Behavior of the Maximum Likelihood in Quantum State Tomography

A talk I gave at the annual March meeting of the American Physical Society. Released under SAND2017-2633 C.

Transcript

1. Sandia National Laboratories is a multi-mission 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 Behavior of the Maximum Likelihood in 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 2017 March 13 Based on arXiv: 1609.04385 (w/ Robin Blume-Kohout)
2. The “curse of dimensionality” makes full-scale state tomography of QIPs difficult.
3. 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 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 Sandia National Laboratories is a multi-mission 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 Behavior of the Maximum Likelihood in 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 2017 March 13 Based on arXiv: 1609.04385 (w/ Robin Blume-Kohout) We’ve been thinking about using model selection to help mitigate this problem.
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 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 We’ve been thinking about using model selection to help mitigate this problem. Sandia National Laboratories is a multi-mission 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 Behavior of the Maximum Likelihood in 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 2017 March 13 Based on arXiv: 1609.04385 (w/ Robin Blume-Kohout) End goal: Find an effective state-space dimension!
5. We use a simplifying ansatz for state tomography: Tomographic data = sequential measurements on some true state. Unconstrained maximum likelihood estimates are a sufficient statistic. Fisher information is proportional to the Hilbert-Schmidt metric. (“spherical cow” approximation) I = ✏2I ⇢⌦N 0 ✏2 ⇠ 1/N Data $ ˆ ⇢
6. Under these assumptions: ⇢0 ˆ ⇢ ˆ ⇢ ⇠ N(⇢0, ✏2I) L ( ⇢ ) / Exp ⇥ Tr[( ⇢ ˆ ⇢ ) 2 ] / 2 ✏2 ⇤ (1) Unconstrained estimates are Gaussian-distributed about true state (2) Likelihood function is isotropic Gaussian around the unconstrained estimates
7. Under these assumptions, the constrained MLE has a simple form. ˆ ⇢d L ( ⇢ ) / Exp ⇥ Tr[( ⇢ ˆ ⇢ ) 2 ] / 2 ✏2 ⇤ ˆ ⇢d = argmax ⇢2D(Hd) Tr[( ⇢ ˆ ⇢ ) 2 ] ⇢0 ˆ ⇢ Fix some Hilbert spaceHd
8. Under these assumptions, the constrained MLE has a simple form. ˆ ⇢d ⇢0 ˆ ⇢ How big of a Hilbert space do I need to accurately model my data?? Fix some Hilbert spaceHd L ( ⇢ ) / Exp ⇥ Tr[( ⇢ ˆ ⇢ ) 2 ] / 2 ✏2 ⇤ ˆ ⇢d = argmax ⇢2D(Hd) Tr[( ⇢ ˆ ⇢ ) 2 ]
9. To compare Hilbert space dimensions, we use the loglikelihood ratio statistic. ( d, d + 1) = 2 log ⇣ L(D(Hd)) L(D(Hd+1)) ⌘ The statistic is defined as Under our assumptions, it simplifies to ˆ ⇢d+1 (*Plus an additional conjecture.) ˆ ⇢d ⇢0 ˆ ⇢ ✏2 = Tr ⇥ (ˆ ⇢d ˆ ⇢)2 (ˆ ⇢d+1 ˆ ⇢)2 ⇤ =) = Tr ⇥ (ˆ ⇢d+1 ⇢0)2 (ˆ ⇢d ⇢0)2 ⇤ * (Mean-squared error!)
10. To choose a Hilbert space dimension, we need to set a threshold for the statistic. ⇢0 Start with simplest model (qubit): Compare qubits to qutrits: (2, 3) If larger than threshold, increment and iterate! D(H2) (2 , 3) > T,2 = ) Reject D ( H2) = ) Compute (3 , 4) What’s the threshold?
11. In the absence of boundaries, thresholds are easy - the exact distribution can be calculated. 1 ↵ = R 1 ↵ Pr( ) d Invoke the Wilks Theorem: Thresholds are given by confidence levels (d, d + 1) ⇠ 2 2d+1
12. Quantum state space has boundaries; the distribution is hard to calculate - use expected value. T,d = h (d, d + 1)i Under Wilks Theorem: Goal: calculate (would be a ~50% confidence level) h (d, d + 1)i = 2d + 1
13. ✏2h i = hTr[(ˆ ⇢d+1 ⇢0)2]i hTr[(ˆ ⇢d ⇢0)2]i ˆ ⇢ ⇠ N(⇢0, ✏2I) In our work, we have computed an approximation to the expected value of λ… Where ˆ ⇢d = argmax ⇢2D(Hd) Tr[( ⇢ ˆ ⇢ ) 2 ] ˆ ⇢d ⇢0 ˆ ⇢
