QuantumFirmware: Engineeringerrorresistanceatthe - - PowerPoint PPT Presentation

Quantum
Firmware:


Engineering
error
resistance
at
the
 physical
level
for
quantum
computation


Support


Michael
J.
Biercuk


Quantum
Control
Laboratory


Centre
for
Engineered
Quantum
Systems
 School
of
Physics,
The
University
of
Sydney


Outline


• Motivation
(quickly)


• Quantum
Firmware
for
error
suppression


at
the
physical
level


• Noise-filtering:
An
Experiment-Friendly


analytical
approach


• Experiments
–
quantum
firmware
in
the
lab

• Noise
filtering
in
nontrivial
gates


Decoherence
due
to
environmental
noise:
 A
Fundamental
Limit


Dephasing
 Relaxation


How
do
we
deal
with
this
error?


• Closed-loop
feedback
control



⇒
Quantum
Error
Correction


• Open-loop
control



⇒Dynamical
Error
Suppression
 Aim:
Improve
QEC
performance
by
driving
 down
physical
error
rates 


Spin
Echo
to
Dynamical
Decoupling


Hahn
1950,
NMR
 Simple
application
of
control
pulses
suppresses
errors…
 Significant
flexibility
in
pulse
number,
pulse
timing,
pulse
types…
 The
art
is
in
efficient
sequencing


PRA
58 58,
2733
(1998).

PRA
59, 59,
4178
(1999).

PRL
95 95,
180501
(2005).



Vision:
“Quantum
Firmware”


• Efficient
physical-layer
error
evasion
strategy

• Simple
implementation

• Absorbed
into
machine-language


abstraction:
invisible
to
programmer


• Useful
for
*any*
quantum
technology

• Similar
to
DRAM
refreshes


What
are
our
interests:


• Making
dynamical
error
suppression
accessible


– i.e.
Please
don’t
speak
group
theory
to
me


• Calculating
error
rates
in
real
environments


instead
of
assuming
some
p


• Considering
realistic
constraints
imposed
by


hardware


• Designing
quantum
control
approaches


compatible
with
these
constraints



Noise
Filtering


Calculating
the
effect
of
fluctuating
noise


Net
Phase
is
convolution
of
noise
and
toggling
frame
 β(t)
represents
dephasing
noise
 is
the
error


Capturing
the
average
behavior


Time
Domain
Convolution

Fourier
Domain
Product


Uhrig Uhrig
et
al. et
al.,
 ,
PRL
 PRL
98,
100504
(2007);
 98,
100504
(2007);
Cywinski Cywinski
et
al.,
 
et
al.,
PRB
 PRB
77,
174509
(2008).
 77,
174509
(2008).
 Biercuk
 Biercuk
et
al et
al.,
 .,
Nature Nature
458,
996
(2009).

Biercuk
et
al.,
 ,
996
(2009).

Biercuk
et
al.,
J.
Phys.
B J.
Phys.
B
44,
154002
(2011).
 ,
154002
(2011).


Capturing
the
average
behavior


Coherence
preserved
if
 F
small
for
dominant
 frequencies


Time
Domain
Convolution

Fourier
Domain
Product


Uhrig Uhrig
et
al. et
al.,
 ,
PRL
 PRL
98,
100504
(2007);
 98,
100504
(2007);
Cywinski Cywinski
et
al.,
 
et
al.,
PRB
 PRB
77,
174509
(2008).
 77,
174509
(2008).
 Biercuk
 Biercuk
et
al et
al.,
 .,
Nature Nature
458,
996
(2009).

Biercuk
et
al.,
 ,
996
(2009).

Biercuk
et
al.,
J.
Phys.
B J.
Phys.
B
44,
154002
(2011).
 ,
154002
(2011).


Coherence:

Fully
characterize
any
sequence
by
F


Biercuk
et
al.,
 Biercuk
et
al.,
J.
Phys.
B J.
Phys.
B
44,
154002
(2011).
 ,
154002
(2011).


Filter Function Normalized Frequency []

Passband 
 Stopband 
 Rolloff
~ 
 Filter
Order 


Fully
characterize
any
sequence
by
F


Plug-In High Pass Filter PHP-1000+

Typical Performance Curves

Insertion Loss

10 20 30 40 50 60 70 80 90 100 1 10 100 1000 10000

Frequency (MHz) Insertion Loss (dB)

Biercuk
et
al.,
 Biercuk
et
al.,
J.
Phys.
B J.
Phys.
B
44,
154002
(2011).
 ,
154002
(2011).


Timing
Constraints
Reduce
Performance


(Daniel
didn’t
tell
the
whole
story…there
are
other
prices
to
pay)


Dimensionless Angular Frequency ( 2 Hz ) Filter Function

10
GHz
clocking
in
1
ms
experiment
 reduces
performance
by
80
dB
@
ωτ = 1



Biercuk,
Doherty,
Uys,
J.
Phys.
B
44 44,
154002
(2011).


