Taking the Quantum Leap with Machine Learning

Introduction Quantum Algorithms Current Research Conclusion

Zack Barnes University of Washington

Bellevue College Mathematics and Physics Colloquium Series [email protected]

January 15, 2019

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Overview

1 Introduction What is Quantum Computing? What is Machine Learning? Quantum Power in Theory

2 Quantum Algorithms HHL Quantum Recommendation

3 Current Research Quantum Supremacy(?)

4 Conclusion

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

What is Quantum Computing?

“Quantum computing focuses on studying the problem of storing, processing and transferring information encoded in quantum mechanical systems.“ [Ciliberto, Carlo et al., 2018]

Unit of quantum information is the qubit, or quantum binary integer.

Zack Barnes University of Washington UW Quantum Machine Learning Supervised Uses labeled examples to predict future events Unsupervised Not classiﬁed or labeled

Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

“Machine learning is the scientiﬁc study of algorithms and statistical models that computer systems use to progressively improve their performance on a speciﬁc task.“ (Wikipedia)

Zack Barnes University of Washington UW Quantum Machine Learning Uses labeled examples to predict future events Unsupervised Not classiﬁed or labeled

Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

“Machine learning is the scientiﬁc study of algorithms and statistical models that computer systems use to progressively improve their performance on a speciﬁc task.“ (Wikipedia)

Supervised

Zack Barnes University of Washington UW Quantum Machine Learning Unsupervised Not classiﬁed or labeled

Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

“Machine learning is the scientiﬁc study of algorithms and statistical models that computer systems use to progressively improve their performance on a speciﬁc task.“ (Wikipedia)

Supervised Uses labeled examples to predict future events

Zack Barnes University of Washington UW Quantum Machine Learning Not classiﬁed or labeled

Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

“Machine learning is the scientiﬁc study of algorithms and statistical models that computer systems use to progressively improve their performance on a speciﬁc task.“ (Wikipedia)

Supervised Uses labeled examples to predict future events Unsupervised

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

“Machine learning is the scientiﬁc study of algorithms and statistical models that computer systems use to progressively improve their performance on a speciﬁc task.“ (Wikipedia)

Supervised Uses labeled examples to predict future events Unsupervised Not classiﬁed or labeled

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

What is Machine Learning?

Image Source: MathWorks

Zack Barnes University of Washington UW Quantum Machine Learning Q.C. can quickly solve what a C.C. also solves quickly BQP * P There are things Q.C. can do quickly that C.C. can’t BQP ⊆ EXP EXP * BQP NP?

Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP

BQP

Zack Barnes University of Washington UW Quantum Machine Learning BQP * P There are things Q.C. can do quickly that C.C. can’t BQP ⊆ EXP EXP * BQP NP?

Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP Q.C. can quickly solve what a C.C. also solves BQP quickly P

Zack Barnes University of Washington UW Quantum Machine Learning There are things Q.C. can do quickly that C.C. can’t BQP ⊆ EXP EXP * BQP NP?

Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP Q.C. can quickly solve what a C.C. also solves BQP quickly P BQP * P

Zack Barnes University of Washington UW Quantum Machine Learning BQP ⊆ EXP EXP * BQP NP?

Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP Q.C. can quickly solve what a C.C. also solves BQP quickly P BQP * P There are things Q.C. can do quickly that C.C. can’t

Zack Barnes University of Washington UW Quantum Machine Learning EXP * BQP NP?

Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP Q.C. can quickly solve what a C.C. also solves BQP quickly P BQP * P There are things Q.C. can do quickly that C.C. can’t BQP ⊆ EXP

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Quantum Power in Theory

Polynomial Time Hierarchy EXP P ⊆ BQP Q.C. can quickly solve what a C.C. also solves BQP quickly P BQP * P There are things Q.C. can do quickly that C.C. can’t BQP ⊆ EXP EXP * BQP NP?

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

HHL Harrow, Hassidim, Lloyd Algorithm (2008)

Given an n × n matrix A and a vector ~b, we must ﬁnd (or approximately ﬁnd) the vector ~x such that A~x = ~b.

On a classical computer this takes O(nc ), for a constantc.

HHL algorithm takes O((log n)2).

A common optimization subroutine technique is to minimize (A~x − ~b)2.

