Tailored Quantum Error CorrectionDaniel 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.

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.

Motivation (Project Summary)

Hardware Tools: "Quantum state tomography""Quantum process tomography"Adaptive tomography & error-correction

Quantum Computer Scientists

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,...

Dramatis Personae

TQECPart I -- the experimental effort

Overview of experimental projects-- photons lab

Generation of 3-photon entangled states by postselected nonunitary operations

Two-photon quantum tomographyBell-state filter diagnosis

Lin. Opt. Implementation of the Deutsch-Jozsa algorithm in a DFS

Two-photon exchange effects in pair absorption (Franson/Sipe)

Two-photon switch & controlled-phase gateBell-state determination, etc.

adaptive tomography + QEC

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.

POVM discrimination of non-orthogonal states… applications to QI protocols? More qbits?

Theory & experiment on extracting weak values of joint observables on postselected systems

Study of which-path info and complementarity

Overview of experimental projects-- atoms lab

Quantum state tomography on atoms in an optical lattice (W, Q, , etc.)

Quantum process tomography in optical lattices

Using superoperator for further characterisation of noise (Markovian/not, etc...)

Bang-bang QEC (pulse echo)

Learning loops for optimized QEC

Completed; article in preparation

Completed; submitted for preparation.

Functioning; this talk.

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 superoperatorsOne photon: H or V.State: two coefficients ()CHCV

( )CHH

CHV

CVH

CVV

Density matrix: 2x2=4 coefficientsMeasure

intensity of horizontalintensity of verticalintensity of 45ointensity 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?

HWP

HWP

HWP

HWP

QWP

QWPQWP

QWPPBS

PBS

Argon Ion Laser

Beamsplitter"Black Box" 50/50

Detector B

Detector ATwo waveplates per photonfor state preparation

Two waveplates per photon for state analysis

SPDC source

Two-photon Process Tomography

Superoperator after transformationto correct polarisation rotations:

Dominated by a single peak;residuals allow us to estimatedegree of decoherence andother errors.

Superoperator provides informationneeded to correct & diagnose operationMeasured superoperator,in Bell-state basis:

The ideal filter would have a single peak.Leading Kraus operator allowsus to determine unitary error.

GOALS: more efficient extraction of information for better correction of errorsiterative search for optimal encodings in presence of collective noise;...

Sometimes-Swap

Consider an optical system withstray reflections – occasionally aphoton-swap occurs accidentally:

Two DFSs (one 1D and one3D exist):

Consider different strategies for identifying a 2D decoherence-free subspace...

Search strategies (simulation)

randomtomography

standardtomography

adaptivetomography

Best known2-D DFS (averagepurity).

# of input states used

operation: “sometimes swap” in random basis.averages

Searchlight algorithmreconstructing a do-nothing op

Underway: application to sometimes-swap; comparison of different algorithms for DFS-search.

Rb atom trapped in one of the quantum levelsof a periodic potential formed by standing

light field (30GHz detuning, 10s of K depth)

Tomography in Optical Lattices

Complete characterisation ofprocess 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>)

left inground band

tunnels outduring adiabaticlowering

(escaped duringpreparation)

initial state displaced delayed & displaced

|c0|2 |c0 + c1 |2 |c0 + i c1 |2

|c1|2

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

Data:"W-like" [Pg-Pe](x,p) for a mostly-excited incoherent mixture

(For 2-level subspace, can alsochoose 4 particular measurementsand directly extract density matrix)

Extracting a superoperator:prepare a complete set of input states and measure each output

0

0.5

1

1.5

2

0 50 100 150 200 250

comparing oscillations for shift-backs applied after time t

1/(1+2)

t(10us)

0 500 s 1000 s 1500 s 2000 s

Towards bang-bang error-correction:pulse echo indicates T2 ≈ 1 ms...

0° 60°

t

t = 0

900 s

measurement

time0° 60°

t

shift-back delayt = 0

900 s

pulse

measurement

(shift-back, , shift) single shift-back

"Bang" pulse for QEC

(Roughly equivalent to a momentum shift – in a periodic potential,a better approximation to a -pulse than a position shift – but as

we shall see, it may work better than expected...)

5 12580 200

5 80 12550

35 50 8060

5 50 8035

806050 70

marks the times at which echo amplitudes were compared.

marks the time at whichmaximum echo amplitude was found.

x0 x1 x2 x3

R = 0.618 ; S = 1.0 – Rx1 = S x3 + R x0

x2 = S x3 + R x1

xmin = 5 s

Golden Section Search algorithmWhat is the optimal pulse duration?

0.06

0.08

0.1

0.12

0.14

20 40 60 80 100 120 140

echo amplitudes

amp

wave-packet amplitude

shift-back delay (us)s)

Sample data

time ( microseconds)

single shift-back echo

(shift-back, delay, shift) echo

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200 1400 1600

Echo amplitude for a single shift-back vs. a pulse (shift-back, delay, shift) at 900 us

Single shift-backpulse

ground state ratio

Echo from optimized pulse

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 forKLM-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!

[See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002);J. Fiurásek, Phys. Rev. A 65, 053818 (2002)]˘

Practical schemes?

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, andvery weak coherent states from a laser, such that if we detectthree photons, we can be pretty sure we got only one from thelaser and only two from the down-conversion...

SPDC

laser

|0> + |2> + O(2)

|0> + |1> + O(2)

|3> + O(2) + O(2) + terms with <3 photons

Trick #2

How to combine three non-orthogonal photons into one spatial mode?

Yes, it's that easy! If you see three photonsout one port, then they all went out that port.

"mode-mashing"

Trick #3But 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 apolarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be anywhere...

(or nothing)

(or <2 photons)

(or nothing)

-

+ei3φ

HWP

QWP

Phaseshifter

PBS

DCphotons PP

Dark ports

Ti:sa

to analyzer

The basic optical scheme

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

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 generate3-photon entanglement, for Heisenberg-limitedinterferometry, and as a resource for LOQC.

Upcoming goals:Develop resource-efficient algorithms for finding

DFS's, and study scaling properties; teston photonic system.

Test learning-loop algorithms for optimizing pulse sequences for QEC and other operationson atomic system. Improve coherence ofreal-world system with unknown noise sources!- adaptive error correction

Extend 3-photon-generation techniques, using single-photon sources developed by other teams.