SITAN CHEN UC BERKELEY BRICE HUANG MIT JERRY LI MSR ALLEN LIU MIT MARK SELLKE IAS LEARNING FROM QUANTUM DATA Learner How much data is needed to learn about the physical world Do practical limitations eg memory constraints impose any inherent statistical overhead ID: 1036809
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1. TIGHT BOUNDS FOR INCOHERENT MEASUREMENTSSITAN CHEN UC BERKELEYBRICE HUANGMITJERRY LIMSRALLEN LIUMITMARK SELLKEIAS
2. LEARNING FROM QUANTUM DATA Learner How much data is needed to learn about the physical world?Do practical limitations (e.g. memory constraints) impose any inherent statistical overhead?
3. LEARNING FROM QUANTUM DATALearner is given copies of unknown state Different ways of interacting: powerful,impracticalweak,practicalPick POVM+ measure outcome Learner computes output based on measurement outcome “Coherent measurements” (can load in all copies at once)
4. LEARNING FROM QUANTUM DATALearner is given copies of unknown state Different ways of interacting: powerful,impracticalweak,practicalPick POVM+ measure Pick POVM+ measure outcome outcome ...“Coherent measurements” (can load in all copies at once)(can only load in one copy of at a time) “Incoherent measurements” Learner computes output based on measurement outcomes Choice of measurements can be adaptive
5. REPRESENTATIVE TASKSGoalClassical version
6. REPRESENTATIVE TASKSGoalClassical versionState tomographyOutput s.t. DistributionlearningGoalClassical versionState tomographyDistributionlearning
7. REPRESENTATIVE TASKSGoalClassical versionState tomographyOutput s.t. DistributionlearningState certificationGiven , decide whether or IdentitytestingGoalClassical versionState tomographyDistributionlearningState certificationIdentitytestingMany more examples: shadow tomography, hypothesis selection, spectrum testing, process tomography, gate set tomography, randomized benchmarking, quantum query complexity, etc.
8. TIGHT BOUNDS FOR STATE TOMOGRAPHYWITH INCOHERENT MEASUREMENTSSITAN CHEN UC BERKELEYBRICE HUANGMITJERRY LIMSRALLEN LIUMITMARK SELLKEIAS
9. CLASSICAL ANALOGUEQ: Given i.i.d. samples from a probability distribution over , output an estimate for which [Folklore]: samples are necessary and sufficientUpper bound: output empirical distributionLower bound: packing argument / Fano’s method
10. CLASSICAL ANALOGUEQ: Given i.i.d. samples from a probability distribution over , output an estimate for which [Folklore]: samples are necessary and sufficient
11. PRIOR WORKCoherent measurements:[O’Donnell-Wright ’15b], [Haah-Harrow-Ji-Wu-Yu ‘15]: copies are necessary and sufficient Upper bound: weak-Schur sampling + Keyl’s method or PGM Lower bound: packing argument / Fano’s method / Holevo
12. PRIOR WORKCoherent measurements:[O’Donnell-Wright ’15b], [Haah-Harrow-Ji-Wu-Yu ‘15]: copies are necessary and sufficient
13. PRIOR WORKIncoherent measurements:[Kueng-Rauhut-Terstiege ‘14]: copies sufficient[Haah-Harrow-Ji-Wu-Yu ‘15]: copies necessary if measurements are nonadaptive[Lowe-Nayak ‘21]: copies necessary if measurements have limited adaptivity or come from set of size How many copies are needed for tomography with general incoherent measurements?
14. OUR RESULTSTheorem [C-Huang-Li-Liu-Sellke ’22]: For state tomography with general incoherent measurements, copies are necessary and sufficient Adaptivity doesn’t help at all for state tomography in trace distance Nonadaptive algorithm based on random measurements is optimalProof is only 4 pages!
