A Quantum of Computing using the Incomprehensible to solve the Intractable Dave McMahon [email protected] @mcmahond http://davemcmahon81.wordpress.com http://www.nxtgenug.net Your Speaker

15 years in Software Industry 9 years Software Systems Architect at Ridgian Co-Founder The Next Generation User Group Spoken at User Groups, DDDs, TechEd First Class Honours Degree in Theoretical Physics From York University 1983 http://www.nxtgenug.net The Aim To show you how Quantum Theory and Quantum Effects are being used to bring about future generations of computers http://www.nxtgenug.net

The Agenda Limits of the Von Neumann Machine What can Quantum Computing Offer? The QuBit A Basic Quantum Algorithm A Quantum Search Algorithm Quantum Computers Today http://www.nxtgenug.net Von Neumann Machine http://www.nxtgenug.net Von Neumann Machine

http://www.nxtgenug.net Limits of the Von Neumann Machine Classical Limitation s Quantum Limitation s Relativisti c Limitation

s http://www.nxtgenug.net Classical Limitations Heat http://www.nxtgenug.net Classical Limitations Heat is the main reason that: Clock speeds are not increasing Multi-core architectures are becoming the norm Heat will (probably) :

Signal the end of the CMOS chip Result in other technologies replacing it However, let us assume that we do solve the heat problem ... http://www.nxtgenug.net Quantum Limitations Measurement and Manufacture http://www.nxtgenug.net Quantum Limitations Quantum Limitations will:

Limit the accuracy to which chips can be made with current etching technology Limit the size to which individual transistor gates can be miniaturised Quantum Limitations are a natural limitation to evermore miniaturisation Lets assume however that we overcome these issues ... http://www.nxtgenug.net Relativistic Limitations Timing and Energy

http://www.nxtgenug.net Relativistic Limitations Consider a 3 GHz processor (off todays shelf) In 1 cycle a signal can propagate at most c/(3 GHz) = ~ 10cm For a 1-cycle round-trip to cache memory and back the cache location can be at most ~5 cm away. Electrical signals travel at ~0.5 c in typical materials therefore in practice, a 1-cycle memory can be at most ~2.5 cm away As clock speeds increase architectures must be increasingly local. http://www.nxtgenug.net

Relativistic Limitations Computer systems are powered at 10V which equates to 10 eV. Corresponds to a max clock speed of 9.7 PHz ~ 5 million times faster than todays CPUs ~ 100,000 times faster than todays superconducting logics http://www.nxtgenug.net Where do we go from here? perhaps Quantum Computing http://www.nxtgenug.net

What Can Quantum Computing Offer? Different Modes of Computatio n Reversible Computatio n Speed of Computatio n http://www.nxtgenug.net

So How can you use Quantum Effects to do Computation? consider the classic Double Slit experiment http://www.nxtgenug.net The Double Slit Experiment Part 1 Beam of Light Barrier with a Single Slit Light waves fan out like water waves

Detector Screen Shows Blurred pattern http://www.nxtgenug.net The Double Slit Experiment Part 2 Beam of Light Barrier with a Double slit

Light waves fan out like water waves Detector Screen Shows Interference Pattern http://www.nxtgenug.net The Double Slit Experiment Part 2 A Beam of Light

Barrier with a double slit Light waves fan out like water waves A Detector Screen ... but Light Now shows an Interference Beams

pattern are NOT made of waves . They are made of particles called Photons ... http://www.nxtgenug.net The Double Slit Experiment Part 3 Beam of Light Barrier with a Double slit Photons

Interfere? Detector Screen Shows Interference Pattern http://www.nxtgenug.net The Double Slit Experiment Part 4 Beam of Light Barrier with a Double slit

Photons Interfere? Detector Screen Still Shows Interference Pattern http://www.nxtgenug.net The Double Slit Experiment Part 5 Beam of Light

Barrier with a double slit Detect Photon? Detector Screen No Interference Pattern http://www.nxtgenug.net The Double Slit Experiment

Questions: How does the Photon know where to end up? If there is interference, what does the Photon interfere with? Why does trying to detect what is happening affect the outcome? There are many different Quantum Theory Interpretations which scientists 100 years after the discovery of quantum nature of light are still arguing about . http://www.nxtgenug.net Quantum Theory Interpretation Quantum Theory Interpretations try to explain what is happening

