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

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

Jan 28, 2024 545 likes β€’931 views

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

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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

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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

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Motivation

qMac 𝑰𝑻

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Motivation

𝑰π‘ͺ

qMac 𝑰𝑻

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Motivation

𝑰π‘ͺ

qMac 𝑰𝑻

𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ

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Motivation

𝑰π‘ͺ

qMac 𝑰𝑻 QEC + FT

𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ

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Motivation

𝑰π‘ͺ

qMac 𝑰𝑻 Dynamical Decoupling QEC + FT

𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ

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Motivation

𝑰π‘ͺ

qMac 𝑰𝑻 Dynamical Decoupling QEC + FT

𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ 𝑰𝑻π‘ͺ

𝑰′𝑻π‘ͺ 𝑰′𝑻π‘ͺ 𝑰′𝑻π‘ͺ 𝑰′𝑻π‘ͺ

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ο€  [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

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ο€  [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

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ο€  [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

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DD DD DD DD DD Ng,Lidar,Preskill PRA 84, 012305(2011)

  • Enhanced fidelity of physical gates via appended DD sequences

ο€  [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

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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.

ο€  [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

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DD DD DD DD DD [[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

Ng,Lidar,Preskill PRA 84, 012305(2011)

  • Enhanced fidelity of physical gates via appended DD sequences
  • Order of decoupling N cannot be arbitrarily large.

οƒΌ Unless 𝐼𝑇𝐢 has restricted locality οƒ  β€˜Local-bath assumption’

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DD DD DD DD DD

< FT

[[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

Ng,Lidar,Preskill PRA 84, 012305(2011)

  • Enhanced fidelity of physical gates via appended DD sequences
  • Order of decoupling N cannot be arbitrarily large.

οƒΌ Unless 𝐼𝑇𝐢 has restricted locality οƒ  β€˜Local-bath assumption’

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DD DD DD DD DD Ng,Lidar,Preskillhas restricted locality β€˜Local-bath assumption’

  • Enhanced fidelity of physical gates via appended DD sequences
  • Order of decoupling N cannot be arbitrarily large.

οƒΌ Unless 𝐼 𝑇𝐢 𝑇𝐢 𝐢 𝑇𝐢 has restricted locality οƒ  β€˜Local-bath assumption’

< FT

n –qubit Pauli basis as decoupling group οƒ  No β€˜local bath assumption’

  • Length of sequence exponential in 2n
  • Pulses look like errors to the code οƒ  limits possible integration with other schemes

[[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

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DD DD DD DD DD Ng,Lidar,Preskillhas restricted locality β€˜Local-bath assumption’

  • Enhanced fidelity of physical gates via appended DD sequences
  • Order of decoupling N cannot be arbitrarily large.

οƒΌ Unless 𝐼 𝑇𝐢 𝑇𝐢 𝐢 𝑇𝐢 has restricted locality οƒ  β€˜Local-bath assumption’

< FT

n –qubit Pauli basis as decoupling group οƒ  No β€˜local bath assumption’

  • Length of sequence exponential in 2n
  • Pulses look like errors to the code οƒ  limits possible integration with other schemes

[[n,k,d]] QEC code

𝐼 = π½βŠ—πΌπΆ + 𝐼𝑇𝐢 οƒ  𝑉 Ο„π‘›π‘—π‘œ = π‘“βˆ’π‘—(𝐼 Ο„π‘›π‘—π‘œ)

πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔 π‘ˆ+𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 )

πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1

πœƒπΈπΈ < πœƒ0

Desiderata for DD +QEC: I. No extra locality assumptions

  • II. Pulses in the code
  • III. Shorter sequences than full decoupling approach.