Digital
modulation
for
large
systems


Hayes,
Khodjasteh,
Viola,
Biercuk,
arXiv:
1109.6002,
to
appear
in
PRA


Time Bin Normalized Pulse Location Walsh Index, n

For
more
see
K.
Khodjasteh’s
talk,
Thursday
2:40
PM 
 The
Walsh
Functions


Other
benefits:
ease
of
sequencing,
unified
framework,
long-time
storage…


Quantum
Firmware
and
Noise
 Filtering
in
the
lab


Experimental
Quantum
System


Crystal
of
Beryllium
ions


Ions
in
a
Penning
Trap


B=4.5
T
 B=4.5
T
 νc ~ 7.6
MHz 7.6
MHz νz ~ 600
kHz
 600
kHz
 νm ~ 20
kHz 20
kHz

9Be+


Field
Sensitive
 124
GHz


Biercuk
et
al,.
Nature
458 458,
996
(2009).
Biercuk
et
al.,
Quant.
Info.
Comp.
9,
920
(2009).



Fluorescence
 Cooling


9Be+
at
4.5T


F=1
 F=2


DD
suppresses
decoherence


Biercuk
et
al.,
Quant.
Info.
Comp.
9,
920
(2009).


1.0 0.8 0.6 0.4 0.2 0.0

Population !"#!

4 3 2 1

Ramsey Precession Time (ms)

T2~2.5
ms


DD
suppresses
decoherence


Biercuk
et
al.,
Quant.
Info.
Comp.
9,
920
(2009).
 1.0 0.8 0.6 0.4 0.2 0.0

Probability !"!#

60 50 40 30 20 10

Free-Precession Time (ms)

Free-Induction
 Decay
 10-pulse
CPMG


Coherence
time
increases
~10X
 (for
10
pulses)


The
filter
function
works!


2.5 2.0 1.5 1.0 0.5 0.0 50 40 30 20 10 Total Free Precession Time (ms) n=4! n=5! n=6! n=8! n=10!

Solid:
CPMG,
Open:
UDD
(Different
Pulse
Spacings)


Error
Probability


Biercuk
et
al.,
Nature
458 458,
996
(2009).
Biercuk
et
al.,
Phys.
Rev.
A
79 79
062324
(2009).


The
filter
function
works!


2.5 2.0 1.5 1.0 0.5 0.0 50 40 30 20 10 Total Free Precession Time (ms) n=4! n=5! n=6! n=8! n=10!

Solid:
CPMG,
Open:
UDD
(Different
Pulse
Spacings)


Error
Probability


Biercuk
et
al.,
Nature
458 458,
996
(2009).
Biercuk
et
al.,
Phys.
Rev.
A
79 79
062324
(2009).


Noise
Injection:
Modeling
Other
Qubits


10

• 3

10

• 2

10

• 1

10 10

1

10

2

10

3

10

4

10

5 6 8

10

2 2 4 6 8

10

3 2 4 6 8

10

4

10

• 2

10

• 1

10 10

1

10

2

10

3

10

4

10

5 6 8

10

2 2 4 6 8

10

3 2 4 6 8

10

4

! (2"Hz)

1/ω


Biercuk
et
al.,
Nature
458,
996
(2009).
Biercuk
et
al.,
Phys.
Rev.
A
79 79
062324
(2009).


UDD
can
outperform
CPMG


Error
Probability


MJB
et
al.,
Nature
458 458,
996
(2009).
MJB
et
al.,
Phys.
Rev.
A
79 79
062324
(2009).,
MJB
et
al
QIC
9,
920
(2009)


2.5 2.0 1.5 1.0 0.5 0.0 Error Probability 4 3 2 1 Free Precession Time (ms)

VN= 0.9 V 0.7 V 0.5 V 0.3 V 0.1 V

Red=UDD,
Black=CPMG
 Noise
Strength
 UDD
 Better
 CPMG
 Better


10

• 5

10

• 2

10

1

10

4

S() (rad/s)

7 8 9

10

2 2 3 4 5 6 7 8 9

10

3 2 3 4 5 6 7 8 9

10

4 2

Angular Frequency (rad/s) (c)

Optimization
by
autonomous
feedback


0.001

2 4

0.01

2 4

0.1

2 4

1 Error Probability 4 3 2 1 Total Free Precession Time (ms)

CPMG UDD LODD

5X
improvement
over
UDD
 8X
improvement
over
CPMG


We
have
realized
improved
performance
by
tailoring
the
filter
function 


Biercuk
et
al.,
Nature
458 458,
996
(2009)


Interlude:
Mike’s
opinion
on
UDD


Dynamical
decoupling
studies
w/ions…


Why
did
this
work?
High-fidelity
control.


1.00 0.95 0.90 0.85 0.80 0.75

Average Fidelity

200 150 100 50

Computational Gate Number

PRL
97 97,
220407
(2006).
PRA
77 77,
012307
(2008).
Biercuk
et
al.,
Quant.
Info.
Comp.
9,
920
(2009).


