Dynamical Decoupling
a tutorial
Daniel Lidar
QEC11
For a great DD tutorial see Lorenza Viola’s talk in
http://qserver.usc.edu/qec07/program.html
Slides & movie.
This tutorial:
• Essential intro material
• High order decoupling
• Decoupling along with computation
Origins: Hahn Spin Echo
Overcoming dephasing via time-reversal
Usain Bolt
Lidar
Time reversal without time travel
http://en.wikipedia.org/wiki/Spin_echo
Modern Hahn Echo experiment (Dieter Suter)
Let’s get serious: the general setting
• Hamiltonian error model
• Joint evolution of system (S) and bath (B); noise Hamiltonian H
“free evolution”
• This talk: all Hamiltonians bounded in the operator norm (largest singular value)
• This assumption is not necessary: norms may diverge (e.g., oscillator bath)
Often it pays to use correlation functions instead.
See, e.g., Mike Biercuk’s and Gonzalo Alvarez’s talks
DD: just a set of interruptions
• Consider a set of instantaneous unitaries 𝑃𝑗 applied to the system only
at times 𝑡𝑗 , inbetween free evolutions:
𝑈DD 𝑇 = 𝑈 τ𝐾 𝑃𝐾 𝑈 τ𝐾−1 𝑃𝐾−1 … 𝑈 τ0 𝑃0
𝑃0 𝑃1
with τ𝑗 = 𝑡𝑗+1 - 𝑡𝑗 .
τ0
𝑃2
τ1
𝑃𝑗
…
τ2
τ𝑗
t
• All DD sequences can be described in this ``bang-bang’’ manner,
disregarding finite pulse-width effects (see, e.g., Lorenza Viola & Dieter Suter’s talks),
• Pulse sequences differ by choice of pulse types 𝑃𝑗 and pulse intervals 𝜏𝑗
• For a qubit typically 𝑃𝑗 ∈ 𝐼, 𝑋, 𝑌, 𝑍 ; other angles and axes are also possible
• Examples:
P DD, S
eriodic
ymmetrized
DD, R
DD, C
andom
oncatenated
DD, U DD, Q
hrig
uadratic
DD, N U DD
ested
hrig
How good does it get?
At the end of the pulse sequence:
𝑃0 𝑃1
τ0
𝑃2
τ1
𝑃𝐾
𝑃𝑗
τ2
𝑡0
𝑈DD 𝑇 = exp[−𝑖𝑇𝐻∅ +
= 𝑈DD 𝑇
τ𝑗 …
𝑇
α 𝐻α,eff
t
𝑂(𝑇𝑁α +1 )]
𝐻∅ is the component of 𝐻 that commutes with a𝐥𝐥 pulses
𝐻α,eff are the remaining errors; they can be computed using, e.g., the
Magnus or Dyson series
𝑁α is the ``decoupling order’’ of the ``α–type’’ error
The fundamental min-max problem of DD:
Maximize 𝑁 = min𝛼 𝑁α ’s while minimizing 𝐾
Magnus & Dyson
so lv e
d
U ( t )   iH ( t )U ( t )
dt
su b je ct to U (0 )  I
Wilhelm Magnus
1907-1990
Freeman Dyson
1923-

U ( t )  e x p [  ( t )],
 (t) 


n
U (t )  I 
(t)
n1

 1 (t )   i
 2 (t )  
n
(t )
n1
t
0
1
dt 1 H ( t 1 )

2
 3 (t )   i
S
1
t
0

6
dt 1

t
0
dt 1
t
1
t
1
0
dt 2

t
0
2
 [ H ( t 1 ),[ H ( t 2 ), H ( t 3 )]]  
dt 3 

[
H
(
t
),[
H
(
t
),
H
(
t
)]]


3
2
1
 n ( t )  ... (ex p licit recu rsiv e ex p ressio n k n o w n )
- p re se rv e s u n itarity to all o rd e rs
- co n v e rg e s if


t
0
dt 1 H ( t 1 )

