Troels S. Jensen & Marco Ugo Gambetta April 2021

Overview of quantum machine learning algorithms

Troels S. Jensen & Marco Ugo Gambetta

April 2021 Agenda

1. Machine Learning

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

© 2021 KPMG P/S, a Danish limited liability partnership and a member firm of the KPMG global organisation of independent member firms affiliated with KPMG International Limited, a private English company limited by guarantee. All rights reserved. 3

Document Classification: KPMG Public 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

Document Classification: KPMG Public 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!

Document Classification: KPMG Public Quantum Computing Different architectures for quantum computers are competing to deliver a next generation of compute devices

Superconducting Photonic Rydberg

Trapped Ions Topological Annealing

Document Classification: KPMG Public 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

Document Classification: KPMG Public 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

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

Document Classification: KPMG Public You 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

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

Document Classification: KPMG Public We 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!

Document Classification: KPMG Public The following is an attempt at giving a simplified overview of an area that is still undergoing active research. Please feel free to provide feedback.

Document Classification: KPMG Public QFT Quantum Fourier Transform Don Coppersmith, 1994

Legend:

Expected quadratic speedup

Expected exponential speedup

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QFT 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

Document Classification: KPMG Public Algorithm for Prime Factorization and Discrete Logarithms

135066410865995223349603216278805969938881475605667 What is the problem? 0275244851438515265106048595338339402871505719094417 Given an integer, find its prime factors 9820728216447155137368041970396419174304649658927425 6239341020864383202110372958725762358509643110564073 501508187510676594629205563685529475213500852879416 377328533906109750544334999811150056977236890927563 What is the advantage? 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)

Document Classification: KPMG Public QFT 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

Document Classification: KPMG Public Unstructured database search

What is the problem? 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 Amplitude

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

Expected exponential speedup

Document Classification: KPMG Public Quantum 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

Expected exponential speedup

Document Classification: KPMG Public Quantum 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

Expected exponential QMC speedup Quantum Monte Carlo methods Ashley Montanaro, 2017

Document Classification: KPMG Public Quantum 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.

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation 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

Document Classification: KPMG Public Quantum 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

Document Classification: KPMG Public QPE QFT QAA / Grover’s alg Quantum Phase Estimation / Quantum Fourier Transform Quantum Amplitude Amplification / Grover’s algorithm for eigenvalue estimation Don Coppersmith, 1994 unstructured search Michael Nielsen and Isaac Chuang, 2000 Lov Grover, 1996 HSP / Shor’s alg HHL * 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 Exponential speedup Patrick Rebentrost, Masoud Seth Lloyd, Masoud Mohseni, Ashley Montanaro, 2017 * 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

Document Classification: KPMG Public QML 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

Document Classification: KPMG Public There 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…

Document Classification: KPMG Public Q-uestions?

Document Classification: KPMG Public Contact Information

Troels Steenstrup Jensen [email protected]