Inductive Supervised Quantum Learning (why you are doing it almost - - PowerPoint PPT Presentation
Inductive Supervised Quantum Learning (why you are doing it almost right) Gael Sents University of Siegen QTML Workshop Verona, November 8, 2017 Alex Monrs, GS, Peter Wittek, PRL 118, 190503 (2017) Inductive supervised learning Binary
Alex Monràs, GS, Peter Wittek, PRL 118, 190503 (2017)
Binary classification: attach labels to data
Training
We are given: Algorithm:
Classification inductive!
Binary classification: attach labels to data We are given: Algorithm:
Training Classification classical variable
Binary classification: attach labels to data We are given: Algorithm:
Training Classification
QM implies...
Binary classification: attach labels to data We are given: Algorithm:
Training Classification
OK, but what advantage do coherent measurements provide?
QM implies...
Previous works: Semi-classical (C) strategies vs fully coherent (Q) ones
States Approach Task C vs Q [1] Pure qubits Bayesian template matching C < Q [2,3,4] Mixed qubits Bayesian classification C = Q [5] Mixed qubits Minimax (LAN) classification C < Q [6] Coherent states Bayesian classification C < Q
[1] M. Sasaki, A. Carlini, PRA 66, 022303 (2002) [2] J.A. Bergou, M. Hillery, PRL 94 160501 (2005) [3] GS, E. Bagan, J. Calsamiglia, R. Muñoz-Tapia, PRA 82, 042312 (2010) [4] GS, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, Sci. Rep. 2, 708 (2012) [5] M. Guţă, W. Kotłowski, NJP 12, 123032 (2010) [6] GS, M. Guţă, G. Adesso, EJP Quantum Technology 2015 2:17 Semi-classical strategy:
Given a training set and test instances , a learning protocol is a stochastic map
Given a training set and test instances , a learning protocol is a stochastic map Its marginal maps are
Given a training set and test instances , a learning protocol is a stochastic map Its marginal maps are We call a learning protocol inductive if the following nonsignalling condition holds:
➔ Each marginal map still depends on A ➔ This definition encompasses the standard assumption of inductive learning
The quality of is measured by the expected risk
Lemma: for every inductive learning protocol , there exists another inductive protocol where are stochastic maps, such that
The quality of is measured by the expected risk
Lemma: for every inductive learning protocol , there exists another inductive protocol where are stochastic maps, such that “Every conceivable inductive learning protocol can be regarded as a two-phase operation, training & test, with no effect on its performance” can be interpreted as applying the marginal protocol
INDUCTIVE = NONSIGNALLING
Nonsignalling is only a function of
Nonsignalling is only a function of
Nonsignalling is only a function of This framework includes several discriminative and generative quantum learning problems:
Error is measured by a symmetrized risk observable ↪A symmetrized channel will have the same error: NONSIGNALLING
If one could apply the conditional map to each of the test instances, one would perform on average as … ...but the map is nonlinear in !!!
Error is measured by a symmetrized risk observable ↪A symmetrized channel will have the same error: THE NO-CLONING THEOREM COMES INTO ACTION NONSIGNALLING
Intuitive solution: The best we can hope for is some form of approximate cloning that distributes the system across identical parties Asymptotic cloning = estimate & prepare ⇒ measure + prepare a clone for each
Intuitive solution: The best we can hope for is some form of approximate cloning that distributes the system across identical parties Asymptotic cloning = estimate & prepare ⇒ measure + prepare a clone for each de Finetti theorem for nonsignalling quantum channels Let be a nonsignalling quantum channel, and let be a local operator. Then, there exists a POVM on and a set of quantum channels such that the quantum channel satisfies Where the coefficient depends on the dimensions of the spaces , , and .
Take-home message: Despite QM in principle allows for much more, the two-phase separation TRAINING / TEST arises naturally for inductive quantum learning protocols from the (reasonable) requirement of being nonsignalling among test instances, for large data sets.
appropriately process to assess the empirical risk → minimize VC bound PRL 118, 190503 (2017)
Take-home message: Despite QM in principle allows for much more, the two-phase separation TRAINING / TEST arises naturally for inductive quantum learning protocols from the (reasonable) requirement of being nonsignalling among test instances, for large data sets.
appropriately process to assess the empirical risk → minimize VC bound the interesting risk empirical risk (?)
PRL 118, 190503 (2017)