Transcript Annual report on our tailored quantum error correction project, May

Tailored Quantum Error Correction
Daniel Lidar (Dept. of Chem., Univ. of Toronto)
Aephraim Steinberg (Dept. of Physics, Univ. of Toronto)
Objective: Design quantum error correction & computation
schemes that are optimized with respect to experimental
constraints and available interactions.
Motivation (Project Summary)
Question: What are main weaknesses of current theoretical
methods for universal quantum computation and quantum error
correction?
Answer: Do not take into account experimental constraints. Rely
on decoherence models that assume specific statistical
correlations and/or time-scales. No natural compatibility with
experiments.
Software Solution: “Tailored Quantum Error Correction”
Use
only naturally available interactions and external controls that are
simple to implement.
Tailor
our treatment to the experimentally measured decoherence. Seek
optimality.
Hardware Tools:
"Quantum state tomography"
"Quantum process tomography"
Adaptive tomography & error-correction
Quantum Computer Scientists
TQEC
Part I -- the experimental effort
Dramatis Personae
U of T quantum optics & laser cooling group:
PI: Aephraim Steinberg
PDFs: Morgan Mitchell (heading  Barcelona)
Marcelo Martinelli (returned  São Paolo)
TBA (contact us!)
Photons labs: Jeff Lundeen
Kevin Resch ( Zeilinger)
Rob Adamson
Masoud Mohseni ( Lidar)
Reza Mir ( real world) Lynden (Krister) Shalm
Atoms labs: Jalani Fox
Ana Jofre ( NIST)
Samansa Maneshi
Outside collaborations:
Stefan Myrskog ( Thywissen)
Mirco Siercke
Chris Ellenor
Janos Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman, Poul Jessen,...
Overview of experimental projects
-- photons lab
Two-photon switch & controlled-phase gate
Bell-state determination, etc.
Two-photon quantum tomography
Bell-state filter diagnosis
adaptive tomography + QEC
Generation of 3-photon entangled states
by postselected nonunitary operations
Lin. Opt. Implementation of the
Deutsch-Jozsa algorithm in a DFS
POVM discrimination of non-orthogonal
states… applications to QI protocols?
More qbits?
Two-photon exchange effects in pair
absorption (Franson/Sipe)
Theory & experiment on extracting weak
values of joint observables on postselected
systems
Study of which-path info and complementarity
Complete.
Considering new applications.
Discussed at last review
In progress; this talk
Completed; this talk
Complete.
Completed; extensions
under consideration.
To appear.
Published.
In preparation.
Overview of experimental projects
-- atoms lab
Quantum state tomography on atoms
in an optical lattice (W, Q, r, etc.)
Completed; article in preparation
Quantum process tomography in
optical lattices
Completed; submitted for preparation.
Bang-bang QEC (pulse echo)
Functioning; this talk.
Learning loops for optimized QEC
Using superoperator for further
characterisation of noise
(Markovian/not, etc...)
In progress; this talk.
Continuing
OUTLINE OF TALK (expt. part)
Review (density matrices & superoperators)
Adaptive tomography / DFS-search for entangled photons
- review tomography experiment
- strategies for efficiently identifying decoherence-free
subspaces
- preliminary data on adaptive tomography
Error correction in optical lattices
- review process tomography results
- pulse echo (bang-bang correction)
- preliminary data on adaptive bang-bang QEC
3-photon entanglement via non-unitary operations
Summary
Density matrices and superoperators
()
( )
One photon: H or V.
State: two coefficients
CH
CV
Density matrix: 2x2=4 coefficients
CHH CVH
CHV
CVV
Measure
intensity of horizontal
intensity of vertical
intensity of 45o
intensity of RH circular.
Propagator (superoperator): 4x4 = 16 coefficients.
Two photons: HH, HV, VH, HV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.
But is all this information needed? Is it all equally valuable?
Is it all equally expensive?
Two-photon Process Tomography
Two waveplates per photon
for state preparation
HWP
QWP
HWP
Detector A
PBS
QWP
SPDC source
"Black Box" 50/50
Beamsplitter
QWP
HWP
QWP
PBS
HWP
Detector B
Argon Ion Laser
Two waveplates per
photon for state analysis
Superoperator provides information
needed to correct & diagnose operation
Measured superoperator,
in Bell-state basis:
The ideal filter would have a
single peak.
Leading Kraus operator allows
us to determine unitary error.
Superoperator after transformation
to correct polarisation rotations:
Dominated by a single peak;
residuals allow us to estimate
degree of decoherence and
other errors.
GOALS: more efficient extraction of information for better correction of errors
iterative search for optimal encodings in presence of collective noise;...
Sometimes-Swap
Consider an optical system with
stray reflections – occasionally a
photon-swap occurs accidentally:
Two DFSs (one 1D and one
3D exist):
Consider different strategies for identifying a 2D
decoherence-free subspace...
Search strategies (simulation)
random
tomography
standard
tomography
adaptive
tomography
# of input states used
Best known
2-D DFS
(average
purity).
averages
operation:
“sometimes swap” in
random basis.
Searchlight algorithm
reconstructing a do-nothing op
Underway: application to sometimes-swap;
comparison of different algorithms for DFS-search.