HHL algorithm gave birth to quantum machine learning.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

HHL Important Caveats

1 QRAM 2 Apply unitary transformation 3 Matrix must be invertible and well conditioned

4 Solution is given in quantum superposition |xi i

“To summarize, HHL is not exactly an algorithm for solving a system of linear equations in logarithmic time... its an algorithm for approximately preparing a quantum superposition of the form ~ |xi i, where ~x is the solution to a linear system A~x = b...“ [Aaronson, 2015].

Zack Barnes University of Washington UW Quantum Machine Learning If the preference matrix is known prior, the solution would be simply to recommend the highest value entry for that user.

We must use matrix reconstruction by using samples of the preference matrix to ﬁnd a good low-rank approximation [Tang, 2018].

Introduction Quantum Algorithms Current Research Conclusion

Quantum Recommendation Netﬂix & Complex

Given past purchases or ratings of n products and m users. This data is then given as an m × n preference matrix.

Zack Barnes University of Washington UW Quantum Machine Learning We must use matrix reconstruction by using samples of the preference matrix to ﬁnd a good low-rank approximation [Tang, 2018].

Introduction Quantum Algorithms Current Research Conclusion

Quantum Recommendation Netﬂix & Complex

Given past purchases or ratings of n products and m users. This data is then given as an m × n preference matrix.

If the preference matrix is known prior, the solution would be simply to recommend the highest value entry for that user.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Quantum Recommendation Netﬂix & Complex

Given past purchases or ratings of n products and m users. This data is then given as an m × n preference matrix.

If the preference matrix is known prior, the solution would be simply to recommend the highest value entry for that user.

We must use matrix reconstruction by using samples of the preference matrix to ﬁnd a good low-rank approximation [Tang, 2018].

Zack Barnes University of Washington UW Quantum Machine Learning Last year, UW1 Ph.D. candidate, Ewin Tang, developed a classical algorithm inspired by its quantum counterpart that also runs in O(poly(k)polylog(mn))!

Introduction Quantum Algorithms Current Research Conclusion

Quantum Supremacy(?) Current Research

When the quantum recommendation system was given in 2016, the best known classical algorithm was O(poly(mn)), and the quantum algorithm does in O(poly(k)polylog(mn)).

1Go Dawgs Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Quantum Supremacy(?) Current Research

When the quantum recommendation system was given in 2016, the best known classical algorithm was O(poly(mn)), and the quantum algorithm does in O(poly(k)polylog(mn)).

Last year, UW1 Ph.D. candidate, Ewin Tang, developed a classical algorithm inspired by its quantum counterpart that also runs in O(poly(k)polylog(mn))!

1Go Dawgs Zack Barnes University of Washington UW Quantum Machine Learning Tang was able to replicate this implicit representation with sampling from a probability distribution and take sub-samples of larger pieces of data.

Introduction Quantum Algorithms Current Research Conclusion

Quantum Supremacy(?) How?

One way is that the algorithm gives a sample of good recommendations instead of complete list. Another source of the quantum speed up was from uses of quantum mechanics (superposition) to represent data implicitly.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Quantum Supremacy(?) How?

Tang was able to replicate this implicit representation with sampling from a probability distribution and take sub-samples of larger pieces of data.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Quantum Supremacy(?)

Since Tang’s development of a classical recommendation algorithm that matches the quantum speed up, she has published two other papers 1 “Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension“ Nov. 2018 2 “Quantum-inspired classical algorithms for principal component analysis and supervised clustering“ Oct. 2018 Both make use of sampling to hinder the quantum speedup.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Conclusion

Quantum computing is motivating advancement in classical computing. People are “dequantising“ quantum algorithms to be used on classical computers.

The placement of BQP in the polynomial-time hierarchy remains unknown, and thus the true power of this “quantum speedup“ is yet to be seen.

Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

Sources

Ewin Tang (2018) A quantum-inspired classical algorithm for recommendation systems CoRR http://arxiv.org/abs/1807.04271.

Scott Aaronson (2015) Quantum Machine Learning: Read the Fine Print Nature Physics 11, https://scottaaronson.com/papers/qml.pdf.

Vedran Dunjko and Hans J Briegel (2018) Machine learning & artiﬁcial intelligence in the quantum domain: a review of recent progress Prog. Phys. 81, http://iopscience.iop.org/article/10.1088/1361-6633/aab406/pdf.

Ciliberto, Carlo et al. (2018). Quantum machine learning: a classical perspective Proceedings sciences 474(2209), Zack Barnes Universityhttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC5806018/. of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion

The End

Zack Barnes University of Washington UW Quantum Machine Learning