15. ANTICONCENTRATION OF POSTERIOR prior: small Gaussian perturbation of
16. ANTICONCENTRATION OF POSTERIOR : posterior distribution over prior: small Gaussian perturbation of
17. ANTI-CONCENTRATION OF POSTERIORConsider the following prior over states for , conditioned on Fix a tomography algorithm and let denote the distribution over transcripts of measurement outcomesGiven transcript , let denote the posterior distribution over Theorem [CHLLS]: With high probability over ,
18. ANTI-CONCENTRATION OF POSTERIORHigh probability event: sufficiently close to not too large in expectation over sampled from prior Theorem [C-Huang-Li-Liu-Sellke ’22]: With high probability over and transcript , the posterior distribution places mass on the -trace norm ball around
19. ANTI-CONCENTRATION OF POSTERIOR
20. ANTI-CONCENTRATION OF POSTERIOR: trace norm ball of radius around Theorem [C-Huang-Li-Liu-Sellke ’22]: W.h.p. over and , posterior places mass on -trace norm ball around
21. ANTI-CONCENTRATION OF POSTERIOR: trace norm ball of radius around Theorem [C-Huang-Li-Liu-Sellke ’22]: W.h.p. over and , posterior places mass on -trace norm ball around
22. ANTI-CONCENTRATION OF POSTERIOR: trace norm ball of radius around Theorem [C-Huang-Li-Liu-Sellke ’22]: W.h.p. over and , posterior places mass on -trace norm ball around Expectation over is , so with high probability, not too large
23. ANTI-CONCENTRATION OF POSTERIOR: trace norm ball of radius around Theorem [C-Huang-Li-Liu-Sellke ’22]: W.h.p. over and , posterior places mass on -trace norm ball around Integral is over a larger ball than in numerator, so suffices to show that conditional expectation of over this ball is not too small
24. ANTI-CONCENTRATION OF POSTERIOR: trace norm ball of radius around Theorem [C-Huang-Li-Liu-Sellke ’22]: W.h.p. over and , posterior places mass on -trace norm ball around Integral is over a larger ball than in numerator, so suffices to show that conditional expectation of over this ball is
25. ANTI-CONCENTRATION OF POSTERIORExpectation over is , so with high probability, not too large : trace norm ball of radius around
26. ANTI-CONCENTRATION OF POSTERIORIntegral is over a larger ball than in numerator, so suffices to show that conditional expectation of over this ball is not too small : trace norm ball of radius around
27. ANTI-CONCENTRATION OF POSTERIORIntegral is over a larger ball than in numerator, so suffices to show that conditional expectation of over this ball is : trace norm ball of radius around
28. ANTI-CONCENTRATION OF POSTERIORIntegral is over a larger ball than in numerator, so suffices to show that conditional expectation of over this ball is Conditional distribution is unwieldy to work with because is centered around , but is centered around (this mismatch is fine because w.h.p., and Gaussian is “flat” in this vicinity) : trace norm ball of radius around
29. ANTI-CONCENTRATION OF POSTERIORKey Lemma: For any transcript , the expectation of over drawn from Lebesgue measure over the -operator-norm ball centered around is Proof: Consider , i.e. consists of a single measurement outcome. Can assume WLOG measurement operators are rank-1Taylor expand for
30. ANTI-CONCENTRATION OF POSTERIOR1st-order term vanishes upon integrating over 2nd-order term which, by rotational invariance of distribution over , has the same expectation as as desired
31. TIGHT BOUNDS FOR STATE CERTIFICATIONWITH INCOHERENT MEASUREMENTSSITAN CHEN UC BERKELEYBRICE HUANGMITJERRY LIMSRALLEN LIUMIT
32. MIXEDNESS TESTINGImportant special case:YES: NO: Classical Q: Given i.i.d. samples from dist. over , decide YES: NO: “Uniformity testing”
33. MIXEDNESS TESTINGClassical Q: Given i.i.d. samples from dist. over , decide YES: NO: [Paninski ’08]: samples are necessary and sufficientUpper bound: count collisionsLower bound: Le Cam, chi-squared bound between point vs. mixture
34. MIXEDNESS TESTING[Paninski ’08]: samples are necessary and sufficient Classical Q: Given i.i.d. samples from dist. over , decide YES: NO:
35. STATE CERTIFICATIONClassical Q: Given distribution over , and given i.i.d. samples from a probability distribution over , decide:YES: NO: [Valiant-Valiant ‘14]: samples necessary and sufficientUpper bound: Pearson’s chi-squared statisticLower bound: Le Cam, chi-squared bound between point vs. mixture
