Introduction to Quantum Computing

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1 Introduction to Quantum Computing Petros Wallden Lecture 1: Introduction 18th September 2017 School of Informatics, University of Edinburgh

2 Resources 1. Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang 2. Lecture Notes available on 1

3 Moore s Law & Quantum Mechanics The number of transistors in each microchip double every two years Soon we will reach atomic scale Quantum Mechanics govern physical systems at this scale Quantum Fluctuations and Uncertainty will affect classical computations 2

4 Bits Vs Qubits Bit Takes values either 0 or 1 Measurement reveals the value of the bit Can be copied String of bits are described in terms of single bits (local) Qubit Can behave as being simultaneously 0 and 1: α 0 + β 1 Measurement disturbs the system Cannot be copied String of qubits can have properties that cannot be described in terms of single qubits (non-local) Qubits behave as waves and interfere with each other Qubits are physical systems. Many different systems have been used such as: Photons (polarization, number, time-bin encoding), Coherent Light, Electrons (spin, number), Nuclear spin, Optical lattices, Superconductors, etc. 3

5 Quantumness as Resource Nobel laureate Richard Feynman 1982: Quantum Computer is a computer that uses QM to its advantage. It can simulate quantum systems. Great Developments: - Quantum Algorithms can lead to speed-up - Quantum Computers can break classical Cryptosystems such as the RSA - Quantum Cryptogaphy can encrypt messages with Unconditionally Security (not relying in computational assumptions) - Principles of Quantum Computation can be used to simulate and explore physical phenomena at domains that are not accessible from Black Hole thermodynamics to Condense Matter Physics 4

6 Secure Quantum Communication - Many quantum cryptographic protocols: Encryption, secret sharing, digital signatures, coin flipping, Unconditionally secure homomorphic encryption - Implementations of Quantum Key Distribution Networks between cities exist in many countries. QKD systems are provided by commercial companies (e.g. idquantique) Quantum Computers - There exist different models of Quantum Computation: Quantum Circuit, Measurement Based (these two will be covered), Adiabatic QC, Topological QC - Implementations have attempted to used different physical systems. Still not scalable (only few qubits operations e.g. factored 143). Superconductor based, Trapped ion, Optical lattices, Nuclear magnetic resonance, quantum optics NQIT (Networked Quantum Information Technologies) Hub (lead by Oxford, Edinburgh is part of): Q20:20, 20 ion traps of 20 qubits each, connected with photons.

7 Quantum Algorithms Speed-up Deutsch & Jozsa showed the first speed up Given a Boolean function f : {0, 1} n {0, 1} determine if it is constant or balanced f = 1 2 n x {0,1} ( 1) f(x) x n The state for any constant function is orthogonal to the state of any balanced function 5

8 Simon s Problem Given a function f : {0, 1} n {0, 1} n finds a such that f(x + a) = f(x) Shor s Algorithm Given n-bit integer, find the prime factorisation. Breaks the RSA cryptosystem (most currently used public key encryptions are based on this)

9 History 1980s Idea of quantum computation. Paul Benioff, Yuri Manin, Richard Feynman, David Deutsch 1990s Theory of efficient quantum simulation. Seth Lloyd 1994 Peter Shor s algorithms for factoring and discrete log. Quantum computers can break RSA, Diffie-Hellman, El Gamal, Elliptic Curve Cryptography and others 2001 Experiment factors 15 using Shor s algorithm 2010s D-Wave, Google, IBM, NQIT and various universities work on developing quantum computers How serious is the involvement in quantum computation? 6

10 Who invests in Quantum Computing? 7

11 Who invests in Quantum Computing? 8

12 Applications 9

13 Misconceptions 10

14 Misconceptions 11

15 State of Art 2001 Shor s algorithm factors 15 on 7 qubits 2011 Shor s algorithm factors Universal quantum computation on 2 fault tolerant qubits Qubits and gates in silicon chips 2015 D-Wave 2X, 1000 qubits, optimization problems, no fault tolerance 2016 IBM, universal quantum computation on 5 fault tolerant qubits (publicly available) 2020 NQIT, Q20:20, fault tolerant (20 qubits), scalable 12

16 State of Art: Cryptography 13

17 State of Art: Cryptography 14

18 What can you buy 15

19 What can you buy 16

20 Quantum Mechanics Nobel laureate Niels Bohr (photo with Einstein) Anyone who is not shocked by quantum theory has not understood it - Basic resource for QC is the distinct properties of quantum theory - To appreciate this one needs to (attempt to) understand QM - QM has been proven very successful and all so far tested predictions has been verified at a unprecedent level of accuracy - However, the conceptual challenges posed by QM are profound. Classical notions such as locality, non-contextuality, determinism even realism has been challenged 17

21 - The role of the observer and of the measurement are very different - Properties with no classical analogue: Uncertainty, wave-particle duality, no-cloning, indistinguishability of quantum states, teleportation - Also QM is incompatible with the other most successful physical theory General Relativity. This is possible because the former deals with the micro-world while the latter with macro-world. However, for a complete theory of nature one needs to construct a theory that includes both QM and GR and this is probably the greatest challenge for contemporary physics.

22 Content of the Course Basic concepts from Linear Algebra Axioms of Quantum Mechanics Non-locality, Bell s inequalities and the interpretations of QM No-cloning and no-deleting theorem Quantum Computing via the circuit model Quantum complexity Quantum Algorithms Quantum Cryptography Quantum Computing via the measurement-based model 18

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