Copenhagen Interpretation Instrumentalist - Shut up and calculate! Others The Many Universes Interpretation (Multiverse) They dont have to be true. They only have to be consistent. http://www.nxtgenug.net The Multiverse

We are aware of only a part of reality Every object in a universe has a counterpart in another universe http://www.nxtgenug.net The Multiverse Counterparts can behave differently from each other Counterparts can affect each other at the Quantum level Result of effects is Quantum Interference http://www.nxtgenug.net

The Multiverse Across the multiverse, the photon goes through both slits at the same time a phenomenon known as Superposition The Photons experience Quantum Interference with each other between universes The final resulting pattern is a interference pattern across the Multiverse. http://www.nxtgenug.net The QuBit A Multiversal Object http://www.nxtgenug.net The QuBit

The Quantum World Superposition Quantum Interference QuBit http://www.nxtgenug.net The QuBit Physicists measure Expectation Values of Quantum Systems or Observables

Expectation Values of Quantum Observables are: The average outcome of a measurement repeated many times or ... The average outcome of a measurement performed on many copies of a system over a region of the Multiverse. If a Quantum System has Observables which are all defined in terms of Boolean Values that system can be termed a QuBit. http://www.nxtgenug.net The QuBit A Bit : A classical system which can take one of two values one or zero A Boolean Observable: A Quantum Observable

whose spectrum contains two values and which can be either or both values simultaneously. Sharp (Same Value in all Universes) Non-Sharp (Superposition) A QuBit : A physical system, each of whose nontrivial observables is Boolean http://www.nxtgenug.net A Single QuBit System http://www.nxtgenug.net A Single QuBit System An observable Z is deemed to be -1 when going to the North Direction and 1 when going in the East Direction. is the Expectation (Average) Value being measured.

Time 0 1 North East Mirror Beam Splitter Beam Splitter

Mirror http://www.nxtgenug.net A Single QuBit System At Time 1 the Observable Z is in SuperPosition with the photon going in both the North Direction and the East Direction in Different Universes. The Expectation Value of Z(t) is therefore 0. Time 0 1

1 0 North East Mirror Beam Splitter Beam Splitter Mirror http://www.nxtgenug.net

A Single QuBit System At Time 2 the Observable Z is in SuperPosition with the photon going in both the East Direction and the North Direction in Different Universes. The Expectation Value of Z(t) is therefore still 0. Time 0 1 1 0

2 0 North East Mirror Beam Splitter Beam Splitter Mirror http://www.nxtgenug.net

A Single QuBit System Theoretical predictions backed up by experimental observations show that the East detector always picks up the Photon and the North Detector never does. So the Expectation value Z(t) is now 1. Time 0 1 1 0

2 0 3 1 North East Mirror Beam Splitter Beam Splitter

Mirror http://www.nxtgenug.net A Single QuBit System Any attempt to determine an intermediate state results in a final expectation value for Z(t) of 0. In this case photons appear at both detectors. Time 0 1 1

0 2 0 3 0 North East Mirror

Beam Splitter Beam Splitter Mirror http://www.nxtgenug.net A Real QuBit This is how a system can use a single QuBit to measure a specific value of -1 or 1. Time 0

1 1 0 2 0 3 1 North

East Mirror Beam Splitter Beam Splitter Mirror http://www.nxtgenug.net A Real QuBit NOT Gate Whats This? Mirror

Input Input Beam Splitter B N B B N Beam Splitter B

Output Output Mirror New Mode of Computation t=3 t=0 t=1 t=2 Operation is Unit Operator http://www.nxtgenug.net

Quantum Computations are Reversible WARNING! Severe Scientific Reasoning Break ... http://www.nxtgenug.net Quantum Computations are Reversible Gates represented by unitary matrices All unitary matrices have inverse matrices that are also unitary so all quantum gates are reversible gates represent computations so all quantum computations are reversible.

http://www.nxtgenug.net Quantum Computations are Reversible END OF Severe Scientific Reasoning Break ... http://www.nxtgenug.net Quantum Computations are Reversible x x f(x) y

yf(x) Must have same number of outputs as inputs Must preserve the original input Copies the result to the control QuBit http://www.nxtgenug.net A Basic Quantum Algorithm http://www.nxtgenug.net A Basic Quantum Algorithm A Program is a way of preparing a computer to carry out a Computation A Computation is a problem to which an

