Quantum ComputingMAS 725Hartmut KlauckNTU13.2.2012

Organization Lectures: Mo. 10:00, TR+9 Lecturer: Hartmut Klauck Office: SPMS-MAS-05-44 Website: http://www.ntu.edu.sg/home/hklauck/QC12.html

Grading

Homework (biweekly): 40% Final exam: 60%

Homework must be written individually! And handed in on time

Required Background

Linear algebra Some basic probability theory No background in physics required

Textbook

Nielsen/Chuang: Quantum Computation and Quantum Information

(Cambridge)

Recommended Reading

http://homepages.cwi.nl/~rdewolf/qcnotes.pdf http://www.cs.berkeley.edu/~vazirani/s09quantum.html http://www.cs.uwaterloo.ca/~watrous/lecture-notes.html

Quantum Mechanics

Quantum mechanics is one of the basic theories of physics

Quantum mechanics is concerned with states, and how they evolve/change

Includes many “strange” effects that are different from “classical”, Newtonian mechanics: Superposition Entanglement

Such effects usually appear in very small systems

Quantum Mechanics and Computing Moore’s Law: The number of transistors that can be

placed on a chip doubles every two years I.e., the computational power doubles This trend has been approximately true for more

than 50 years Main way to achieve this is by making smaller

transistors! Even today quantum mechanical effects are

important to chip design

Another problem: heat generation in integrated circuits

This heat is the result of erasing information Quantum computations are (for the most part)

reversible Reversible computations (ideally) do not

generate (much) heat

Quantum Mechanics and Computing

Quantum Mechanics and Computing Chip designers nowadays mostly “combat” quantum

effects

Is it possible to make good use of quantum effects?

Quantum Computing

First suggested by Feynman and Benioff in the 1980’s

Feynman’s observation: Simulating quantum systems on classical

computers takes exponential time in the ‘size’ of the quantum system

Conclusion: build universal quantum systems• Quantum systems that can simulate all other quantum

systems (up to a size) I.e., quantum computers

Quantum Computing

Hence reasons for investigating quantum computing are: Making good use of quantum effects instead of

trying to force microscopic system to adhere to classical physics

There are quantum algorithms and protocols that achieve things that seem to be impossible for classical algorithms/protocols

If the world is quantum mechanical, the ultimate limits of computation are determined by quantum physics

Quantum Computing Examples

Some example of tasks that quantum computers can do: Efficiently factor natural numbers Break public key cryptosystems like RSA Search an unordered database in sublinear time Provide cryptographic protocols that are secure

without placing assumptions on the computational power of an eavesdropper

Quantum Computing Models

There are several models of quantum computing

E.g. Deutsch (1985) defined Quantum Turing Machines as a universal model of quantum computation

Another (easier to handle) model are quantum circuits

But first we need to understand some basics about quantum mechanics

Quantum Mechanics

The double slit experiment for light

Quantum Mechanics

Perform the “same” experiment with electrons We observe the same outcome of the experiment Even when single electrons are emitted The wave-like behavior is not just statistical

Quantum Mechanics

The name “Quantum Mechanics” (coined by Planck) derives from the fact that certain quantities can change only by a discrete amount E.g. The smallest unit of electromagnetic

radiation is a photon (a quantum of light) It is possible to emit and detect single photons

Quantum Mechanics

Quantum Mechanics

Some History

Development of quantum mechanics:Planck 1900, Schrödinger, Heisenberg, Bohr, Einstein.....

1930’s: von Neumann’s formalism 1935: Einstein, Podolsky, Rosen describe

“entanglement” in an attempt to show that quantum mechanics is not a “complete” theory of reality (German “Verschränkung”)

Today quantum mechanics is the best established theory in physics

(Quantum) Computer Science

1936: Turing defines a “universal” machine (Church Turing Thesis)

1948: Shannon’s Information Theory 1965: Moore’s Law 1982: Feynman proposes quantum computers (to

simulate quantum systems) 1982 Wiesner: first proposal of quantum

cryptography published (after more than 10 years)

(Quantum) Computer Science

1985: Deutsch finds the first quantum algorithm 1993: quantum teleportation 1994: Shor finds a quantum algorithm for

factorization 1996: Grover’s algorithm finds a marked element in

a database with n elements in time Since then the field is steadily growing…

Quantum States

Quantum mechanics is an abstract theory of states and transformations on states

Can be derived from certain axioms Quantum states are vectors in a Hilbert space Hilbert Space:

A real or complex vector space with an inner product that maps vectors to their length

Must be complete We will only consider finite dimensional spaces Usually either Rn or Cn

Bits

A bit is either the value 0 or the value 1, stored in a register

We will write the state of a bit as |0i, |1i Examples:

A bit stored in the memory of a computer The path that a ball took in a giant double slit

experiment

Bits and Qubits

We identify the states |0i, |1i with the basis vectors of a two dimensional space (say C2):(1,0) and (0,1)

The states of a quantum bit (qubit) are arbitrary unit vectors in C2

Hence all the states of a qubit are:|0i + |1i with ||2 +||2=1

Qubits

|0i, |1i are two basis vectors in C2

Qubits have states: |0i+ |1i with ||2 +||2=1 , are called amplitudes Qubit states are unit vectors under the euclidean norm

Comparison to Probability Theory Suppose we have a random bit (say a coin flip) Then we need to specify the probability of 1 and 0

(coin may not be fair) For example

0 has probability p, 1 has probability 1-p) probability distributions on bits are unit vectors under 1-Norm

Qubits: , are complex numbers,possibly negative!

