Tailored Quantum Error Correction

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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.
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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
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Quantum Computer Scientists
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TQECPart 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 Stefan Myrskog (? Thywissen) Ana Jofre (?
NIST) Mirco Siercke Samansa Maneshi Chris
Ellenor Outside collaborations Janos Bergou,
Mark Hillery, John Sipe, Paul Brumer, Howard
Wiseman, Poul Jessen,...
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Overview of experimental projects-- photons lab
Two-photon switch controlled-phase gate
Complete. Considering new applications. Discuss
ed at last review In progress this
talk Completed this talk Complete. Completed
extensions under consideration. To
appear. Published. In preparation.
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
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Overview of experimental projects-- atoms lab
Completed article in preparation Completed
submitted for preparation.
Bang-bang QEC (pulse echo) Learning loops for
optimized QEC
Functioning this talk.
In progress this talk.
Using superoperator for further
characterisation of noise (Markovian/not,
etc...)
Continuing
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OUTLINE OF TALK (expt. part)
Review (density matrices superoperators) Adapti
ve 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
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Density matrices and superoperators
But is all this information needed? Is it all
equally valuable? Is it all equally expensive?
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Two-photon Process Tomography
Two waveplates per photon for state preparation
Detector A
HWP
HWP
PBS
QWP
QWP
SPDC source
QWP
QWP
PBS
HWP
HWP
Detector B
Argon Ion Laser
Two waveplates per photon for state analysis
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Superoperator provides informationneeded to
correct diagnose operation
GOALS more efficient extraction of information
for better correction of errors iterative search
for optimal encodings in presence of collective
noise...
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(No Transcript)
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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...
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Search strategies (simulation)
standard tomography
random tomography
adaptive tomography
of input states used
Best known 2-D DFS (average purity).
operation sometimes swap in random basis.
averages
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Searchlight algorithmreconstructing a do-nothing
op
Underway application to sometimes-swap
comparison of different algorithms for
DFS-search.
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Tomography in Optical Lattices
Complete characterisation of process on arbitrary
inputs?
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First task measuring state populations
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Time-resolved quantum states
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Atomic state measurement(for a 2-state lattice,
with c00gt c11gt)
initial state
displaced
delayed displaced
left in ground band

tunnels out during adiabatic lowering
(escaped during preparation)
c0 i c1 2
c02
c0 c1 2
c12
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Data"W-like" Pg-Pe(x,p) for a mostly-excited
incoherent mixture
(For 2-level subspace, can also choose 4
particular measurements and directly extract
density matrix)
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Extracting a superoperatorprepare a complete
set of input states and measure each output
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Superoperator for resonant drive
Operation ?x (resonantly couple 0 and 1 by
modulating lattice periodically) Measure
superoperator to diagnose single-qubit
operation (and in future, to correct for errors
and decoherence)
Observed Bloch sphere
Upcoming goals generate tailored pulse sequences
to preserve coherence determine whether
decoherence is Markovian et cetera.
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Towards bang-bang error-correctionpulse echo
indicates T2 1 ms...
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"Bang" pulse for QEC
(shift-back, ? , shift)
single shift-back
(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...)
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Golden Section Search algorithm
What is the optimal pulse duration?
x0
x1
x2
x3
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Sample data
(ms)
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Echo from optimized pulse
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Highly number-entangled states("low-noon"
experiment) .
The single-photon superposition state 1,0gt
0,1gt, 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 2,0gt 0,2gt.
States such as n,0gt 0,ngt ("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 ngt2.... until now!
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Practical schemes?
Important factorisation
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Non-unitary operations in the popular press
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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...
0gt e 2gt O(e2)
?? 3gt O(?2) O(? 2) terms with lt3 photons
0gt ? 1gt O(?2)
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Trick 2
How to combine three non-orthogonal photons into
one spatial mode?
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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...
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The basic optical scheme
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More detailed schematic of experiment
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It works!
Singles
Coincidences
Triple coincidences
Triples (bg subtracted)
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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 interferomet
ry, 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 single- photon sources
developed by other teams.