πœƒπΈπΈ < πœƒ0

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The magic is in the decoupling group

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The magic is in the decoupling group

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

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The magic is in the decoupling group

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

  • Mutually Orthogonal Operator (generator) Set = {Ω𝑗}𝑗=1,…,𝐿

(Ω𝑗)2 = 𝐽 Ω𝑗 β„¦π‘˜ = βˆ’1 𝑔(𝑗,π‘˜)β„¦π‘˜ Ω𝑗; 𝑔(𝑗, π‘˜) = {0,1} Ω𝑗 β„¦π‘˜ β‰  Ω𝑙

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The magic is in the decoupling group

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

  • Mutually Orthogonal Operator (generator) Set = {Ω𝑗}𝑗=1,…,𝐿

(Ω𝑗)2 = 𝐽 Ω𝑗 β„¦π‘˜ = βˆ’1 𝑔(𝑗,π‘˜)β„¦π‘˜ Ω𝑗; 𝑔(𝑗, π‘˜) = {0,1} Ω𝑗 β„¦π‘˜ β‰  Ω𝑙 Concatenated Dynamical Decoupling (CDD) [Khodjasteh and Lidar, Phys. Rev. Lett. 95, 180501 (2005)] Pulses οƒ  <MOOS> (2𝐿)𝑂 pulses Nested Uhrig Dynamical Decoupling (NUDD) [Wang and Liu, Phys. Rev. A 83, 022306 (2011)] Pulses οƒ  MOOS (𝑂 + 1)𝐿 pulses

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What we propose…

  • Stabilizer generators = {S𝑗}𝑗=1,…,𝑅

MOOS = {S𝑗}𝑗=1,…,𝑅

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What we propose…

  • Stabilizer generators = {S𝑗}𝑗=1,…,𝑅

MOOS = {S𝑗}𝑗=1,…,𝑅 MOOS = {S𝑗}𝑗=1,…,𝑅 ⋃{ X𝑗

(𝑀), Z𝑗 (𝑀)}𝑗=1,…,𝑙

  • Logical operators (Pauli basis) = { X𝑗

(𝑀), Z𝑗 (𝑀)}𝑗=1,…,𝑙

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What we propose…

  • Stabilizer generators = {S𝑗}𝑗=1,…,𝑅

𝑉𝐸𝐸 π‘ˆ = π‘“βˆ’π‘—(πΌβˆ…,𝑓𝑔𝑔𝑃 π‘ˆ +𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃(π‘ˆπ‘‚+1)) πΌβˆ…,𝑓𝑔𝑔 ∝ {S𝑗}𝑗=1,…,𝑅

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

MOOS = {S𝑗}𝑗=1,…,𝑅 MOOS = {S𝑗}𝑗=1,…,𝑅 ⋃{ X𝑗

(𝑀), Z𝑗 (𝑀)}𝑗=1,…,𝑙

  • Logical operators (Pauli basis) = { X𝑗

(𝑀), Z𝑗 (𝑀)}𝑗=1,…,𝑙

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What do we gain ?

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

𝑰𝑻π‘ͺ 1 - 0

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What do we gain ?

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

𝑰𝑻π‘ͺ 1 - 0 2 - 0

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What else do we gain ?

  • Shorter sequences than full decoupling approach:

For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators 𝐷𝐸𝐸(<Ω𝑗>,𝑂) οƒ  2 π‘œ+π‘™βˆ’π‘• 𝑂 < 22π‘œπ‘‚ 𝑂𝑉𝐸𝐸({Ω𝑗,𝑂}) οƒ  (𝑂 + 1)π‘œ+π‘™βˆ’π‘• < (𝑂 + 1)2π‘œ

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What else do we gain ?

  • Shorter sequences than full decoupling approach:

For stabilizer/subsystem codes: MOOS : n-k-g stabilizers + 2k logical operators 𝐷𝐸𝐸(<Ω𝑗>,𝑂) οƒ  2 π‘œ+π‘™βˆ’π‘• 𝑂 < 22π‘œπ‘‚ 𝑂𝑉𝐸𝐸({Ω𝑗,𝑂}) οƒ  (𝑂 + 1)π‘œ+π‘™βˆ’π‘• < (𝑂 + 1)2π‘œ e.g. [[ π‘œ2, 1, π‘œ]] Bacon Shor code: Stabilizer generators: 2(n-1) Logical generators: 2 SXDD Full decoupling D(MOOS) 2 π‘œ 2 π‘œ2 NUDD (𝑂 + 1)2 π‘œ (𝑂 + 1)2 π‘œ2 CDD 22 π‘œ 𝑂 22 π‘œ2𝑂

𝑰𝑻π‘ͺ 3 - 0

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πœ½π‘¬π‘¬ < 𝜽𝟏 ?

