Overview of quantum machine learning algorithms - Troels S. Jensen & Marco Ugo Gambetta April 2021 - IEEE ...
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Overview of quantum machine learning algorithms Troels S. Jensen & Marco Ugo Gambetta April 2021
Agenda
1. Machine Learning
2. Quantum Computing
3. Quantum Machine Learning
4. Q&A
Who are we?
Troels S. Jensen Marco Ugo Gambetta
Senior Manager, Head of Consultant, Quantum expert
Machine Learning and KPMG NewTech
Quantum Technologies
KPMG NewTech
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Machine Learning
Original definition
”Machine Learning is the subfield of computer
science that gives computers the ability to learn
without being explicitly programmed.”
- Arthur Lee Samuel, 1959
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Machine Learning is arguably one of the most compute
resource hungry fields that exists
Often, the ML algorithm of choice is the one that scales the best as more data is added…
… making it natural to look for quantum-based algorithmic speed-up!
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Quantum Computing
Different architectures for quantum computers are
competing to deliver a next generation of compute devices
Superconducting Photonic Rydberg
Trapped Ions Topological Annealing
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We can solve a new class of problems, the BQP!
Quantum computers (QC) are faster for some problems
• QC exploit quantum phenomena to have a computational
advantage over classical computers for some problems
• There exist known algorithms that are proven to be better
than classical counterparts
• QC can bring an advantage to certain classes of problems
that are hard
In a nutshell:
• We want to exploit quantum computers for problems
intractable or inefficient on classical computers
• We do not expect quantum computers to replace
classical computers in general
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Quantum Machine Learning
Quantum computers can process all the data in one run…
• One can create a massive superposition of data and
apply simultaneously the computation to each element
• With n qubits you can represent 2n bit strings
• For example with 3 qubits you can represents 8 bit
strings
000, 001, 010, 011, 100, 101, 110, 111
for example creating the following state:
• With 50 qubits => 1015 bit strings
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Document Classification: KPMG Public…but you need to read out the result to gain a speed-up
Heisenberg Uncertainty Principle
You cannot access all the information contained in a
quantum system
Holevo Bound
Given n qubits, although they can "carry" a larger amount
of (classical) information, the amount of classical
information that can be retrieved when measured, can be
only up to n classical bits.
Summarizing
If you are not smart about it, quantum computers will not
give you a speed-up
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Document Classification: KPMG PublicYou can compress classical data with exponential efficiency…
• Only log(n) qubits are needed to map data n data points to a quantum state
• This is called quantum RAM or qRAM
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Document Classification: KPMG Public…but you need to prepare and read the quantum state
• Only log(n) qubits are needed to map data n data points to a quantum state
• This is called quantum RAM or qRAM
• qRAM is hard to build
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Document Classification: KPMG PublicWe need clever ways to access the result we are interested in
How do we solve these challenges?
• One needs to find subroutines where quantum computations can
give the desired result with high probability
• Luckily, research in quantum algorithms has been an active field
for many years
• Several major breakthroughs have been made, some of which we
will now dive into!
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Document Classification: KPMG PublicThe following is an attempt at giving a simplified overview
of an area that is still undergoing active research.
Please feel free to provide feedback.
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Document Classification: KPMG PublicQFT
Quantum Fourier Transform
Don Coppersmith, 1994
Legend:
Expected quadratic
speedup
Expected exponential
speedup
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Document Classification: KPMG PublicQuantum algorithm for the Fourier Transform
What is the problem?
The Fourier transform is used as a tool in for instance
signal processing, but interestingly it can also serve as
a fundamental subcomponent of other algorithms
What is the advantage?
What do you need?
Exponential speedup compared to classical
One of the fundamental algorithms of
quantum computing.
Reference
https://arxiv.org/pdf/quant-ph/0201067.pdf
Requirements and limitations
The algorithm is not error robust, it requires error
correction
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Document Classification: KPMG PublicQFT
Quantum Fourier Transform
Don Coppersmith, 1994
HSP / Shor’s alg
Hidden subgroup problem / Legend:
Shor’s algorithm for factorization
and discrete logarithm
Expected quadratic
Peter Shor, 1994 speedup
Expected exponential
speedup
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Document Classification: KPMG PublicAlgorithm for Prime Factorization and Discrete Logarithms
What is the problem?