tn 1
0
dt n H ( t n )
dt 2 [ H ( t 1 ), H ( t 2 )]
0

Sn (t )  (  i)
n
t
0
dt 1 H ( t 1 )  
re late d , e .g .:
 1 ( t )  S1 ( t )
 2 (t )  S2 (t ) 
1
2
2
S1 ( t )
- e asy to w rite d o w n
- n o re strictio n o n H ( t ) fo r co n v e rg e n ce
relevant for DD after transformation to ``toggling frame” (rotates with pulse Hamiltonian)
(small piece of) The DD pulse sequence zoo
eriodic
DD
the payoff 𝑁
1
DD
≤ 8 (twice PDD)
2
DD
𝑂(4𝑁 )
𝑁
U DD
𝑂(𝑁) (single error type only)
𝑁
Q
𝑂(𝑁 2 )
𝑁
S
ymmetrized
C
oncatenated
hrig
uadratic
DD
sequence length & min decoupling order
P
the price 𝐾 for one qubit
≤4
PDD: first order decoupling & group averaging
free evolution:
f  exp(  iH  )
Apply pulses via a unitary symmetrizing group G  { g j } Kj 01
†
†
( g K  1 f g K  1 )( g K  2 f g K  2 )
†
†
( g 1 f g 1 )( g 0 f g 0 )
repeat: “periodic DD”
PDD: first order decoupling & group averaging
free evolution:
f  exp(  iH  )
Apply pulses via a unitary symmetrizing group G  { g j } Kj 01
†
†
†
†
( g K  1 f g K  1 )( g K  2 f g K  2 )
( g 1 f g 1 )( g 0 f g 0 )
PK  1
P1
repeat: “periodic DD”
Pj  g j g j  1 ; g K  g 0
†
pulses
PDD: first order decoupling & group averaging
f  exp(  iH  )
free evolution:
Apply pulses via a unitary symmetrizing group G  { g j } Kj 01
†
†
( g K  1 f g K  1 )( g K  2 f g K  2 )
( g 1 f g 1 )( g 0 f g 0 )  exp (  iT
†
†
g

K
†
j
Pj  g j g j  1 ; g K  g 0
†
pulses
K
j1
P1
PK  1
1
H g j  O ( T ))
2
PDD: first order decoupling & group averaging
f  exp(  iH  )
free evolution:
Apply pulses via a unitary symmetrizing group G  { g j } Kj 01
†
†
( g K  1 f g K  1 )( g K  2 f g K  2 )
( g 1 f g 1 )( g 0 f g 0 )  exp (  iT
†
†
H 
H
1
K
g

