1 / 100

J. Eisert

Optimizing linear optics quantum gates. J. Eisert. University of Potsdam, Germany. Entanglement and transfer of quantum information Cambridge, September 2004 . Quantum computation with linear optics. Effective non-linearities.

randyr
Download Presentation

J. Eisert

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related