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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 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 © 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 © 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. 5 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! © 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. 6 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 © 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. 8 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 © 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. 9 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 © 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. 11 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 © 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. 12 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 © 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. 13 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 © 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. 14 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! © 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. 15 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. © 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. 16 Document Classification: KPMG Public QFT Quantum Fourier Transform Don Coppersmith, 1994 Legend: Expected quadratic speedup Expected exponential speedup © 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. 17 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 © 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. 18 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 © 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. 19 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) © 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. 20 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
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