K
†
j
H g j  O ( T ))
j1
P1
PK  1
1
K

K
†
gi H gi
commutes with all the pulses:
“G-symmetrization”
j1
O (T )  N  1
2
higher order terms:
first order decoupling
 [ g i H g i , g j H g j ]  ...
†
i j
†
2
Example 0: Hahn echo revisited –
suppressing single-qubit dephasing
n o ise : H e rr  X  B X  Y  B Y  Z  B Z
f  exp(  iH  )
d e co u p lin g g ro u p : G  { I , X }
Pj  g j g j  1 ; g K  g 0 
†
P1  X I  X , P2  IX  X
p u lse se q u e n ce : f X f X
X
X
𝜏
0
 U D D ( T )  e x p [  iT X  B X  O ( T )( X  B X  Y  B Y  Z  B Z )]
𝜏
𝑇 = 2𝜏
2
t
H
commutes with G;
undecoupled
'
'
'
H X ,eff
H Y ,e ff
H Z ,eff
anti-commute with G;
decoupled to 1st order;
``detected” by G
Example 1: ``Universal decoupling group” –
suppressing general single-qubit decoherence
n o ise : H e rr  X  B X  Y  B Y  Z  B Z
f  exp(  iH  )
d e co u p lin g g ro u p : G  { I , X , Y , Z }
Pj  g j g j  1 ; g K  g 0 
†
P1  X I  X , P2  Y X  Z , P3  Z Y  X , P4  IZ  Z
p u lse se q u e n ce : f X f Z f X f Z
H
Z
X
𝜏
0
Z
𝜏
 U D D ( T )  e x p [  iT I  B I
X
𝜏
 O ( T )( X  B X  Y  B Y  Z  B Z )]
2
𝜏
𝑇 = 4𝜏
'
'
'
H X ,eff
H Y ,e ff
H Z ,e ff
t
decoupled to 1st order;
``detected” by G
(small piece of) The DD pulse sequence zoo
eriodic
DD
the payoff 𝑁
1
DD
≤ 8 (twice PDD)
2
DD
𝑂(4𝑁 )
𝑁
U DD
𝑂(𝑁) (single error type only)
𝑁
Q
𝑂(𝑁 2 )
𝑁
S
ymmetrized
C
oncatenated
hrig
uadratic
DD
sequence length & min decoupling order
P
the price 𝐾 for one qubit
≤4
(small piece of) The DD pulse sequence zoo
eriodic
DD
the payoff 𝑁
1
DD
≤ 8 (twice PDD)
2
DD
𝑂(4𝑁 )
𝑁
U DD
𝑂(𝑁) (single error type only)
𝑁
Q
𝑂(𝑁 2 )
𝑁
S
ymmetrized
C
oncatenated
hrig
uadratic
DD
sequence length & min decoupling order
P
the price 𝐾 for one qubit
≤4
Any palindromic (time-reversal symmetric) pulse sequence is automatically
2nd order wrt the base sequence: all even terms in the Magnus series vanish if
𝐻 𝑡 = 𝐻(𝑇 − 𝑡)
Example 2: Palindromic suppression of general
single-qubit decoherence to second order
n o ise : H e rr  X  B X  Y  B Y  Z  B Z
f  exp(  iH  )
d e co u p lin g g ro u p : G  { I , X , Y , Z }
p u lse se q u e n ce f X f Z f X f Z , Z f X f Z f X f Z e cn e u q e s e slu p
=fX fZfX ffX fZfX fZ
Z
X
𝜏
Z
𝜏
X
X
𝜏
2𝜏
X
Z
𝜏
𝜏
Z
𝜏
𝜏
t
𝑇 = 8𝜏
0
 U D D ( T )  e x p [  iT I  B I  O ( T )( X  B X  Y  B Y  Z  B Z )]
3
decoupled to 2nd order:
'
'
'
H X ,e ff
H Y ,e ff
H Z ,e ff
The quest for high order
How do we go systematically beyond second order decoupling?
Two general techniques:
• Concatenation (CDD)
• Pulse interval optimization (UDD, QDD, NUDD)
Concatenated DD
n o ise : H err  X  B X  Y  B Y  Z  B Z
f  exp(  iH  )
(0)
d e co u p lin g g ro u p : G  { I , X , Y , Z }
p u lse se q u e n ce : p 1  f X f Z f X f Z
Z
X
𝜏
0
Z
𝜏
 U D D ( T )  e x p [  iT I  B I
(1)
X
𝜏
H
(1)
 O ( T )( X  B X  Y  B Y  Z  B Z )]
2
𝜏
𝑇
(1)
(1)
t
(1)
H e rr
(1)
Concatenated DD
f  exp(  iH  )
n o ise : H err  X  B X  Y  B Y  Z  B Z
(0)
d e co u p lin g g ro u p : G  { I , X , Y , Z }
p u lse se q u e n ce : p 1  f X f Z f X f Z
Z
X
Z
H
 U D D ( T )  e x p [  iT I  B I
(1)
X
(1)
 O ( T )( X  B X  Y  B Y  Z  B Z )]
2
𝑇
0
Z
X
Z
X
(1)
(1)
(1)
t
(1)
H e rr
Same as the original problem, so apply 𝑝1 again, keeping T fixed, shrinking 𝜏:
p 2  p 1 X p 1Z p 1 X p 1Z

U D D ( T )  e x p [  iT I  B I
(2)
(2)
 O ( T ) H err ]
3
(2)
Concatenated DD
f  exp(  iH  )
n o ise : H err  X  B X  Y  B Y  Z  B Z
(0)
d e co u p lin g g ro u p : G  { I , X , Y , Z }
p u lse se q u e n ce : p 1  f X f Z f X f Z
Z
X
Z
H
 U D D ( T )  e x p [  iT I  B I
(1)
X
(1)
 O ( T )( X  B X  Y  B Y  Z  B Z )]
2
Z
X
Z
X
(1)
(1)
t
𝑇
0
(1)
(1)
H e rr
Same as the original problem, so apply 𝑝1 again, keeping T fixed, shrinking 𝜏:
p 2  p 1 X p 1Z p 1 X p 1Z

U D D ( T )  e x p [  iT I  B I
(2)
(2)
 O ( T ) H err ]
3
(2)
…
p k  p k 1 X p k  1Z p k  1 X p k  1Z

U D D ( T )  e x p [  iT I  B I
(k)
(k)
 O (T
k 1
(k)
) H e rr ]
Concatenated DD
f  exp(  iH  )
n o ise : H err  X  B X  Y  B Y  Z  B Z
(0)
d e co u p lin g g ro u p : G  { I , X , Y , Z }
p u lse se q u e n ce : p 1  f X f Z f X f Z
Z
X
Z
H
 U D D ( T )  e x p [  iT I  B I
(1)
X
(1)
 O ( T )( X  B X  Y  B Y  Z  B Z )]
2
Z
X
Z
X
(1)
(1)
t
𝑇
0
(1)
(1)
H e rr
Same as the original problem, so apply 𝑝1 again, keeping T fixed, shrinking 𝜏:
p k  p k 1 X p k  1Z p k  1 X p k  1Z

