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Dynamical Decoupling
a tutorial
Daniel LidarQEC11
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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
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Origins: Hahn Spin Echo
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Overcoming dephasing via time-reversal
Lidar
Usain Bolt
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Time reversal without time travel
http://en.wikipedia.org/wiki/Spin_echo
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Modern Hahn Echo experiment (Dieter Suter)
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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
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DD: just a set of interruptions• Consider a set of instantaneous unitaries applied to the system
only at timesinbetween free evolutions:
…
with - .
• 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 and pulse intervals
• For a qubit typically ; other angles and axes are also possible
• Examples: PeriodicDD, SymmetrizedDD, RandomDD, ConcatenatedDD, UhrigDD, QuadraticDD, NestedUhrigDD
t
…
𝑃0𝑃1 𝑃2 𝑃 𝑗
τ 0τ 1 τ 2 τ 𝑗
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How good does it get?
At the end of the pulse sequence:
is the component of that commutes with a
are the remaining errors; they can be computed using, e.g., the Magnus or Dyson series
is the ``decoupling order’’ of the ``α–type’’ error
t
…
𝑃0𝑃1 𝑃2 𝑃 𝑗
τ 0τ 1 τ 2 τ 𝑗
𝑃 𝐾
𝑇𝑡 0
¿𝑈 DD (𝑇 )
The fundamental min-max problem of DD:Maximize ’s while minimizing
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Magnus & Dyson
Wilhelm Magnus1907-1990
Freeman Dyson1923-
solve ( ) ( ) ( )
subject to (0)
dU t iH t U t
dtU I
1
( ) exp[ ( )], ( ) ( )nn
U t t t t
1
( ) ( )nn
U t I S t
1
1 2
1 1 10
2 1 2 1 20 0
1 2 3
3 1 2 30 0 0
3 2 1
( ) ( )
1( ) [ ( ), ( )]
2
[ ( ),[ ( ), ( )]]1( )
[ ( ),[ ( ), ( )]]6
( ) ... (explicit recursive expression known)
t
t t
t t t
n
t i dt H t
t dt dt H t H t
H t H t H tt i dt dt dt
H t H t H t
t
1 10
- preserves unitarity to all orders
- converges if ( )tdt H t
1
1 10 0( ) ( ) ( ) ( )nt tn
n n nS t i dt H t dt H t
- easy to write down
- no restriction on ( ) for convergenceH t
1 1
22 2 1
related, e.g.:
( ) ( )
1( ) ( ) ( )
2
t S t
t S t S t
relevant for DD after transformation to ``toggling frame” (rotates with pulse Hamiltonian)
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(small piece of) The DD pulse sequence zoo
the price for one qubit the payoff PeriodicDD 1
SymmetrizedDD (twice PDD) 2
ConcatenatedDD
UhrigDD (single error type only)
QuadraticDD
seq
uen
ce le
ng
th &
min
deco
up
ling
ord
er
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PDD: first order decoupling & group averaging
free evolution:
†
2 1
† † †1 1 2 1 0 0( ) ( ))( )(
KK K K g gg g g g g gf f f f
exp( )iH f
Apply pulses via a unitary symmetrizing group 10{ }K
j jG g
repeat: “periodic DD”
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PDD: first order decoupling & group averaging
free evolution:
11
† † †
1 2 2 1 1 0
†1 0( )( ) ( )( )
K
K K K
PP
K g g g g g gg gf f f f
Apply pulses via a unitary symmetrizing group
†1 0; j j j KP g g g g
10{ }K
j jG g
repeat: “periodic DD”
pulses
exp( )iH f
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PDD: first order decoupling & group averaging
free evolution:
11
† † †
1 2 2 1 1 0
†
11
†0
2( )( ) ( )( ) exp(1
( ))
K
K K K
PP
K
j jj
K g g g g g g iT g HgK
g g O Tf f f f
Apply pulses via a unitary symmetrizing group 10{ }K
j jG g
†1 0; j j j KP g g g g
pulses
exp( )iH f
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PDD: first order decoupling & group averaging
free evolution:
Apply pulses via a unitary symmetrizing group 10{ }K
j jG g
†
1
1 K
i ij
H H g HgK
commutes with all the pulses: “G-symmetrization”
11
† † †
1 2 2 1 1 0
†
11
†0
2( )( ) ( )( ) exp(1
( ))
K
K K K
PP
K
j jj
K g g g g g g iT g HgK
g g O Tf f f f
2( ) 1O NT first order decoupling
† †[ , ] ...i i j ji j
g Hg g Hghigher order terms:
exp( )iH f
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Example 0: Hahn echo revisited – suppressing single-qubit dephasing
errnoise: X Y ZH X B Y B Z B
decoupling grou {p: , }G I X
