Inaccuracy in Quantum State Tomography - What is to Be Done?

A talk I gave at the 2015 APS Four Corners meeting in Tempe, AZ.

Screencast available at https://youtu.be/D1BVh1sHmXU

Transcript

1. Inaccuracy in Quantum State Tomography - What is to Be Done? Travis L Scholten @Travis_Sch CQuIC @ UNM CCR @ SNL APS Four Corners Meeting ASU @ Tempe, AZ 2015 October 16 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
2. Let’s talk about characterizing quantum devices. Characterization Control Computation What Architectures/Platforms?
3. Let’s talk about characterizing quantum devices. Characterization = Control = Computation superconducting circuit ion trap quantum dot
4. Photonic devices would be really, really useful… …if we can characterize quantum states of light.
5. Light is big, in terms of Hilbert space. A bit too big. ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A Estimating quantum state of light requires infinite number of parameters! Density Matrix Wigner function
6. Estimation problems come up a lot in physics… Let’s fit this data…how?
7. …and infinity is a big number! ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A Few parameters A lot of parameters! Pretty well understood Not well understood
8. Is there a finite dimensional estimate which is a good model? Depends on what you mean by “good”.
9. Let’s find a good model using model selection. Let’s fit this data…how?
10. Model selection chooses “best” model. Which of these fits is better?
11. Which is “best” depends on how you score the model. Several tools… …such as information criteria… …like the Akaike Information Criterion
12. The only math formula we need: Goodness of fit Complexity Each useless parameter can contribute 1 unit of goodness of fit AIC (model) = 2 log Pr (Data seen | model) + 2(# of parameters)
13. Useless parameters aren’t necessary, but can be fit. Why not a cubic?
14. What about boundaries? What about constraints? Each useless parameter can contribute 1 unit of goodness of fit The relationship between uselessness and goodness is subtle.
15. Constraints mean parameters not independent. AIC overestimates complexity penalty (p1, 1 p1, 0) (p1, p2, 1 (p1 + p2)) AIC should not be used
16. We need new model selection tools for tomography. Notion of effective number of parameters… …which may depend on the true parameter! 0 1 1 2
17. 0 1 1 2 Characterization Control Computation
18. Images: Wigner function: By Gerd Breitenbach (dissertation) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY- SA-3.0 (http://creativecommons.org/licenses/by-sa/ 3.0/)], via Wikimedia Commons Laser: ”Military laser experiment" by US Air Force - This Image was released by the United States Air Force with the ID 090809-F-5527s-0001 gmon transmon qubit - Michael Fang, John Martinis group, Google; http://web.physics.ucsb.edu/ ~martinisgroup/photos.shtml