Need
to
develop
new
 Need
to
develop
new
flexible flexible
high-fidelity
 
high-fidelity
 control
hardware,
and
understand
gate
errors control
hardware,
and
understand
gate
errors 


99.92%
Fidelity/Gate


1 2.0 1.5 1.0 0.5

Pulse Length (ms) !"=185 !s

Dynamical
protection
beyond
Memory


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Dynamical
protection
beyond
Memory


“Decouple-then-compute”
ignores
noise
during
operations
 Key
Question:
 How
do
we
accurately
calculate
and
improve
gate
error
(“p”)
 in
the
presence
of
experimental
noise?


Noise filtering in nontrivial quantum logic gates

Todd Green,1 Hermann Uys,2 and Michael J. Biercuk1, ∗

1Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006 Australia 2National Laser Centre, Council for Scientific and Industrial Research, Pretoria, South Africa

(Dated: November 1, 2011) Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Dephasing
effect
on
a
gate


Detuning
during
a
control
operation
 gives
under-rotation
and
phase
shift 


• General
depolarization
errors
can
arise
from
pure
dephasing
noise

• Time
variation
in
detuning
generally
difficult
to
treat
analytically

• Sources
include
environmental
fluctuations
and
master-oscillator
instability


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


H(t)
defined
piecewise
 Control
propagator
 Effective
Hamiltonian
 Lowest
order
Magnus


Effective
Hamiltonian
Theory


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Noise
filtering
for
nontrivial
logic


Any
piecewise
constant
control
function
can
be
treated
in
this
way.


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


is
the
Filter
Function
 Given
an
effective
Hamiltonian,
calculate
the
gate
fidelity


Filter
functions
for
piecewise
 constant
control


Sequence
of
interest
 Closed-form
solutions
for
the
filter
function
are
possible


This
approach
covers
DD
(with
nonzero
pulses),
primitive
gates,
and
many
DCGs


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Filter
functions
for
piecewise
 constant
control


Sequence
of
interest
 Closed-form
solutions
for
the
filter
function
are
possible


This
approach
covers
DD
(with
nonzero
pulses),
primitive
gates,
and
many
DCGs


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Dephasing


Filter
functions
for
piecewise
 constant
control


Sequence
of
interest
 Closed-form
solutions
for
the
filter
function
are
possible


Polarization
 Damping


This
approach
covers
DD
(with
nonzero
pulses),
primitive
gates,
and
many
DCGs


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Dephasing


Filter
functions
for
piecewise
 constant
control


Sequence
of
interest
 Closed-form
solutions
for
the
filter
function
are
possible


Polarization
 Damping


This
approach
covers
DD
(with
nonzero
pulses),
primitive
gates,
and
many
DCGs


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Dephasing


Dynamically
protected
gates


Time


Dynamically
 corrected
“NOT”
gate 


Khodjasteh
and
Viola,
PRA
80,
032314
(2009).



Time


Filter
Functions
for
DCGs


10

5

10

• 5

10

• 15

10

• 25

Filter Function

10

• 6

10

• 4

10

• 2

10 10

2

x

Primitive DCG

X X X

• X X1/2

5

(a)

Effect
of
 extended
time 
 Effect
of
dynamic
 protection 


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Validating
the
Filter
Function
(Numerics)


10 10

• 3

10

• 6

10

• 9

1 10 100 x [x

(Min) ]

Analytical: Primitive X Numeric: Primitive X Analytical: DCG X Numeric: DCG X

(c)

S = -1; = 1e-1 [10-5, 10-2] 2 / x

(Min)

Filter
Function
approach
agrees
with
brute-force
numerics


Error
Probability


Green,
Uys,
&
Biercuk,
arXiv:1110.6686


Next:
exploit
flexible
control


Next:
exploit
flexible
control


Summary


• Approach:
physical-layer
error
evasion


strategies
–
Quantum
Firmware


– New
analytical
approach
based
on
noise
filters
 – Approaches
compatible
with
large-scale
systems
 – Filtering
during
gate
operations
 – Experimental
demonstrations
at
a
small
scale


• We’re
bringing
a
“30,000
foot”
viewpoint
to


these
analyses.


• QC
isn’t
the
only
application…


Acknowledgements
&
Collaborators


Todd
Green,
 Andrew
Doherty,
 Stephen
Bartlett
 Lorenza
Viola
 Kaveh
Khodjasteh
 Hermann
Uys


Quantum
Firmware
Collaboration


Michael
J.
Biercuk


www.physics.usyd.edu.au/~mbiercuk


Lorenza
Viola
 Amir
Yacoby


yacoby.physics.harvard.edu/
 www.dartmouth.edu/~viola/
 Support


PhD
opportunities
and
postdoctoral
 fellowships
available
in
my
Group
 michael.biercuk@sydney.edu.au