Applying Model Selection to Quantum State Tomography: Choosing Hilbert Space Dimension

A talk from the 2015 APS March meeting.

Screencast available via https://www.youtube.com/watch?v=_31FztfVyZA

Transcript

1. 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
2. We do tomography to make the unknown knowable Tomography is hard. State tomography Process tomography Gate Set tomography
3. Doing so in infinite dimensional Hilbert space is harder. From measurements on a continuous variable system, we estimate… 484 Simulated Heterodyne Measurement Outcomes
4. Doing so in infinite dimensional Hilbert space is harder. 484 Simulated Heterodyne Measurement Outcomes …a Wigner function…
5. Doing so in infinite dimensional Hilbert space is harder. 484 Simulated Heterodyne Measurement Outcomes …a Wigner function… ˆ ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A ˆ ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A …or a density matrix
6. Doing so in infinite dimensional Hilbert space is harder. ˆ ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A Ad-hoc smoothing/binning not always reliable Don’t use an infinite matrix Infinite parameters! estimated from finite data!
7. Let’s make it easier. ˆ ⇢ = 0 B @ ⇢00 ⇢01 · · · ⇢10 ⇢11 · · · . . . . . . ... 1 C A Don’t use an infinite matrix… ˆ ⇢ = ✓ ⇢00 ⇢01 ⇢10 ⇢11 ◆ …use a smaller one instead! Finite parameters! estimated from finite data.
8. Let’s make it easier. This is where we will do tomography. ⇢ 2 D(H2) model is (3 parameters) ⇢ 2 D(H3) model is (8 parameters) ⇢ 2 D(Hd) model is d2 1 parameters) ( D(Hd) is set of density operators ˆ ⇢ = 0 B B B @ ⇢00 ⇢01 ⇢02 · ·· ⇢10 ⇢11 ⇢12 · ·· ⇢20 ⇢21 ⇢22 · ·· . . . . . . . . . ... 1 C C C A |0i |1i |2i Low energy states can be modeled using a truncated Fock basis: Hd = span(|0i, · · · , |d 1i) |d 1i
9. Let’s make it easier. We have an algorithm for deciding which d is best. ⇢ 2 D(H2) model is (3 parameters) ⇢ 2 D(H3) model is (8 parameters) ⇢ 2 D(Hd) model is d2 1 parameters) ( D(Hd) is set of density operators ˆ ⇢ = 0 B B B @ ⇢00 ⇢01 ⇢02 · ·· ⇢10 ⇢11 ⇢12 · ·· ⇢20 ⇢21 ⇢22 · ·· . . . . . . . . . ... 1 C C C A 1) Take data 2) For d in [2, 3, 4, …], compare model d to d + 1 3) If model d + 1 fits significantly better, reject d Otherwise, d is where we stop.
10. Let’s make it easier. We can quantify “fitting significantly better”. ( d1, d2) = 2 log ✓ L ( d1) L ( d2) ◆ = 2 log 0 @ max ⇢2D(Hd1 ) L ( ⇢ ) max ⇢2D(Hd2 ) L ( ⇢ ) 1 A Goodness of fit = loglikelihood ratio statistic: ⇢ 2 D(H2) model is (3 parameters) ⇢ 2 D(H3) model is (8 parameters) ⇢ 2 D(Hd) model is d2 1 parameters) ( D(Hd) is set of density operators ˆ ⇢ = 0 B B B @ ⇢00 ⇢01 ⇢02 · ·· ⇢10 ⇢11 ⇢12 · ·· ⇢20 ⇢21 ⇢22 · ·· . . . . . . . . . ... 1 C C C A
11. Let’s make it easier. The precise algorithm for “fitting significantly better”. ( d1, d2) = 2 log ✓ L ( d1) L ( d2) ◆ = 2 log 0 @ max ⇢2D(Hd1 ) L ( ⇢ ) max ⇢2D(Hd2 ) L ( ⇢ ) 1 A 1) Take data 2) Compute 3) If Otherwise, d is where we stop. (d, d + 1) for d in [2, 3, 4, …] (d, d + 1) greater than threshold, reject d
12. Let’s make it easier. An example demonstrates use of loglikelihood ratio statistic. Can you tell which model fits the best?
13. Let’s make it easier. An example demonstrates use of loglikelihood ratio statistic. Can you tell which model fits the best?
14. Let’s make it easier. An example demonstrates use of loglikelihood ratio statistic. Can you tell which model fits the best? ! What if you did not know the true state?
15. Let’s make it easier. A threshold decides when smaller models fit worse. Qubit/qutrit does not well model 4 dimensional system…statistic grows with sample size. REJECT REJECT ACCEPT
16. Let’s make it easier. When smaller model fits well, all larger models do too. This is the focus of my current work. Use numerics to investigate. ACCEPT