135066410865995223349603216278805969938881475605667
0275244851438515265106048595338339402871505719094417
Given an integer, find its prime factors
9820728216447155137368041970396419174304649658927425
6239341020864383202110372958725762358509643110564073
501508187510676594629205563685529475213500852879416
What is the advantage? 377328533906109750544334999811150056977236890927563
Shor’s algorithm provides an exponential speedup over
What do you need?
the fastest known classical algorithms – a milestone in
Shor’s algorithm uses:
quantum computing
• FFT
Reference
Requirements and limitations https://arxiv.org/pdf/quant-
Currently, quantum computers are not sufficiently large ph/9508027.pdf
to tackle real-life problems (need some 2k logical qubits
to break RSA-2024)
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Document Classification: KPMG PublicQFT QAA / Grover’s alg
Quantum Fourier Transform Quantum Amplitude Amplification
/ Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Lov Grover, 1996
HSP / Shor’s alg
Hidden subgroup problem / Legend:
Shor’s algorithm for factorization
and discrete logarithm
Expected quadratic
Peter Shor, 1994 speedup
Expected exponential
speedup
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Document Classification: KPMG PublicUnstructured database search
What is the problem?
Amplitude
Given a system of N = 2n states and certain condition
that is satisfied for a unique element of the system x,
find that element x
Elements
What is the advantage?
Grover’s algorithm provides a quadratic speedup over What do you need?
classical solutions Grover’s algorithm uses:
• An initial superposition
• Inversion about average routine
Requirements and limitations Reference
You need the condition to be verified only for a unique https://arxiv.org/pdf/quant-
element in the original version, but this has since been ph/9605043.pdf
relaxed
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HSP / Shor’s alg
Hidden subgroup problem / Legend:
Shor’s algorithm for factorization
and discrete logarithm
Expected quadratic
Peter Shor, 1994 speedup
Expected exponential
speedup
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Document Classification: KPMG PublicQuantum algorithm for eigenvalue estimation
What is the problem?
Estimating the eigenvalues of an eigenvector of a
unitary operator (quantum states cannot be simply read
out)
?
What is the advantage?
Problem proper of the quantum realm – efficient What do you need?
implementation QPE uses:
• QFT
Reference
https://arxiv.org/pdf/1809.09697.pdf
Requirements and limitations
The (quantum) computational cost depends on the
desired accuracy – scales logarithmically
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HSP / Shor’s alg QAE
Hidden subgroup problem / Quantum Amplitude Estimation Legend:
Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Michele Mosca, Alain Tapp, 2002 Expected quadratic
Peter Shor, 1994 speedup
Expected exponential
speedup
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Document Classification: KPMG PublicQuantum Algorithm for Amplitude Estimation
What is the problem?
Estimate coefficients for specific components of Recover a specific coefficient from a state,
quantum vectors (quantum states cannot be simply read for instance -2/3 from the below |10> state:
out)
What is the advantage?
This is used for instance as a final component of
quantum algorithms in order to effectively read out a What do you need?
result with a desired accuracy. The speed up is QAE uses:
quadratic • Quantum Amplitude Amplification
• Quantum Fourier Transform
Requirements and limitations?
The coefficient is not read out exactly, but to within a Reference
certain error – square root improvement over classical https://arxiv.org/pdf/quant-
implementation ph/0005055.pdf
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HSP / Shor’s alg QAE
Hidden subgroup problem / Quantum Amplitude Estimation Legend:
Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Michele Mosca, Alain Tapp, 2002 Expected quadratic
Peter Shor, 1994 speedup
Expected exponential
QMC speedup
Quantum Monte Carlo methods
Ashley Montanaro, 2017
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Document Classification: KPMG PublicQuantum Monte Carlo methods
What is the problem?
Monte Carlo methods are a widely used technique for
estimating hard to compute quantities.
What is the advantage?
Quantum Monte Carlo can obtain quadratic speed-up What do you need?
over classical Markov chain Monte Carlo methods. QMC uses:
• Quantum Amplitude Amplification
• Quantum Amplitude Estimation
Requirements and limitations?
Current implementations of Quantum Monto Carlo Reference
methods are limited by the depth / coherence time of https://arxiv.org/pdf/1504.06987.pdf
current hardware.
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QMC speedup
Quantum Monte Carlo methods
Ashley Montanaro, 2017
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Document Classification: KPMG PublicQuantum Algorithm for Solving Linear System of Equations
What is the problem?
Given a matrix A and a vector b, find a vector x such
that Ax=b
What is the advantage?
The quantum algorithm HHL provides an exponential What do you need?
speedup over classical solutions HHL uses:
• representation of data as quantum
state amplitudes
• Quantum Fourier transform
Requirements and limitations • Phase estimation algorithm
• A needs to be s-sparse
Reference
• You do not get the solution vector as classical
https://arxiv.org/pdf/0811.3171.pdf
output, but you can access properties of the solution
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QMC speedup
Support Vector Machine Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014
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Document Classification: KPMG PublicQuantum Support Vector Machine
What is the problem?