οƒΌ Recall our (effective) noise rates: πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 οƒΌ Are an overestimation: οƒ  bounds obtained without using the QEC code structure. (work in progress)

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πœ½π‘¬π‘¬ < 𝜽𝟏 ?

οƒΌ Recall our (effective) noise rates: πœƒ0 = 𝐼𝑇𝐢 Ο„π‘›π‘—π‘œ πœƒπΈπΈ(𝑂) = 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 οƒΌ Are an overestimation: οƒ  bounds obtained without using the QEC code structure. (work in progress) ο€  How to compute 𝐼𝑇𝐢,𝑓𝑔𝑔 𝑃 π‘ˆπ‘‚+1 ?? οƒ  NLP results (Eqs. 152-164) Recursive relations for 𝐼𝑇𝐢,𝑓𝑔𝑔(π‘Ÿ) and πΌβˆ…,𝑓𝑔𝑔(π‘Ÿ) at every degree of concatenation q. 𝐼𝑇𝐢,𝑓𝑔𝑔(π‘Ÿ) ≀ π‘†π‘Ÿ(π‘Ÿ+3)/2 𝑑 𝐼𝑇𝐢 + 𝐼𝐢 Ο„0 π‘Ÿβˆ’1 𝐼𝑇𝐢 π‘ˆ(π‘Ÿ) = π‘†π‘ŸΟ„π‘›π‘—π‘œ where 𝑆 = 2𝐸(𝑁𝑃𝑃𝑇) and 𝑑 ~1

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πœ½π‘¬π‘¬ < 𝜽𝟏

[[9,1,3]] – BS code: πœπ‘›π‘—π‘œ = 1 ;𝐸 𝑁𝑃𝑃𝑇 = 4 + 2 N=1 N=2 N=3 π½βŠ—πΌπΆ = 𝐾0 𝐼𝑇𝐢 = 𝐾𝑇𝐢

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πœ½π‘¬π‘¬ < 𝜽𝟏

[[9,1,3]] – BS code: πœπ‘›π‘—π‘œ = 1 ;𝐸 𝑁𝑃𝑃𝑇 = 4 + 2 N=1 N=2 N=3 π½βŠ—πΌπΆ = 𝐾0 𝐼𝑇𝐢 = 𝐾𝑇𝐢

𝑰𝑻π‘ͺ 4 - 0

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πœ½π‘¬π‘¬ < 𝜽𝟏

[[9,1,3]] – BS code: πœπ‘›π‘—π‘œ = 1 ;𝐸 𝑁𝑃𝑃𝑇 = 4 + 2 N=1 N=2 N=3 π½βŠ—πΌπΆ = 𝐾0 𝐼𝑇𝐢 = 𝐾𝑇𝐢

𝑰𝑻π‘ͺ 4 - 0 4 - 1

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Beyond 𝐼𝑇 = 0

οƒΌ 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)]

𝑰𝑻π‘ͺ 5 - 1

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Conclusions

  • We have shown how to integrate dynamical decoupling and quantum error

correction codes in a β€˜natural’ way. οƒΌ No extra locality conditions οƒΌ Pulses in the code. οƒΌ Shorter sequences than full decoupling approach. οƒΌ Improve effective error rates AND deal where Hamiltonians QEC fails.

  • The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance

friendly ο€ 

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36

Conclusions

  • We have shown how to integrate dynamical decoupling and quantum error

correction codes in a β€˜natural’ way. οƒΌ No extra locality conditions οƒΌ Pulses in the code. οƒΌ Shorter sequences than full decoupling approach. οƒΌ Improve effective error rates AND deal where Hamiltonians QEC fails.

  • The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance

friendly ο€  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)

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SLIDE 37

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Conclusions

  • We have shown how to integrate dynamical decoupling and quantum error

correction codes in a β€˜natural’ way. οƒΌ No extra locality conditions οƒΌ Pulses in the code. οƒΌ Shorter sequences than full decoupling approach. οƒΌ Improve effective error rates AND deal where Hamiltonians QEC fails.

  • The pulses of the DD are bitwise and therefore EXPERIMENTALLY/fault-tolerance

friendly ο€  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 ?