Algorithm is a solution An Algorithm is a Hardware-independent specification of a Computation http://www.nxtgenug.net A Basic Quantum Algorithm Question: Can we find out what the Boolean function f(x) is without looking inside the Black Box? x x f y

yf(x) Unit Operator NOT Operator f(x) = 1 OR f(x) = -1 Constant 1 Constant -1 http://www.nxtgenug.net A Basic Quantum Algorithm x=1 x=1 x=-1

f y=1 x=-1 f f(1) y=1 f(-1) Must Run System Twice True for both classical and quantum versions http://www.nxtgenug.net

A Basic Quantum Algorithm x=1 x=1 x=-1 f y=1 x=-1 f f(1)

y=1 f(-1) What about does f(1) = f(-1)? Same as f(1)f(-1) Classically we must still run the system twice. http://www.nxtgenug.net A Basic Quantum Algorithm Q1 Q2 |1>

|1> H NOT |-1> H |1>|-1> |-1>|1> |1>|-1>

f H |1> f(|1>|-1>) = f(|1>)f(|-1>) We prepare our two inputs using a NOT Gate We put our QuBit into a State of Superposition using Hadamard Gate (Superposition state represented by |1>|-1>) Run the Calculation f f(|1>)(f|-1>) appears in ONE run of the system Demonstrates Quantum Parallelism http://www.nxtgenug.net

A Quantum Search Algorithm http://www.nxtgenug.net A Quantum Search Algorithm Exhaustive Search Algorithm Example in C# Programming Language public Boolean Is151InAList(Ilist aList) { for (Int32 i=0 ; i < aList.Count() ; i++) { if (aList[i] == 151) {return true;} } return false; }

Classically this would take on average half the number of items in the list i.e N/2. It could take N-1 times. http://www.nxtgenug.net A Quantum Search Algorithm F(x) is -1 when X is the searched for value F(X) is 1 when X is not the searched for value. X is in the range 0 -> N where N = 2L L QuBits x 1 QuBit

y F x yF(x) http://www.nxtgenug.net A Quantum Search Algorithm H puts L QuBits in Superposition Place four Quantum Gates M H B H in succession MHBH is the Grover Operator L QuBits

x Aux QuBit y H M H G B

H http://www.nxtgenug.net A Quantum Search Algorithm Expectation Value Superposition State (H) The Grover Operator Expectation Value Mark Target (M) T(arget) Grover

Diffusion Operator (HBH) (Invert about Average) T Expectation Value http://www.nxtgenug.net A Quantum Search Algorithm Consider the combined QuBit states as a vector The Grover Operator rotates vector towards target Target State Initial State

http://www.nxtgenug.net A Quantum Search Algorithm Grover Search Algorithm = Hadamard Gate + Repeated Grover Operators. Maths shows ~ N Grover Operations needed L QuBits x Aux QuBit y

H G G G x yF(x) http://www.nxtgenug.net A Quantum Search Algorithm At 100 million Grover operations per second Searching 1030 items

~4 months http://www.nxtgenug.net A Quantum Search Algorithm At 100 million Classical Operations per second Searching 1030 items ~Age of Universe http://www.nxtgenug.net A Quantum Search Algorithm Using a Quantum Computer you would be running 1030 operations in parallel.

Thats more operations than if you took all the Silicon on Earth and made it into transistors and ran them all in parallel! http://www.nxtgenug.net Finally Quantum Computers Today back to reality http://www.nxtgenug.net Quantum Computers Today Quantum Encryption Devices http://www.magiqtech.com/

University of Innsbruck 14 Qubit Calculation (1 Apr 2011) D-Wave Machine http://www.dwavesys.com http://scottaaronson.com/blog 128 Qubit calculation? http://www.nxtgenug.net Quantum Computers Today http://www.nxtgenug.net Summary

We will reach a limit on current CMOS technology soon Quantum Computing makes use of the Quantum Multiverse Trying to observe what happens destroys Quantum Interference Some known Quantum Algorithms will make certain intractable problems do-able A viable Universal Quantum Computer is yet to be built. http://www.nxtgenug.net A Quantum of Computing Qs & As Thank You

Dave McMahon [email protected] @mcmahond http://davemcmahon81.wordpress.com http://www.nxtgenug.net