Tomography in Optical Lattices
Rb atom trapped in one of the quantum levels
of a periodic potential formed by standing
light field (30GHz detuning, 10s of mK depth)
Complete characterisation of
process on arbitrary inputs?
First task: measuring state
populations
Time-resolved quantum states
Atomic state measurement
(for a 2-state lattice, with c0|0> + c1|1>)
initial state
displaced
delayed & displaced
left in
ground band
tunnels out
during adiabatic
lowering
(escaped during
preparation)
|c0|2
|c1|2
|c0 + c1 |2
|c0 + i c1 |2
Data:"W-like" [Pg-Pe](x,p) for
a mostly-excited incoherent mixture
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
(For 2-level subspace, can also
choose 4 particular measurements
and directly extract density matrix)
Extracting a superoperator:
prepare a complete set of input states and measure each output
Towards bang-bang error-correction:
pulsecomparing
echooscillations
indicates
T2 ≈ 1 ms...
for shift-backs
applied after time t
2
1/(1+2)
1.5
1
0.5
0
00
50
500
ms
100
1000
ms
t(10us)
150
1500
ms
200
2000
ms
250
"Bang" pulse for QEC
time
(shift-back,  , shift)
single shift-back
0°
0°
60°
60°
900 ms
900 ms
pulse
shift-back delay 
t=0
t=0
measurement
measurement
t
t
(Roughly equivalent to a momentum shift – in a periodic potential,
a better approximation to a p-pulse than a position shift – but as
we shall see, it may work better than expected...)
Golden Section Search algorithm
What is the optimal pulse duration?
x0
x1
x2
x3
5
80
125
200
80
125
50
5
5
35
35
50
80
50 60
80
50 60 70 80
R = 0.618 ; S = 1.0 – R
x1 = S x3 + R x0
x2 = S x3 + R x1
Dxmin = 5 ms
marks the times at which echo
amplitudes were compared.
marks the time at which
maximum echo amplitude
was found.
Sample data
amp
echo amplitudes
0.14
wave-packet amplitude
0.12
0.1
0.08
0.06
20
40
60
80
100
shift-back delay (ms)
(us)
120
140
Echo from optimized pulse
Single shift-back
pulse
Echo amplitude for a single shift-back vs.
a pulse (shift-back, delay, shift) at 900 us
1
ground state ratio
0.9
single shift-back echo
0.8
0.7
0.6
(shift-back, delay, shift) echo
0.5
0.4
0.3
0
200
400
600
800
1000 1200 1400 1600
time ( microseconds)
Highly number-entangled states
("low-noon" experiment) .
The single-photon superposition state |1,0> + |0,1>,
which may be regarded as an entangled state of two
fields, is the workhorse of classical interferometry.
The output of a Hong-Ou-Mandel interferometer is |2,0> + |0,2>.
States such as |n,0> + |0,n> ("high-noon" states, for n large) have
been proposed for high-resolution interferometry – related to
"spin-squeezed" states.
Multi-photon entangled states are the resource required for
KLM-like efficient-linear-optical-quantum-computation schemes.
A number of proposals for producing these states have been made,
but so far none has been observed for n>2.... until now!
Practical schemes?
[See for example
H. Lee et al., Phys. Rev. A 65, 030101 (2002);
J. Fiurásek,
Phys. Rev. A 65, 053818 (2002)]
˘
Important factorisation:
+
=
A "noon" state
A really odd beast: one 0o photon,
one 120o photon, and one 240o photon...
but of course, you can't tell them apart,
let alone combine them into one mode!
Trick #1
Okay, we don't even have single-photon sources.
But we can produce pairs of photons in down-conversion, and
very weak coherent states from a laser, such that if we detect
three photons, we can be pretty sure we got only one from the
laser and only two from the down-conversion...
SPDC
|0> + e |2> + O(e2)
laser
|0> +  |1> + O(2)
e |3> + O(2) + O(e 2)
+ terms with <3 photons
Trick #2
How to combine three non-orthogonal photons into one spatial mode?
"mode-mashing"
Yes, it's that easy! If you see three photons
out one port, then they all went out that port.
Trick #3
But how do you get the two down-converted photons to be at 120o to each other?
More post-selected (non-unitary) operations: if a 45o photon gets through a
polarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be
anywhere...
(or nothing)
(or nothing)
(or <2 photons)
The basic optical scheme
+ e i3
Dark ports
PBS
DC
photons
HWP
to
analyzer
PP
Phase
shifter
QWP
Ti:sa
It works!
Singles:
Coincidences:
Triple
coincidences:
Triples (bg
subtracted):
SUMMARY (expt'l TQEC)
Adaptive process tomography / error correction
being studied in both photonic and atomic systems,
and both theoretically and experimentally.
Non-unitary operations successfully used to generate
3-photon entanglement, for Heisenberg-limited
interferometry, and as a resource for LOQC.
Upcoming goals:
Develop resource-efficient algorithms for finding
DFS's, and study scaling properties; test
on photonic system.
Test learning-loop algorithms for optimizing
pulse sequences for QEC and other operations
on atomic system. Improve coherence of
real-world system with unknown noise sources!
- adaptive error correction
Extend 3-photon-generation techniques, using singlephoton sources developed by other teams.