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.

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 LundeenKevin Resch (  Zeilinger) Rob AdamsonMasoud Mohseni (  Lidar) Reza Mir (  real world) Lynden (Krister) Shalm Atoms labs: Jalani FoxStefan Myrskog (  Thywissen) Ana Jofre (  NIST)Mirco Siercke Samansa ManeshiChris Ellenor Outside collaborations: Janos Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman, Poul Jessen,... Dramatis Personae TQEC Part I -- the experimental effort

Overview of experimental projects -- photons lab Generation of 3-photon entangled states by postselected nonunitary operations Two-photon quantum tomography Bell-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 gate Bell-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 superoperators But is all this information needed? Is it all equally valuable? Is it all equally expensive?

HWP QWP PBS Argon Ion Laser Beamsplitter "Black Box" 50/50 Detector B Detector A Two waveplates per photon for state preparation Two waveplates per photon for state analysis SPDC source Two-photon Process Tomography

Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors. 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. 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 Best known 2-D DFS (average purity). # of input states used operation: “sometimes swap” in random basis. averages

Searchlight algorithm reconstructing a do-nothing op Underway: application to sometimes-swap; comparison of different algorithms for DFS-search.

Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of  K depth) Tomography in Optical Lattices 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 c 0 |0> + c 1 |1>) left in ground band tunnels out during adiabatic lowering (escaped during preparation) initial statedisplaceddelayed & displaced |c 0 | 2 |c 0 + c 1 | 2 |c 0 + i c 1 | 2 |c 1 | 2

Data:"W-like" [P g -P e ](x,p) for a mostly-excited incoherent mixture (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

Observed Bloch sphere Bloch sphere predicted from truncated harmonic-oscillator plus decoherence as measured previously. 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) Upcoming goals: generate tailored pulse sequences to preserve coherence; determine whether decoherence is Markovian; et cetera.

0 500  s 1000  s 1500  s 2000  s Towards bang-bang error-correction: pulse echo indicates T2 ≈ 1 ms...

0°0° 60 ° t t =  s measurement time 0°0° 60 °  t shift-back delay t =  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...)

marks the times at which echo amplitudes were compared. marks the time at which maximum echo amplitude was found. x0x0 x1x1 x2x2 x3x3 R = ; S = 1.0 – R x 1 = S x 3 + R x 0 x 2 = S x 3 + R x 1  x min = 5  s Golden Section Search algorithm What is the optimal pulse duration?

 s) Sample data

time ( microseconds ) single shift-back echo (shift-back, delay, shift) echo 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 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!

[See for example H. Lee et al., Phys. Rev. A 65, (2002); J. Fiurásek, Phys. Rev. A 65, (2002)] ˘ Practical schemes? Important factorisation: =+ A "noon" state A really odd beast: one 0 o photon, one 120 o photon, and one 240 o photon... but of course, you can't tell them apart, let alone combine them into one mode!

Non-unitary operations in the popular press

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 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 photons out one port, then they all went out that port. "mode-mashing"

Trick #3 But how do you get the two down-converted photons to be at 120 o to each other? More post-selected (non-unitary) operations: if a 45 o photon gets through a polarizer, it's no longer at 45 o. If it gets through a partial polarizer, it could be anywhere... (or nothing) (or <2 photons) (or nothing)

The basic optical scheme

More detailed schematic of experiment

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 single- photon sources developed by other teams.