 Are an overestimation: bounds obtained without using the QEC code structure. (work in progress) 푁+1 How to compute 퐻푆퐵,푒푓푓 푂 푇 ?? NLP results (Eqs. 152-164) Recursive relations for 퐻푆퐵,푒푓푓(푞) and 퐻∅,푒푓푓(푞) at every degree of concatenation q. 푞(푞+3)/2 푞−1 퐻푆퐵,푒푓푓(푞) ≤ 푅 푐 퐻푆퐵 + 퐻퐵 τ0 퐻푆퐵 푞 푇(푞) = 푅 τ푚푖푛 퐷(푀푂푂푆) where 푅 = 2 and 푐 ~1 30 휼푫푫 < 휼ퟎ N=1 N=2 N=3 퐼⊗퐻퐵 = 퐽0 퐻푆퐵 = 퐽푆퐵 [[9,1,3]] – BS code: 휏푚푖푛 = 1 ; 퐷 푀푂푂푆 = 4 + 2 휼푫푫 < 휼ퟎ N=1 N=2 N=3 퐼⊗퐻퐵 = 퐽0 퐻푆퐵 = 퐽푆퐵 4 - 0 푯푺푩 [[9,1,3]] – BS code: 휏푚푖푛 = 1 ; 퐷 푀푂푂푆 = 4 + 2 휼푫푫 < 휼ퟎ N=1 N=2 N=3 퐼⊗퐻퐵 = 퐽0 퐻푆퐵 = 퐽푆퐵 4 - 10 푯푺푩 [[9,1,3]] – BS code: 휏푚푖푛 = 1 ; 퐷 푀푂푂푆 = 4 + 2 Beyond 퐻푆 = 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)] 5 - 1 푯푺푩 34 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.