U D D ( T )  e x p [  iT I  B I
(k)
Alternatively: keep 𝜏 fixed, then 𝑇 = 4𝑘 𝜏

(k)
 O (T
k 1
(k)
) H er r ]
optimal concatenation level:

k o p t   lo g 4  H e rr  H B  


(small piece of) The DD pulse sequence zoo
eriodic
DD
the payoff 𝑁
1
DD
≤ 8 (twice PDD)
2
DD
𝑂(4𝑁 )
𝑁
U DD
𝑂(𝑁) (single error type only)
𝑁
Q
𝑂(𝑁 2 )
𝑁
S
ymmetrized
C
oncatenated
hrig
uadratic
DD
sequence length & min decoupling order
P
the price 𝐾 for one qubit
≤4
More for Less
CDD requires exponential number of pulses for given decoupling order.
Can we do better?
At the end of the pulse sequence:
𝑃0 𝑃1
τ0
𝑃2
τ1
𝑃𝐾
𝑃𝑗
τ2
𝑡0
𝑈DD 𝑇 = exp[−𝑖𝑇𝐻∅ +
τ𝑗 …
= 𝑈DD 𝑇
𝑇
t
𝑁α +1 )]
𝐻
𝑂(𝑇
α α,eff
The optimization problem:
Maximize the smallest decoupling order min(𝑁𝛼 ) while minimizing the number
of pulses K.
Or: what is the smallest number of pulses such that the first N terms in the
Dyson series of 𝑈DD (𝑇) vanish, for an arbitrary bath?
Answer: N for pure dephasing, 𝑁 2 for general single-qubit decoherence
Uhrig DD: choose those intervals well
H  Z  BZ  I  BI
Suppresses single-axis decoherence to Nth order with only N pulses
Optimal for ideal pulses, sharp high-frequency cutoff
= X pulse
divide semicircle into N+1 equal angles
j
t j  T sin
2 j
T
𝑡𝑁
for j  1,
0
𝑇
𝑗𝜋
𝑡𝑗 = (1 − cos
)
2
𝑁+1
U D D ( T )  e x p [  iT H  ]  Z  B Z T
'
N 1
j
2
2( N  1)
,N
,
How about general qubit decoherence?
H  X  B X  Y  BY  Z  B Z  I  B I
Quadratic DD (QDD):
a nesting of two types (e.g., X and Z) of UDD sequences.
How about general qubit decoherence?
H  X  B X  Y  BY  Z  B Z  I  B I
Quadratic DD (QDD):
a nesting of two types (e.g., X and Z) of UDD sequences.
divide semicircle
into 𝑁2 + 1 equal
angles
X
T
0
How about general qubit decoherence?
H  X  B X  Y  BY  Z  B Z  I  B I
Quadratic DD (QDD):
a nesting of two types (e.g., X and Z) of UDD sequences.
divide semicircle
into 𝑁2 + 1 equal
angles
divide each small
semicircle into
𝑁1 + 1 equal
angles
X
Z
T
0
How about general qubit decoherence?
H  X  B X  Y  BY  Z  B Z  I  B I
Quadratic DD (QDD):
a nesting of two types (e.g., X and Z) of UDD sequences.
Uses (N1 +1)(N2 +1) pulses to
remove the first min(N1 , N2)
orders in Dyson series
 Proof: talk by Liang Jiang
(Wed. 2:40)

X
Z
T
0
How about general qubit decoherence?
H  X  B X  Y  BY  Z  B Z  I  B I
Quadratic DD (QDD):
a nesting of two types (e.g., X and Z) of UDD sequences.
Decoupling order of each error type :
Uses (N1 +1)(N2 +1) pulses to
remove the first min(N1 , N2)
orders in Dyson series
 Proof: talk by Liang Jiang
(Wed. 2:40), poster by WanJung Kuo

U D D ( T )  e x p [  iT H  ] 

