Quantum Computing
MAS 725
Hartmut Klauck
NTU
13.2.2012
Organization
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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
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Homework (biweekly): 40%
Final exam: 60%
Homework must be written individually!
And handed in on time
Required Background
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Linear algebra
Some basic probability theory
No background in physics required
Textbook
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Nielsen/Chuang: Quantum Computation and
Quantum Information
(Cambridge)
Recommended Reading
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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
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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
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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
Quantum Mechanics and
Computing
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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
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Chip designers nowadays mostly “combat” quantum
effects
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Is it possible to make good use of quantum effects?
Quantum Computing
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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)
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I.e., quantum computers
Quantum Computing
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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
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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
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There are several models of quantum computing
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E.g. Deutsch (1985) defined Quantum Turing
Machines as a universal model of quantum
computation
Another (easier to handle) model are quantum
circuits
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But first we need to understand some basics about
quantum mechanics
Quantum Mechanics
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The double slit experiment for light
Quantum Mechanics
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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
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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
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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
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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
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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
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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
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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
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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
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|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
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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
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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 product
h (vi) | (wi) i = i=1…k vi* wi
x*: complex conjugate
Dirac Notation
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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
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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
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For Hilbert spaces H, K, with dimensions dH and dK
their tensor product HK 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 HK: all |xi|yi =|xyi where |xi,|yi are
basis vectors of H,K
Not all vectors in HK are tensor products of vectors
in H and K
Example
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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?
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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
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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 =I
U*: take the complex conjugate of all entries
Uy = (U*)T
Unitary transformation preserve the euclidean length of
vectors
Transformations in QM: unitary
(i.e., reversible and length preserving)
Examples
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On one qubit:
classical transformations: identity, negation
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Hadamard Transformation:
Applying Hadamard
Applying Hadamard
Applying Hadamard
Unitary Transformations
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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 xy
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Tensor product for matrices:
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A B=
Unitary Transformations
Hadamard Transformation
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x,z2{0,1}n and x¢z= xizi
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For any x we have Hn |xi=
1/2n/2 (|0i +(-1)x(1) |1i)(|0i +(-1)x(n) |1i)
Applying many unitary
transformations
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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
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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
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Measuring the state
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Will result in the outcome 0 or 1, each with
probability ½
If we measured 1, the resulting state after the
measurement will be
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Overview
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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