14. Behavior of the Maximum Likelihood in Quantum State Tomography Travis L Scholten and Robin Blume-Kohout Center for Computing Research (CCR), Sandia National Labs and University of New Mexico (Dated: September 14, 2016) Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. Model selection can be used to tailor the number of fit parameters to the data, but quantum tomography violates some common assumptions that underly canonical model selection techniques arXiv:1609.04385 h (⇢0 , Md)i ⇡ 2rd r2 + rq2 + N(N + q2) ⇡ ✓ ⇡ 2 sin 1 ✓ q 2 p N ◆◆ q(q2 + 26N) 24⇡ p 4N q2 (10) where q is given in Equation (8), N = d r, and r = Rank(⇢0 ). Equation (10) is our main result. To test its validity, we compare it to numerical simulations for d = 2, . . . , 30 and r = 1, . . . , 10, in Figure 3. The prediction of the Wilks Theorem is wildly incorrect for r ⌧ d. In contrast, Equation 10 is almost perfectly accurate when r ⌧ d, but it does begin to break down (albeit fairly gracefully) as r becomes comparable to d. We conclude that our analysis [and Equation (10)] correctly models tomography if the and H5 erodyn MLEs merical d, we a an emp pair. N = d r r = Rank(⇢0) hTr[(ˆ ⇢d ⇢0)2]i ght: The nu- of ˆ ⇢ into two s = ˆ ⇢ ⇢0 , simplify the onstraint and -dimensional mation when ement of the n the “L” do ained by the rom classical has Gaussian jk i = 1. As “L”, h i L = e the bound- Here, we turn e for finding he eigenbasis pendix I for a more detailed discussion of this series of approximations.) To proceed with truncation, we observe that the j are symmetrically distributed around  = 0, so half of them are negative. Therefore, with high probabil- ity, Tr [Trunc(ˆ ⇢)] > 1, and so we will need to subtract q1l from ˆ ⇢ before truncating. The appropriate q solves Tr [Trunc(ˆ ⇢ q1l)] = 1. This equation can be solved us- ing the ansatz established so far, and some series expan- sions (see Appendix I) yield the solution: q ⇡ 2 p N (240r⇡)2/5 4 N1/10 + (240r⇡)4/5 80 N 3/10. (8) Now that we know how much to subtract o↵ in the truncation process, we can compute h i kite . Defining (x)+ = max(x, 0): h i kite = * r X j=1 [⇢jj (pj q)]2 + N X j=1 ⇥ (j q)+ ⇤2 + Z p Not the same as Wilks Theorem! …by approximating the expected mean-squared error. Depends on true state! (See paper for techniques used.)
15. The threshold based off of this result is much lower than the Wilks Theorem… …meaning less data is needed to justify an extra dimension.
16. Next steps: how to improve this result by using the rank of the tomographic estimates. Condition on rank of estimate h (d, d + 1, rd, rd+1)i Thresholds will (basically)coincide, due to concentration
17. What I didn’t tell you… * How to take the expected value and come up with a confidence interval. * How we to use this to tackle information criteria. Read the paper, or Come grab me if you’re interested! * The compressed sensing connection, via the statistical dimension.
18. Thank you! @Travis_Sch Behavior of the Maximum Likelihood in Quantum State Tomography Travis L Scholten and Robin Blume-Kohout Center for Computing Research (CCR), Sandia National Labs and University of New Mexico (Dated: September 14, 2016) Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. Model selection can be used to tailor the number of fit parameters to the data, but quantum tomography violates some common assumptions that underly canonical model selection techniques based on ratios of maximum likelihoods (loglikelihood ratio statistics), due to the nature of the state space boundaries. Here, we study the behavior of the maximum likelihood in di↵erent Hilbert space dimensions, and derive an expression for a complexity penalty – the expected value of the loglikelihood ratio statistic (roughly, the logarithm of the maximum likelihood) – that can be used to make an appropriate choice for d. arXiv:1609.04385