36. STATE CERTIFICATIONClassical Q: Given distribution over , and given i.i.d. samples from a probability distribution over , decide:YES: NO: “Identity testing”[Valiant-Valiant ‘14]: samples necessary and sufficient “Instance-optimal identity testing”
37. PRIOR WORKCoherent measurements:[O’Donnell-Wright ’15a], [Badescu-O’Donnell-Wright ‘17]: copies sufficient[O’Donnell-Wright ’15a]: copies necessary for mixedness testing
38. PRIOR WORKIncoherent measurements:[Bubeck-C-Li ‘20] (mixedness testing): copies necessary and sufficient with nonadaptive copies necessary
39. PRIOR WORKIncoherent measurements:[Bubeck-C-Li ‘20] (mixedness testing): copies necessary and sufficient with nonadaptive copies necessary[C-Li-O’Donnell ‘20] (state certification) copies necessary and sufficient w/ nonadaptivecopies necessary
40. OUR RESULTSTheorem [C-Huang-Li-Liu ’22]: For state certification with general incoherent measurements, copies are necessary and sufficient. For mixedness testing, we also show how to remove the log factors Adaptivity doesn’t help* for state certification in trace distance Nonadaptive algorithm based on random measurements is optimal
41. OUR RESULTSTheorem [C-Huang-Li-Liu ’22]: For mixedness testing, copies are necessary and sufficient. We also obtain nearly tight bounds for state certification, see paperAdaptivity doesn’t help* for state certification in trace distance Nonadaptive algorithm based on random measurements is optimal
42. POINT-VERSUS-MIXTURE PROBLEMYES: NO:
43. POINT-VERSUS-MIXTURE PROBLEMYES: NO: for Haar-random
44. POINT-VERSUS-MIXTURE PROBLEMYES: NO: for “trace-centered” Gaussian matrix given by sampling GOE matrix , defining
45. Le Cam: suffices to show that for insufficiently large sample sizes ,Suffices to show that with high probability over LIKELIHOOD RATIO
46. LIKELIHOOD RATIO Again assume WLOG that measurements are rank-1, so is a function of a sequence of unit vectors :
47. LIKELIHOOD RATIO (complicated expression involving perfect matchings among ’s) with -th vector removed Let’s pretend all the ’s on the right are the same...
48. LIKELIHOOD RATIO (complicated expression involving perfect matchings among ’s) with -th vector removed Let’s pretend all the ’s on the right are the same...
49. LIKELIHOOD RATIO
50. LIKELIHOOD RATIO matrix martingale with fluctuations
51. LIKELIHOOD RATIOWith each new measurement, changes by factor of Cumulatively over N steps, changes by factor of As long as ,
52. LIKELIHOOD RATIO (complicated expression involving perfect matchings among ’s) with -th vector removed Let’s pretend all the ’s on the right are the same...
53. LIKELIHOOD RATIO (complicated expression involving perfect matchings among ’s) with -th vector removed Circular: cannot pretend all the ’s on the right are the same... (Fix with bootstrapping argument)
54. OPEN QUESTIONS TAKEAWAYS lower bound for tomography when is rank-?Spectrum estimation?Intermediate levels of entanglement / quantum memory?Instance-optimal state certification with entangled measurements?Simpler proof of lower bound for mixedness testing with entangled measurements using Gaussian instance? Tight bounds for sample complexity of tomography and state certification with incoherent measurementsAdaptivity does not help, and simple algorithms based on random measurements are optimalGaussian priors easier to analyze than Haar-random ones:Tomography: simple proof based on anticoncentration of posteriorState certification: recursive structure of likelihood ratioThanks!
55. OPEN PROBLEMS lower bound for tomography when is rank-?Spectrum estimation?Intermediate levels of entanglement / quantum memory?Instance-optimal state certification with entangled measurements?Process tomography?Simpler proof of lower bound for mixedness testing with entangled measurements using Gaussian instance?
56. TAKEAWAYSWe proved tight lower bounds for the sample complexity of state tomography and state certification with incoherent measurementsOur bounds show that for these tasks, adaptivity does not help, and simple algorithms based on random measurements are optimalGaussian priors easier to analyze than Haar-random ones:Proof for tomography was a surprisingly simple argument based on anticoncentration of the posteriorProof for state certification was based on the recursive structure of the likelihood ratio