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J. Eisert

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  1. Optimizing linear optics quantum gates J. Eisert University of Potsdam, Germany Entanglement and transfer of quantum information Cambridge, September 2004

  2. Quantum computation withlinear optics

  3. Effective non-linearities • Photons are relatively prone to decoherence, precise state control is possible with linear optical elements • Universal quantum computation can be done using optical systems only • The required non-linearities can be effectively obtained … Input Output Opticalnetwork

  4. Effective non-linearities • Photons are relatively prone to decoherence, precise state control is possible with linear optical elements • Universal quantum computation can be done using optical systems only • The required non-linearities can be effectively obtained … Input Output Opticalnetwork

  5. ? Effective non-linearities • Photons are relatively prone to decoherence, precise state control is possible with linear optical elements • Universal quantum computation can be done using optical systems only • The required non-linearities can be effectively obtained … Input Output Opticalnetwork

  6. Effective non-linearities • Photons are relatively prone to decoherence, precise state control is possible with linear optical elements • Universal quantum computation can be done using optical systems only • The required non-linearities can be effectively obtained … Input Output Linear opticsnetwork Auxiliary modes, Auxiliary photons Measurements • by employing appropriate measurements

  7. KLM scheme Knill, Laflamme, Milburn (2001): Universal quantum computation is possible with • Single photon sources • linear optical networks • photon counters, followed by postselection and feedforward Input Output Linear opticsnetwork Auxiliary modes, Auxiliary photons Measurements E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) TB Pittman, BC Jacobs, JD Franson, Phys Rev A 64 (2001) JL O’ Brien, GJ Pryde, AG White, TC Ralph, D Branning, Nature 426 (2003)

  8. Non-linear sign shifts • At the foundation of the KLM contruction is a non-deterministic gate, - the non-linear sign shift gate, acting as • Using two such non-linear sign shifts, one can construct a control-sign and a control-not gate NSS NSS

  9. Success probabilities • At the foundation of the KLM contruction is a non-deterministic gate, - the non-linear sign shift gate, acting as • Using teleportation, the overall scheme can be uplifted to a scalable scheme with close-to-unity success probability, using a significant overhead in resources • To efficiently use the gates, one would like to implement them with as high a probability as possible

  10. Central question of the talk • How well can the elementary gates be performed with - static networks of arbitrary size, - using any number of auxiliary modes and photons, - making use of linear optics and photon counters, followed by postselection? • Meaning, what are the optimal success probabilities of elementary gates?

  11. Central question of the talk • How well can the elementary gates be performed with - static networks of arbitrary size, - using any number of auxiliary modes and photons, - making use of linear optics and photon counters, followed by postselection? • Meaning, what are the optimal success probabilities of elementary gates? Seems a key question for two reasons: • Quantity that determines the necessary overhead in resources • For small-scale applications such as quantum repeaters, high fidelity of the quantum gates may often be the demanding requirement of salient interest (abandon some of the feed-forward but rather postselect)

  12. Networks for the non-linear sign shift Input: Output:

  13. Networks for the non-linear sign shift Input: Output: • Success probability (obviously, as thenon-linearity is not available) Network of linear optics elements

  14. Networks for the non-linear sign shift Input: Output: Photon counter Auxiliary mode • Success probability (the relevant constraints cannot be fulfilled) Network of linear optics elements

  15. Networks for the non-linear sign shift Input: Output: Photon counters Auxiliary modes • Success probability (the best known scheme has this success probability Network of linear optics elements

  16. Networks for the non-linear sign shift Input: Output: E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) Alternative schemes: S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A68 (2003) TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A 65 (2001) • Success probability (the best known scheme has this success probability

  17. Networks for the non-linear sign shift Input: Output: Photon counters Auxiliary modes • Success probability Network of linear optics elements

  18. Networks for the non-linear sign shift Input: Output: Photon counters Auxiliary modes • Success probability Network of linear optics elements

  19. Short history of the problem for the non-linear sign-s • Knill, Laflamme, Milburn/Ralph, White, Munro, Milburn, Scheel, Knight (2001-2003): Construction of schemes that realize a non-linear sign shift with success probability 1/4 • Knill (2003): Any scheme with postselected linear optics cannot succeed with a higher success probability than 1/2 • Reck, Zeilinger, Bernstein, Bertani (1994)/ Scheel, Lütkenhaus (2004): Network can be written with a single beam splitter communicating with the input Conjectured that probability 1/4 could already be optimal • Aniello (2004) Looked at the problem with exactly one auxiliary photon E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A 65 (2001) S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A 68 (2003) M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

  20. (A late) overview over the talk • Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics • Why is this a difficult problem? J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