The task of SVM is to find an optimal separating
hyperplane between the two classes of data such that
the margins from the data to the hyperplane are
maximized
What is the advantage?
QSVM provides an exponential speedup when classical What do you need?
sampling algorithms require polynomial time QSVM uses:
• Phase estimation
• HHL
• qRAM
Requirements and limitations? Reference
Similar requirements as HHL https://arxiv.org/pdf/1307.0471.pdf
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
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Document Classification: KPMG PublicQuantum Principal Components Analysis
What is the problem?
Compute the largest eigenvalues of the covariance
matrix to for instance help identifying dominating sets of
features in a dataset.
What is the advantage?
The QPCA provides an exponential speedup over What do you need?
classical algorithms QPCA uses:
• Phase estimation algorithm
• qRAM or quantum data to create the
Requirements and limitations? covariance matrix
• Works best when the system has a few dominating Reference
eigenvalues https://arxiv.org/pdf/1307.0401.pdf
• QPCA can only reveal a fraction of the full
information required to describe the system
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA
Gaussian Processes
Zhikuan Zhao, Jack Fitzsimons
and Joseph Fitzsimons, 2015
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Document Classification: KPMG PublicQuantum Algorithm for Gaussian Processes regression
What is the problem?
Gaussian Process regression is an interesting
technique that estimates both mean and variance, but it
is unfortunately a computationally expensive method
What is the advantage?
QGPA provides an exponential speedup over classical What do you need?
solutions QGPA uses:
• HHL
Reference
Requirements and limitations? https://arxiv.org/pdf/1512.03929.pdf
QGPA uses HHL HHL requirements are needed
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS
Gaussian Processes Recommendation Systems
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam
and Joseph Fitzsimons, 2015 Prakash, 2016
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Document Classification: KPMG PublicQuantum Recommendation System
What is the problem?
Identify a preference matrix that indicates whether a
product is a good match for a user based on data on the
user and on other users as well
What is the advantage?
QRS has an exponential speedup over known
What do you need?
algorithms…until a quantum inspired algorithm was
found that performs similar to it (more on this later) QRS uses:
• QPE
Reference
https://arxiv.org/pdf/1603.08675.pdf
Requirements and limitations?
There is no sparsity requirement on the matrix
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Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS QRL
Gaussian Processes Recommendation Systems Reinforcement Learning
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam Vedran Dunjko, Jacob Taylor and Hans Briegel, 2016
and Joseph Fitzsimons, 2015 Prakash, 2016 Vedran Dunjko, Yi-Kai Liu, Xingyao Wu and Jacob Taylor, 2017
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Document Classification: KPMG PublicQuantum Reinforcement Learning Learn
What is the problem?
Intelligent agents take actions in an environment in
order to maximize a cumulative reward
Observe Act
What is the advantage?
Based on the quantum subroutine used the QRS gains
What do you need?
an exponential or quadratic speedup over classical
algorithms QRL can use:
• HHL
• Grover
Reference
Requirements and limitations?
• https://arxiv.org/pdf/1610.08251.pdf
Based on the subroutine it has HHL or Grover’s • https://arxiv.org/pdf/1710.11160.pdf
requirements
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40
Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS QRL
Gaussian Processes Recommendation Systems Reinforcement Learning
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam Vedran Dunjko, Jacob Taylor and Hans Briegel, 2016
and Joseph Fitzsimons, 2015 Prakash, 2016 Vedran Dunjko, Yi-Kai Liu, Xingyao Wu and Jacob Taylor, 2017
QuGAN
Generative Adversarial Network
Seth Lloyd and Christian
Weedbrook, 2018
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Document Classification: KPMG PublicQuantum Generative Adversarial Network
Generator Data
What is the problem?
A generator creates artificial data with patterns that
statistically match real data and a discriminator needs to
understand if an observation is artificial or real
True
Discriminator
Fake
What is the advantage?
QuGAN may provide exponential speedup over What do you need?
classical GAN QuGAN uses:
• HHL (optional)
• Matrix exponentiation techniques
Requirements and limitations?
There are different types of QuGAN, with discriminator, Reference
generator and data that can be quantum or classical. https://arxiv.org/pdf/1804.09139.pdf
Classical data with quantum generator and
discriminator is expected to provide an exponential
speed up
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42
Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS QRL
Gaussian Processes Recommendation Systems Reinforcement Learning
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam Vedran Dunjko, Jacob Taylor and Hans Briegel, 2016
and Joseph Fitzsimons, 2015 Prakash, 2016 Vedran Dunjko, Yi-Kai Liu, Xingyao Wu and Jacob Taylor, 2017
QuGAN QHNN
Generative Adversarial Network Hopfield Neural Network
Seth Lloyd and Christian Patrick Rebentrost, Thomas Bromley,
Weedbrook, 2018 Christian Weedbrook and Seth Lloyd, 2018
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43
Document Classification: KPMG PublicQuantum Hopfield Neural Network
What is the problem?
