Towards a Model Selection Rule for Quantum State Tomography

A talk I gave at the annual March meeting of the American Physical Society.

Transcript

1. 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. CCR Center for Computing Research Towards a Model Selection Rule for Quantum State Tomography Travis L Scholten @Travis_Sch Center for Quantum Information and Control, UNM Center for Computing Research, Sandia National Labs APS March Meeting 2016 March 14
2. Characterization of quantum devices gets hard as we scale them up. One qubit Two qubits N qubits Three qubits ˆ ⇢ = ✓ ⇢00 ⇢01 ⇢10 ⇢11 ◆ ˆ ⇢ = 0 B B @ ⇢00 ⇢01 ⇢02 ⇢03 ⇢10 ⇢11 ⇢12 ⇢13 ⇢20 ⇢21 ⇢22 ⇢23 ⇢30 ⇢31 ⇢32 ⇢33 1 C C A ˆ ⇢ = 0 B B B B B B B B B B @ ⇢00 ⇢01 ⇢02 ⇢03 ⇢04 ⇢05 ⇢06 ⇢07 ⇢10 ⇢11 ⇢12 ⇢13 ⇢14 ⇢15 ⇢16 ⇢17 ⇢20 ⇢21 ⇢22 ⇢23 ⇢24 ⇢25 ⇢26 ⇢27 ⇢30 ⇢31 ⇢32 ⇢33 ⇢34 ⇢35 ⇢36 ⇢37 ⇢40 ⇢41 ⇢42 ⇢43 ⇢44 ⇢45 ⇢46 ⇢47 ⇢50 ⇢51 ⇢52 ⇢53 ⇢54 ⇢55 ⇢56 ⇢57 ⇢60 ⇢61 ⇢62 ⇢63 ⇢64 ⇢65 ⇢66 ⇢67 ⇢70 ⇢71 ⇢72 ⇢73 ⇢74 ⇢75 ⇢76 ⇢77 1 C C C C C C C C C C A n = 3 parameters n = 15 parameters n = 63 parameters n = 4N 1 parameters
3. Continuous-variable tomography helps us think about this problem. Finite data…infinite parameters! How do we find a small, yet good model? ˆ ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A
4. Applying Model Selection to Quantum State Tomography: Choosing Hilbert Space Dimension Travis L Scholten APS March Meeting 5 March 2015 Tomography is hard 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. Let’s make it easier… Doing so in infinite dimensional Hilbert space is harder We have investigated how model selection tools can be applied.
5. We have investigated how model selection tools can be applied. Applying Model Selection to Quantum State Tomography: Choosing Hilbert Space Dimension Travis L Scholten APS March Meeting 5 March 2015 Tomography is hard 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. Let’s make it easier… Doing so in infinite dimensional Hilbert space is harder ˆ ⇢ = 0 B B B @ ⇢00 ⇢01 ⇢02 · · · ⇢10 ⇢11 ⇢21 · · · ⇢20 ⇢21 ⇢22 · · · . . . . . . . . . ... 1 C C C A Md = {⇢ | ⇢ 2 B(Hd), Hd = Span(|0i, · · · , |d 1i)}
6. We have investigated how loglikelihood ratios can be applied. Compare model to truth with loglikelihood ratios ( ⇢0, Md) = 2 log 0 @ L ( ⇢0) max ⇢2Md L ( ⇢ ) 1 A ⇢0
7. Helpfully, the Wilks Theorem predicts the distribution of the statistic. The Wilks Theorem: We want to check these numbers! N Compare model to truth with loglikelihood ratios ( ⇢0, Md) = 2 log 0 @ L ( ⇢0) max ⇢2Md L ( ⇢ ) 1 A h i ⇢0 2 Md =) ⇠ 2 d2 1 ⇢0
8. Sadly, the Wilks Theorem doesn’t apply for state tomography. Many true states, dimensions
9. Why?
10. Wilks computes a Taylor series. Let’s start there. Statistic depends on observed information (H) and fluctuations (F). (Ignore first-order term) (⇢0 , Md) ⇡ 1 2 DD ⇢0 ˆ ⇢d) @2 @2⇢ ⇢0 ˆ ⇢d EE = Tr(HF)
11. Without boundaries, we recover the Wilks prediction. Wilks: Information and fluctuations align & saturate (classical) Cramer-Rao bound Where does this go wrong in tomography? h i ⇡ Tr(hHFi) ⇡ Tr(hHihFi) ⇡ Tr(hHihHi 1) ⇡ d2 1
12. With boundaries, fluctuations are distorted. h i ⇡ Tr(hHFi) ⇡ Tr(hHihFi) ⇡ ??? Reality: Information and fluctuations do not align & do not saturate (classical) Cramer-Rao bound We have to respect state-space boundaries!
13. Let’s rescale the Fisher information by the fluctuations. What are these numbers? Can we make sense of them? Qubits & Wilks: 0 B B @ 0 1 1 1 1 C C A ! 0 B B @ .5 .5 1 1 .5 .5 1 C C A ! ✓ .5 1 1 .5 ◆ h i ⇡ Tr(hHFi) ⇡ Tr(hHihFi) = Tr( p hFihHi p hFi | {z } )
14. Different matrix units have different contributions. ⇢0 = |0ih0| d = 2 1 1 1 C C A ! 0 B B @ .5 .5 1 1 .5 .5 1 C C A ! ✓ .5 1 1 .5 ◆
15. Different matrix units have different contributions.
16. We’ve built a simple model to explain these results. Coherent and diagonal matrix elements dominate 0 @ 1 A 0 @ 1 A + 2rd r(r + 1) S(d, r) Depends on rank of true state… but unitarily invariant. h (⇢0, Md)i ⇡ h (⇢0, Md)i ⇡ + r = rank(⇢0) Reduces to Wilks in certain cases.
17. Our model does better than Wilks at predicting the expected value.
18. Our model does better than Wilks at predicting the expected value.
19. Where does this leave us?
20. Three key takeaways: Use of Wilks Theorem/AIC not advised for some model selection problems. We found a building block for a replacement. Different parameters = different contributions (Sample-size dependent) 0 @ 1 A 0 @ 1 A + h (⇢0, Md)i ⇡
21. Only so much structure… let’s model that well! We’re building a new tool for model selection, advancing the state-of-the-art in device characterization.
22. @Travis_Sch Thank you!
23. Image credits: 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 gmon qubit: By Michael Fang, John Martinis group. http:// web.physics.ucsb.edu/~martinisgroup/photos.shtml NIST Ion Trap: http://phys.org/news/2006-07-ion-large-quantum.html YouTube Logo: https://commons.wikimedia.org/wiki/ File:YouTube_Logo.svg