  21. (A late) overview over the talk • Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics • Why is this a difficult problem? • Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

  22. (A late) overview over the talk • Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics • Why is this a difficult problem? • Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools • Formulate strategy: will developa general recipe to giverigorous bounds on success probabilities • Look at more general settings, work in progress J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

  23. (A late) overview over the talk • Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics • Why is this a difficult problem? • Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools • Formulate strategy: will developa general recipe to giverigorous bounds on success probabilities • Look at more general settings, work in progress • Finally: stretch the developed ideas a bit further: • Experimentally accessible entanglement witnesses for imperfect photon detectors • Complete hierarchies of tests for entanglement J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

  24. Quantum gates Input: Output: • These are the quantum gates we will be looking at in the following (which include the non-linear sign shift)

  25. Quantum gates Input: Output: Arbitrary number of additional fieldmodesauxiliary photons (Potentially complex) networks of linear optics elements

  26. Quantum gates Input: Output: Arbitrary number of additional fieldmodesauxiliary photons (Potentially complex) networks of linear optics elements

  27. Quantum gates Input: Output: Arbitrary number of additional fieldmodesauxiliary photons (Potentially complex) networks of linear optics elements M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

  28. The input is linked only once to the auxiliary modes Input: Output: • State vector of auxiliary modes “preparation” • “measure- ment” (Potentially complex) networks of linear optics elements M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

  29. Finding the optimal success probability Input: Output: • State vector of auxiliary modes “preparation” • “measure- ment”

  30. Finding the optimal success probability • Single beam splitter, characterized by complex transmittivity • State vector of auxiliary modes “preparation” • “measure- ment”

  31. Finding the optimal success probability • Arbitrarily many ( ) states of arbitrary or infinite dimension • State vector of auxiliary modes “preparation” • “measure- ment”

  32. Finding the optimal success probability • Arbitrarily many ( ) states of arbitrary or infinite dimension • State vector of auxiliary modes “preparation” • “measure- ment” • Weights • Non-convex function (exhibiting many local minima)

  33. The problem with non-convex problems • This innocent-looking problem of finding the optimal success probability may be conceived as an optimization problem, but one which is - non-convex and - infinite dimensional,as we do not wish to restrict the number of - photons in the auxiliary modes - auxiliary modes - linear optical elements

  34. The problem with non-convex problems • This innocent-looking problem of finding the optimal success probability may be conceived as an optimization problem, but one which is - non-convex and - infinite dimensional,as we do not wish to restrict the number of - photons in the auxiliary modes - auxiliary modes - linear optical elements Infinitely many local maxima

  35. The problem with non-convex problems Infinitely many local maxima

  36. The problem with non-convex problems Infinitely many local maxima

  37. The problem with non-convex problems Infinitely many local maxima

  38. The problem with non-convex problems • Somehow, it would be good to arrive from the “other side” Infinitely many local maxima

  39. The problem with non-convex problems • Somehow, it would be good to arrive from the “other side” Infinitely many local maxima

  40. The problem with non-convex problems • Somehow, it would be good to arrive from the “other side” Infinitely many local maxima

  41. The problem with non-convex problems • Somehow, it would be good to arrive from the “other side” • This is what we will be trying to do… Infinitely many local maxima

  42. Convex optimization? Can it help?

  43. Convex optimization problems • What is a convex optimization problem again? • Find the minimum of a convex function over a convex set

  44. Convex optimization problems • What is a convex optimization problem again? • Find the minimum of a convex function over a convex set Function Set

  45. Convex optimization problems • What is a convex optimization problem again? • Find the minimum of a convex function over a convex set Function Set

  46. Semidefinite programs • Class of convex optimization problems that we will make use of - is efficiently solvable (but we are now not primarily dealing with numerics), - and is a powerful analytical tool: • So-called semidefinite programs Function Set

  47. Semidefinite programs • Class of convex optimization problems that we will make use of - is efficiently solvable (but we are now not primarily dealing with numerics), - and is a powerful analytical tool: • So-called semidefinite programs Linear function Vector Minimize the linear multivariate function subject to the constraint Matrices Set • We will see in a second why they are so helpful

  48. Yes, ok, … … but why should this help us to assess the performance of quantum gates in the context of linear optics?

  49. 1. Recasting the problem • Again, the output of the quantum network, depending on preparations and measurements, can be written as • Functioning of the gate requires that for all • Here J Eisert, quant-ph/0409156

  50. 1. Recasting the problem • After all, the (i) success probability should be maximized, (ii) provided that the gate works • Functioning of the gate requires that for all • Here J Eisert, quant-ph/0409156