The squares of the absolute values of the amplitudes form a probability distribution

The Quantum Formalism

Quantum states are vectors in a Hilbert space The Hilbert space corresponds to a register that can

hold a quantum state

Hilbert space here: Ck with the inner producth (vi) | (wi) i = i=1…k vi

* wi

x*: complex conjugate

Dirac Notation

h | “BRA” row vector | i “KET” column vector h | i inner product

(product of a row and a column vector) |ÁihÃ| outer (matrix valued) product

Many Qubits

To hold k qubits we need a Hilbert space of dimension 2k

I.e. 2k basis vectors (corresponding to the 2k values of k bits) First notation: |ii, i=1,...,2k. Unit vectors are of the form

i i |ii; i=1....2k with i |i |2 = 1 Better notation: identify i=1...2k with x2{0,1}k

Basis states are |xi, x2 {0,1}k

Basis states correspond to classical values a register can hold General quantum states are linear combinations of the 2k

classical (basis) states Also called “superpositions”

Tensor Product

For Hilbert spaces H, K, with dimensions dH and dK

their tensor product H K is a Hilbert space of dimension dH¢dK

Tensor product of vectors: (a1,..., al) (b1,...,br)= (a1b1,a1b2,...,a1br,a2 b1,......,albr)

Example: |0i = (10)T; |1i= (01)T

and |01i= |0i |1i = (0100)T

A basis of H K: all |xi |yi =|xyi where |xi,|yi are basis vectors of H,K

Not all vectors in H K are tensor products of vectors in H and K

Example

Basis of C4: |00i, |01i, |10i, |11i Another basis: (|00i + |11i)

(|00i - |11i)(|01i + |10i)(|01i - |10i)(scaled by square root of 2) None of these are tensor products of vectors in

C2

What can we do with one or more qubits? Quantum systems evolve according to the

Schrödinger equation The result can be described as the application of a

unitary transformation to the quantum state Additionally quantum states can be measured

This leads to observable output Need some background from linear algebra…

Linear Algebra

Linear transformations: A(x+y)=Ax + Ay x,y: vectors in Ck, A: k£k matrix (complex entries) Over the reals a linear transformation O is orthogonal, if

OOT=I Over the complex numbers a matrix U is unitary, if

UUy =IU*: take the complex conjugate of all entriesUy = (U*)T

Unitary transformation preserve the euclidean length of vectors

Transformations in QM: unitary(i.e., reversible and length preserving)

Examples

On one qubit:classical transformations: identity, negation

Hadamard Transformation:

Applying Hadamard

Applying Hadamard

Applying Hadamard

Unitary Transformations

Define U |xi for all x2 {0,1}k

) U is defined. The U|xi need to be unit vectors and U|xi? U|yi for all xy

Tensor product for matrices:

A B=

Unitary Transformations

Hadamard Transformation

x,z2{0,1}n and x¢z= xizi

For any x we have H n |xi=

1/2n/2 (|0i +(-1)x(1) |1i) (|0i +(-1)x(n) |1i)

Applying many unitary transformations Later we will construct unitary transformations as

the product of many “simple” unitary transformation

First applying a unitary U, and then a unitary V is the same as applying the product VU.

Note that the product of two unitary matrices is unitary

Careful: matrix multiplication is not commutative! The exact sequence of multiplications matters

Measurements

Quantum states (unit vectors in Ck) can be changes by applying a unitary transformation

Computations on quantum states consist of unitary transformations and measurements

Measurements allow us to access the result of a computation

What happens if we measure i i |ii ? The result will be i with probability |i|2

i |i2|=1 is very helpful now!

After measuring the value i the state “collapses” to |ii

Example

Measuring the state

Will result in the outcome 0 or 1, each with probability ½

If we measured 1, the resulting state after the measurement will be

Overview

Quantum states: unit vectors in a Hilbert space, the log of the dimension corresponds to the number of qubits

States in a Hilbert space of dimension 2k correspond to superpositions of strings of length k and the space is a register of k qubits

Evolution: by applying unity transformations Measurement: i |ii results in output i

with probability |i|2, the state collapses to |ii if i is the measurement result