Recognize neuron patterns given samples in a fully
connected NN (not a feedforward NN topology)
What is the advantage?
QHNN provides an exponential speedup over classical What do you need?
solution and exponential storage advantage QHNN uses:
• HHL
Reference
Requirements and limitations? https://arxiv.org/pdf/1710.03599.pdf
Not a common Neural Network topology
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44
Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS QRL
Gaussian Processes Recommendation Systems Reinforcement Learning
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam Vedran Dunjko, Jacob Taylor and Hans Briegel, 2016
and Joseph Fitzsimons, 2015 Prakash, 2016 Vedran Dunjko, Yi-Kai Liu, Xingyao Wu and Jacob Taylor, 2017
QA QuGAN QHNN
Quantum Annealing Generative Adversarial Network Hopfield Neural Network
Tadashi Kadowaki and Hidetoshi Seth Lloyd and Christian Patrick Rebentrost, Thomas Bromley,
Nishimori, 1998 Weedbrook, 2018 Christian Weedbrook and Seth Lloyd, 2018
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guarantee. All rights reserved.
45
Document Classification: KPMG PublicQuantum Annealing and Boltzmann Machine
What is the problem?
Quantum Unconstrained Binary Optimization Models
(QUBO)
What is the advantage?
QA advantage is still being researched, but it can be
What do you need?
used as a fast heuristic and also for generating samples
from hard distributions (Boltzmann for instance) n.a.
Reference
https://arxiv.org/abs/1912.08480
Requirements and limitations?
Requires quantum annealer hardware
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guarantee. All rights reserved.
46
Document Classification: KPMG PublicQPE QFT QAA / Grover’s alg
Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification
eigenvalue estimation / Grover’s algorithm for
Don Coppersmith, 1994
unstructured search
Michael Nielsen and Isaac
Chuang, 2000 Lov Grover, 1996
HHL HSP / Shor’s alg QAE
*
Solving systems of linear Hidden subgroup problem / Quantum Amplitude Estimation Legend:
equations Shor’s algorithm for factorization
and discrete logarithm Gilles Brassard, Peter Høyer,
Aram Harrow, Avinatan Hassidim Michele Mosca, Alain Tapp, 2002 Expected quadratic
and Seth Lloyd, 2009 Peter Shor, 1994 speedup
Expected exponential
QSVM QPCA QMC speedup
*
Support Vector Machine Principal Component Analysis Quantum Monte Carlo methods
Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017
* Exponential speedup
challenged by Ewin
Tang, et al.
Mohseni, Seth Lloyd, 2014 Patrick Rebentrost, 2014
QGPA QRS QRL
Gaussian Processes
*
Recommendation Systems Reinforcement Learning
Zhikuan Zhao, Jack Fitzsimons Iordanis Kerenidis and Anupam Vedran Dunjko, Jacob Taylor and Hans Briegel, 2016
and Joseph Fitzsimons, 2015 Prakash, 2016 Vedran Dunjko, Yi-Kai Liu, Xingyao Wu and Jacob Taylor, 2017
QA QuGAN QHNN
Quantum Annealing Generative Adversarial Network Hopfield Neural Network
Tadashi Kadowaki and Hidetoshi Seth Lloyd and Christian Patrick Rebentrost, Thomas Bromley,
Nishimori, 1998 Weedbrook, 2018 Christian Weedbrook and Seth Lloyd, 2018
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guarantee. All rights reserved.
47
Document Classification: KPMG PublicQML in the cloud
Pennylane: A cross platform Python library
Open source SDK by IBM with integrated QML library
Forest: Rigetti SDK
D-Wave SDK to access the quantum annealer
Prototyping of hybrid quantum-classical ML models
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48
Document Classification: KPMG PublicThere are challenges for QML
qRAM Logical Qubits Gate Depth
qRAM is often needed to efficiently Algorithms are usually not error Quantum information needs longer
encode classical data in a quantum robust, ca. 100 logical qubits are decoherence time, a gate depth of
state needed for reliable results ca. 100.000 has been estimated
We are not quite there yet…
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49
Document Classification: KPMG PublicQ-uestions?
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50
Document Classification: KPMG PublicContact Information
Troels Steenstrup Jensen
trsjensen@kpmg.comYou can also read
