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Introduction to
Quantum Computing
Petros Wallden∗
Lecture 1: Introduction
18th September 2017
∗School of Informatics, University of Edinburgh
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Resources
1. “Quantum Computation and Quantum In-
formation”
by Michael A. Nielsen & Isaac L. Chuang
2. Lecture Notes available on
http://qcintro.wordpress.com
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Moore’s Law
& Quantum Mechanics
• The number of transistors in each microchip doubleevery two years
• Soon we will reach atomic scale
• Quantum Mechanics govern physical systems at thisscale
• Quantum Fluctuations and Uncertainty will affectclassical computations
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Bits Vs Qubits
Bit QubitTakes values either 0 or 1 Can behave as being simul-
taneously 0 and 1: α |0〉+β |1〉
Measurement reveals thevalue of the bit
Measurement disturbs thesystem
Can be copied Cannot be copiedString of bits are describedin terms of single bits (lo-cal)
String of qubits can haveproperties that cannot bedescribed in terms of singlequbits (non-local)Qubits behave as wavesand interfere with eachother
• Qubits are physical systems. Many different systemshave been used such as:
Photons (polarization, number, time-bin encoding), Co-
herent Light, Electrons (spin, number), Nuclear spin,
Optical lattices, Superconductors, etc.
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Quantumness as ResourceNobel laureate Richard Feynman 1982:
Quantum Computer is a computer that uses QM to itsadvantage. It can simulate quantum systems.
Great Developments:
- Quantum Algorithms can lead to speed-up
- Quantum Computers can break classical Cryptosys-tems such as the RSA
- Quantum Cryptogaphy can encrypt messages with Un-conditionally Security (not relying in computational as-sumptions)
- 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
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Secure Quantum Communication
- Many quantum cryptographic protocols: Encryption,secret sharing, digital signatures, coin flipping, Uncon-ditionally secure homomorphic encryption
- Implementations of Quantum Key Distribution Net-works between cities exist in many countries. QKDsystems are provided by commercial companies (e.g.idQuantique)
Quantum Computers
- There exist different models of Quantum Computa-tion: Quantum Circuit, Measurement Based (these twowill be covered), Adiabatic QC, Topological QC
- Implementations have attempted to used different phys-ical systems. Still not scalable (only few qubits op-erations e.g. factored 143). Superconductor based,Trapped ion, Optical lattices, Nuclear magnetic reso-nance, quantum optics
NQIT (Networked Quantum Information Technologies)Hub (lead by Oxford, Edinburgh is part of): Q20:20, 20ion traps of 20 qubits each, connected with photons.
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Quantum Algorithms
Speed-up
- 1985 Deutsch & Jozsa showed the first speed up
Given a Boolean function f : {0,1}n → {0,1} determineif it is constant or balanced
|f〉 = 1√2n
∑x∈{0,1}n(−1)f(x) |x〉
The state for any constant function is orthogonal to thestate of any balanced function
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- 1994 Simon’s Problem
Given a function f : {0,1}n → {0,1}n finds a such thatf(x+ a) = f(x)
- 1994 Shor’s Algorithm
Given n-bit integer, find the prime factorisation. Breaksthe RSA cryptosystem (most currently used public keyencryptions are based on this)
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History• 1980s Idea of quantum computation. Paul Benioff,
Yuri Manin, Richard Feynman, David Deutsch
• 1990s Theory of efficient quantum simulation. SethLloyd
• 1994 Peter Shor’s algorithms for factoring and dis-crete log. Quantum computers can break RSA,Diffie-Hellman, El Gamal, Elliptic Curve Cryptog-raphy and others
• 2001 Experiment factors 15 using Shor’s algorithm
• 2010s D-Wave, Google, IBM, NQIT and variousuniversities work on developing quantum computers
How serious is the involvement in quantum computa-tion?
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Who invests in
Quantum Computing?
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Who invests in
Quantum Computing?
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Applications
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Misconceptions
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Misconceptions
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State of Art• 2001 Shor’s algorithm factors 15 on 7 qubits
• 2011 Shor’s algorithm factors 21
• 2012 Universal quantum computation on 2 faulttolerant qubits
• 2014-2015 Qubits and gates in silicon chips
• 2015 D-Wave 2X, 1000 qubits, optimization prob-lems, no fault tolerance
• 2016 IBM, universal quantum computation on 5fault tolerant qubits (publicly available)
• 2020 NQIT, Q20:20, fault tolerant (20 qubits),scalable
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State of Art:
Cryptography
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State of Art:
Cryptography
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What can you buy
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What can you buy
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Quantum MechanicsNobel laureate Niels Bohr (photo with Einstein)
“Anyone who is not shocked by quantum theory has notunderstood it”
- Basic resource for QC is the distinct properties ofquantum theory
- To appreciate this one needs to (attempt to) under-stand QM
- QM has been proven very successful and all so fartested predictions has been verified at a unprecedentlevel of accuracy
- However, the conceptual challenges posed by QM areprofound. Classical notions such as locality, non-contextuality,determinism even realism has been challenged
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- The role of the observer and of the measurement arevery different
- Properties with no classical analogue: Uncertainty,wave-particle duality, no-cloning, indistinguishability ofquantum states, teleportation
- Also QM is incompatible with the other most success-
ful 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 chal-
lenge for contemporary physics.
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Content of the Course• Basic concepts from Linear Algebra
• Axioms of Quantum Mechanics
• Non-locality, Bell’s inequalities and the interpreta-tions 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-basedmodel
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