Tailored Quantum Error Correction

1. 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.

2. 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

3. Quantum Computer Scientists

4. 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,...

5. Overview of experimental projects-- photons lab Two-photon switch & controlled-phase gate 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. 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

6. Overview of experimental projects-- atoms lab Quantum state tomography on atoms in an optical lattice (W, Q, r, etc.) Quantum process tomography in optical lattices 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

7. 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

8. Density matrices and superoperators But is all this information needed? Is it all equally valuable? Is it all equally expensive?

9. Two-photon Process Tomography "Black Box" 50/50 Beamsplitter 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

10. Superoperator provides informationneeded to correct & diagnose operation Measured superoperator, in Bell-state basis: Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors. 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;...

12. 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...

13. 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

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

15. 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?

16. First task: measuring state populations

17. Time-resolved quantum states

18. 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 + i c1 |2 |c0|2 |c0 + c1 |2 |c1|2

19. 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)

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

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

22. "Bang" pulse for QEC 0° 60° time 0° 60° 900s 900s pulse  shift-back delay t = 0 t = 0 t t measurement measurement (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...)

23. Golden Section Search algorithm 80 125 200 5 80 125 50 5 R = 0.618 ; S = 1.0 – R x1 = S x3 + R x0 x2 = S x3 + R x1 Dxmin = 5 ms 80 5 35 50 marks the times at which echo amplitudes were compared. 50 80 35 60 marks the time at which maximum echo amplitude was found. 60 70 80 50 What is the optimal pulse duration? x0 x1 x2 x3

24. Sample data (ms)

25. Echo from optimized pulse single shift-back echo (shift-back, delay, shift) echo time ( microseconds)

26. 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!

27. Practical schemes? [See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)] ˘ + = 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! Important factorisation:

28. Trick #1 SPDC laser 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... |0> + e |2> + O(e2)  |3> + O(2) + O( 2) + terms with <3 photons |0> +  |1> + O(2)

29. Trick #2 "mode-mashing" Yes, it's that easy! If you see three photons out one port, then they all went out that port. How to combine three non-orthogonal photons into one spatial mode?

30. Trick #3 (or nothing) (or nothing) (or <2 photons) 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...

31. The basic optical scheme

32. It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted):

33. 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.