pulse sequence: X Xf f
exp( )iH f
†
1 0 1 2; ,
j j j KP g g g g P XI X P IX X
t𝜏
𝑇=2𝜏0
XX
𝜏 2 ' ' 'DD( ) exp[ ( )( )]X X Y ZU T iTX B O T X B Y B Z B
H
commutes with G;undecoupled
,effXH ,effYH
anti-commute with G;decoupled to 1st order;``detected” by G
,effZH
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errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
pulse sequence: X Z X Zf f f f
†
1 0 1 2 3 4; , , ,
j j j KP g g g g P XI X P YX Z P ZY X P IZ Z
Example 1: ``Universal decoupling group” –
suppressing general single-qubit decoherence
𝑇=4𝜏0𝜏
XZ
𝜏 𝜏XZ
𝜏t
DD
2 ' ' '
( ) exp[
( )( )]I
X Y Z
U T iTI B
O T X B Y B Z B
H
,effYH ,effZH
decoupled to 1st order;``detected” by G
,effXH
exp( )iH f
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(small piece of) The DD pulse sequence zoo
the price for one qubit the payoff PeriodicDD 1
SymmetrizedDD (twice PDD) 2
ConcatenatedDD
UhrigDD (single error type only)
QuadraticDD
seq
uen
ce le
ng
th &
min
deco
up
ling
ord
er
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(small piece of) The DD pulse sequence zoo
the price for one qubit the payoff PeriodicDD 1
SymmetrizedDD (twice PDD) 2
ConcatenatedDD
UhrigDD (single error type only)
QuadraticDD
seq
uen
ce le
ng
th &
min
deco
up
ling
ord
er
Any palindromic (time-reversal symmetric) pulse sequence is automatically 2nd order wrt the base sequence: all even terms in the Magnus series vanish if
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errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
pulse sequence ecneuqes eslup
,
=
Z
X Z X Z X Z X Z
X Z X X Z X Z
f f f f f f f f
f f f ff f f f
Example 2: Palindromic suppression of general single-qubit decoherence to
second order
𝑇=8𝜏
3 ' ' 'DD( ) exp[ ( )( )]I X Y ZU T iTI B O T X B Y B Z B
decoupled to 2nd order:,effYH ,effZH
,effXH
0
𝜏ZX
𝜏𝜏XZ
2𝜏t
𝜏XZ
𝜏 𝜏ZX
𝜏
exp( )iH f
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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)
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Concatenated DD (0)
errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
1pulse sequence: X X Zp Zf f f f
exp( )iH f
𝑇0
𝜏XZ
𝜏 𝜏XZ
𝜏t
(1) (1)DD
(1) (1) (12 )
( ) exp[
( )( )]I
X Y Z
U T iTI B
O T X B Y B Z B
H
(1)errH
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Concatenated DD (0)
errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
1pulse sequence: X X Zp Zf f f f
𝑇0
XZ XZ
t
(1) (1)DD
(1) (1) (12 )
( ) exp[
( )( )]I
X Y Z
U T iTI B
O T X B Y B Z B
H
(1)errH
Same as the original problem, so apply again, keeping T fixed, shrinking :
32 1 1 1 1
(2) (2) (2)DD err ( ) exp[ ( ) ]IX Z X Z U T iTI B O T Hp p p p p
XZ XZ
exp( )iH f
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Concatenated DD (0)
errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
1pulse sequence: X X Zp Zf f f f
(1) (1)DD
(1) (1) (12 )
( ) exp[
( )( )]I
X Y Z
U T iTI B
O T X B Y B Z B
H
(1)errH
Same as the original problem, so apply again, keeping T fixed, shrinking :
32 1 1 1 1
(2) (2) (2)DD err ( ) exp[ ( ) ]IX Z X Z U T iTI B O T Hp p p p p
( ) ( ) ( )
DD er1 1 r1
1 1 ( ) exp[ ( ) ]k kk k k
k kk IkX Z X Z U T iTI B O T Hp p p p p
…
𝑇0
XZ XZ
tXZ XZ
exp( )iH f
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Concatenated DD (0)
errnoise: X Y ZH X B Y B Z B
decoupling group { , ,: } ,G I X Y Z
1pulse sequence: X X Zp Zf f f f
(1) (1)DD
(1) (1) (12 )
( ) exp[
( )( )]I
X Y Z
U T iTI B
O T X B Y B Z B
H
(1)errH
Same as the original problem, so apply again, keeping T fixed, shrinking :
( ) ( ) ( )DD er1 1 r
11 1 ( ) exp[ ( ) ]k k
k k kk k
k IkX Z X Z U T iTI B O T Hp p p p p
Alternatively: keep fixed, then optimal concatenation level:
𝑇0
XZ XZ
tXZ XZ
opt 4 errlog Bk H H
exp( )iH f
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(small piece of) The DD pulse sequence zoo
the price for one qubit the payoff PeriodicDD 1
SymmetrizedDD (twice PDD) 2
ConcatenatedDD
UhrigDD (single error type only)
QuadraticDD
seq
uen
ce le
ng
th &
min
deco
up
ling
ord
er
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More for Less
At the end of the pulse sequence:
t
…
𝑃0𝑃1 𝑃2 𝑃 𝑗
τ 0τ 1 τ 2 τ 𝑗
𝑃 𝐾
𝑇𝑡 0
¿𝑈 DD (𝑇 )
CDD requires exponential number of pulses for given decoupling order.Can we do better?