𝑁𝛼 − 1
not both even
   B T
'
N
  X ,Y , Z
X
Z
0
T
Further nesting: NUDD, useful for multi-qubit DD
(small piece of) The DD pulse sequence zoo
eriodic
DD
the payoff 𝑁
1
DD
≤ 8 (twice PDD)
2
DD
𝑂(4𝑁 )
𝑁
U DD
𝑂(𝑁) (single error type only)
𝑁
Q
𝑂(𝑁 2 )
𝑁
S
ymmetrized
C
oncatenated
hrig
uadratic
DD
sequence length & min decoupling order
P
the price 𝐾 for one qubit
≤4
DD sequences battle it out numerically
J. R. West, B. H. Fong, & DAL, PRL 104, 130501 (2010).
D=averaged trace-norm distance between initial and final system-only state.
Initial state is random pure state of system & bath. Bath contains 4 spins.
DD & Computation
Problem: DD pulses interfere with computation – they cancel everything!
How can they be reconciled?
At least three approaches:
• Decouple-while-compute
• Decouple-then-compute
• Dynamically corrected gates (see Lorenza Viola’s talk at 3 today)
DD & Computation
Problem: DD pulses interfere with computation – they cancel everything!
How can they be reconciled?
At least three approaches:
• Decouple-while-compute
• Decouple-then-compute
• Dynamically corrected gates (see Lorenza Viola’s talk at 3 today)
Decouple-while-compute
Need pulses and computation to commute
Solutions:
- Use encoding and stabilizer/normalizer structure
- Use double commutant structure of noiseless subsystems
E.g.:
- DD pulses are the stabilizer generators of a stabilizer code:
𝐻α,eff 𝑂(𝑇𝑁α +1 )]
𝑈DD 𝑇 = exp[−𝑖𝑇𝐻∅ +
α
𝐻∅ consists of the logical operators of the stabilizer code
- DD pulses are collective rotations of all qubits
𝐻∅ consists of Heisenberg exchange interactions;
used, e.g., to demonstrate high fidelity gates for quantum dots
DD & Computation
Problem: DD pulses interfere with computation – they cancel everything!
How can they be reconciled?
At least three approaches:
• Decouple-while-compute
• Decouple-then-compute
• Dynamically corrected gates (see Lorenza Viola’s talk at 3 today)
Consider a fault-tolerant simulation of a circuit
T h e n o ise stre n g th :   H err  0   0 ~ 10
4
 F T sim u latio n p o ss ib le
Now prepend DD: decouple-then-compute
T
𝑈DD 𝑇 = exp[−𝑖𝑇𝐻∅ +
𝑁α +1 )]
𝐻
𝑂(𝑇
α α,eff
T h e n ew n o ise stren g th :  D D  H eff T   0 ~ 10
4
 F T sim u latio n p o ssib le
Noise strengths can be upper-bounded for a
well-behaved bath
 allows us to examine each DD-protected gate separately.
actually this assumption can be relaxed: see Gerardo Paz’s talk, 3:40
DD-protected gates can be better
 DD / 
  H err  H B
H.-K. Ng, DAL, J. Preskill, PRA 84, 012305 (2011)
CDD-protected gates can be even better
 DD / 
( opt )
  H e rr  H B
H.-K. Ng, DAL, J. Preskill, PRA 84, 012305 (2011)
Fighting decoherence with hands tied
Dynamical decoupling is
• A method where one applies fast & strong control pulses to the system
• Open-loop, feedback- and measurement-free
Dynamical decoupling is not
• A stand-alone solution
It cannot, by itself, be made fault-tolerant (see Kaveh Khodjasteh’s talk Thu 2:40)
So, why not use the full power of fault-tolerance?
• Open-loop is technically easier than closed-loop or topological methods
• DD can be used at the lowest (physical) level to improve performance
and reduce overhead of fault tolerance
• DD has been widely experimentally tested, with encouraging results
Essential references for this talk
• L. Viola, S. Lloyd PRA 58, 2733 (1998): first DD paper
• L. Viola, E. Knill, S. Lloyd, PRL 82, 2417 (1999): General theory of DD
• P. Zanardi Phys. Lett. A 258, 77 (1999): General theory of DD, DD as
symmetrization
• K. Khodjasteh, D.A. Lidar, PRL 95, 180501 (2005): first CDD paper
• F. Casas, J. Phys. A 40, 15001 (2007): convergence of Magnus expansion
• G. S. Uhrig, PRL 98, 100504 (2007): first UDD paper
• W. Yang, R.-B. Liu, PRL 101, 180403 (2008): first proof of universality of UDD
• J. R. West, B. H. Fong, D.A. Lidar, PRL 104, 130501 (2010): first QDD paper
• Z. Wang, R.-B. Liu, PRA 83, 022306 (2011): first NUDD paper
• H.-K. Ng, D.A. Lidar, J. Preskill, PRA 84, 012305 (2011): DD and fault
tolerance, derivation of Magnus series; proof of vanishing even orders of
Magnus for palindromic sequences
• W.-J. Kuo, D.A. Lidar, PRA, 84 042329 (2011): first complete proof of
universality of QDD; see Wan’s poster