The optimization problem:Maximize the smallest decoupling order while minimizing the numberof pulses K.
Or: what is the smallest number of pulses such that the first N terms in the Dyson series of vanish, for an arbitrary bath?
Answer: N for pure dephasing, for general single-qubit decoherence
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Uhrig DD: choose those intervals well
Suppresses single-axis decoherence to Nth order with only N pulses
Optimal for ideal pulses, sharp high-frequency cutoff
T 0
2sin ,2( 1)
for 1, ,
j
jt T
N
j N
2 j
j
𝑡 𝑗=𝑇2(1− cos ( 𝑗 𝜋
𝑁+1 ))𝑡𝑁
divide semicircle into N+1 equal angles
Z IH Z B I B
= X pulse
' 1
DD( ) exp[ ] NZU T iTH Z B T
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Quadratic DD (QDD): a nesting of two types (e.g., X and Z) of UDD sequences.
How about general qubit decoherence?
X Y Z IH X B Y B Z B I B
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0T
X
divide semicircle into equal angles
How about general qubit decoherence?
Quadratic DD (QDD): a nesting of two types (e.g., X and Z) of UDD sequences.
X Y Z IH X B Y B Z B I B
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0T
Z
X
divide each small
semicircle into equal angles
divide semicircle into equal angles
How about general qubit decoherence?
Quadratic DD (QDD): a nesting of two types (e.g., X and Z) of UDD sequences.
X Y Z IH X B Y B Z B I B
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0T
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)
How about general qubit decoherence?
Quadratic DD (QDD): a nesting of two types (e.g., X and Z) of UDD sequences.
X Y Z IH X B Y B Z B I B
Z
X
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0T
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 Wan-Jung Kuo
How about general qubit decoherence?
'DD
, ,
( ) exp[ ] N
X Y Z
U T iTH B T
Quadratic DD (QDD): a nesting of two types (e.g., X and Z) of UDD sequences.
Decoupling order of each error type :
𝑁 𝛼−1not both even
Further nesting: NUDD, useful for multi-qubit DD
X Y Z IH X B Y B Z B I B
Z
X
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(small piece of) The DD pulse sequence zoo
the price for one qubit the payoff PeriodicDD 1
SymmetrizedDD (twice PDD) 2
ConcatenatedDD
UhrigDD (single error type only)
QuadraticDD
seq
uen
ce le
ng
th &
min
deco
up
ling
ord
er
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DD sequences battle it out numericallyJ. 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.
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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)
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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)
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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:
consists of the logical operators of the stabilizer code
- DD pulses are collective rotations of all qubits
consists of Heisenberg exchange interactions; used, e.g., to demonstrate high fidelity gates for quantum dots
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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)
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Consider a fault-tolerant simulation of a circuit
4err 0 0The noise strength: FT simulation poss ~10 ibleH
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Now prepend DD: decouple-then-compute
T
4DD eff 0 ~10The new noise strength: FT simulation possibleH T
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Noise strengths can be upper-bounded for a well-behaved bath
actually this assumption can be relaxed: see Gerardo Paz’s talk, 3:40
allows us to examine each DD-protected gate separately.
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DD-protected gates can be better
DD /
err BH HH.-K. Ng, DAL, J. Preskill, PRA 84, 012305
(2011)
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CDD-protected gates can be even better
err BH H
(opt)DD /
H.-K. Ng, DAL, J. Preskill, PRA 84, 012305 (2011)
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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
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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