# a tutorial for computer scientists

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16 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Itay Hen

June 6
th
, 2012

UCSC

Itay

Hen

UCSC

June 6
th

2012

computation

a tutorial for computer scientists

Dept. of Physics, UCSC

Itay Hen

June 6
th
, 2012

UCSC

introduction I
: what is a quantum computer?

introduction II

quantum computing

[AQC and machine learning(?)]

Outline

Itay Hen

June 6
th
, 2012

UCSC

What is a

Quantum Computer?

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

both classical and quantum computers may be viewed as

machines that perform computations on given inputs and produce

outputs. Both inputs and outputs are strings of
bits
(0’s and 1’s).

classical computers are based on
manipulations of bits
.

at any given time the system is in a unique “classical”

configuration (i.e., in a state that is a sequence of 0’s and 1’s).

computer

010011

011

input state

output state

computation

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

a classical algorithm (circuit) looks like this:

at
any given time the system is in a unique “classical”

configuration (i.e., in a state that is a sequence of 0’s and 1’s).

computer

010011

011

input state

output state

computation

0 1 0 1 1

0 1 0 1 0

1 1 1 0 1

1 0 0 1 0

input state

output state

computation

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

quantum computers on the other hand manipulate objects called

quantum bits
or
qubits

for short.

010011

011

input state

output state

computation

computer

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

quantum computers on the other hand manipulate objects called

quantum bits
or
qubits

for short.

010011

011

input state

output state

computation

computer

w
hat are
qubits
?

Itay Hen

June 6
th
, 2012

UCSC

What are
qubits
?

a
qubit

is a generalization on the concept of
bit
.

motivation / idea behind it
: small particles, say electrons, obey

different laws than ‘big’ objects (such as, say, billiard balls).

small particles obey the laws of
Quantum Physics
.

big objects are classical entities

they obey the laws of
Classical

Physics
. (however, big objects are collections of small particles!)

most notably: a quantum particle can be in a
superposition
of

several classical configurations.

classical world

I’m here

I’m there

or

quantum world

I’m here

I’m there

a
nd

Itay Hen

June 6
th
, 2012

UCSC

What are
qubits
?

while a
bit
can be either in the a
|
0

state or a
|
1

state (this is

Dirac’s
|
𝑘 

notation).

a
qubit

can be in a superposition

of these two states, e.g.,

imagine
|
0

and
a
|
1

as being two
orthogonal basis vectors
.

while a classical bit is either of the two, a
qubit

can be an arbitrary

(yet normalized) linear combination of the two.

in order to
access the information stored in a
qubit

one must

perform a
measurement

that “collapses” the
qubit
. extremely non
-

intuitive. important: only 1 bit of information is stored in a
qubit
.

|
𝜑

=
1
2
|
0

+
|
1

Itay Hen

June 6
th
, 2012

UCSC

What are
qubits
?

take for example a system of 3 bits. Classically, we have
2
3
=
8

possible “classical configurations”:

the corresponding
quantum state of a 3
-
qubit system
would be:

this is a (normalized) 8
-
component vector over the complex

numbers (8 complex coefficients).
|
𝑖

enumerates the 8 classical

configurations

listed above.

|
000

,
|
00
1

,
|
0
1
0

,
|
0
11

,

|
1
00

,
|
1
01

,
|
1
10

,
|
1
11

,

|
𝜑

=


𝑖
7
𝑖
=
0
|
𝑖

Itay Hen

June 6
th
, 2012

UCSC

Encoding optimization problems

suppose that we are given a function

and are asked to find its minimum

and the corresponding minimizing
configuration
.

this is the same as having the diagonal matrix:

and
which is the smallest eigenvalue (minimum of
f

) and

corresponding eigenvector (minimizing configuration
)
.

here, the
basis vectors

𝑖

correspond to
|
𝑖

and the elements on

the diagonal are the evaluation of
f
on them.


:
{
0
,
1
}
3


=

Itay Hen

June 6
th
, 2012

UCSC

Encoding optimization problems

in matrix
-
form we can generalize classical optimization problems

to quantum optimization problems by adding off
-
diagonal

elements to the matrix.

we still look for the smallest
eigen
-
pair, i.e.,

but problem now is much harder.
(this is
Dirac’s


|

notation).

in quantum mechanics, the minimal (
eigen
)vector is called the

“ground state”

of the system and the corresponding minimal

(
eigen
)value is called the
“ground state energy”
.

also, the matrix
F

is called the Hamiltonian (usually denoted by
H
).

quantum mechanics is just linear algebra in disguise.

min
|
𝜑

𝜑
|

|
𝜑

=

min
𝜑
𝜑


𝜑

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

quantum computers manipulate
quantum bits
or
qubits

for short.

a quantum algorithm (circuit) may look like this:

010011

011

input state

output state

computation

computer

0 1 0 1 1

0 1 0 1 0

+

0 1 0 0 1

1 0 0 1 0

input state

output state

computation

1 1 0 1 0

+

1 1 0 0 1

+

1 1
1 0
1

Itay Hen

June 6
th
, 2012

UCSC

What is a quantum computer?

space of possible quantum states is huge.

range of operations on
qubits

is huge.

capabilities are greater
(name dropping: superposition,

entanglement, tunneling, interference…). can solve problems faster.

only problem with quantum computers: they do not exist!

theory is very advanced but many technological unresolved

challenges.

010011

011

input state

output state

computation

computer

Itay Hen

June 6
th
, 2012

UCSC

quantum computing

Itay Hen

June 6
th
, 2012

UCSC

Motivation

i
n
what ways quantum computers are
more efficient than
classical computers?

w
hat
problems could be solved more
efficiently on a q
uantum
computer
?

these are two of the most

basic and important questions in the

field of quantum information/computation.

clearly, a quantum computer is a

generalization of a classical computer

Itay Hen

June 6
th
, 2012

UCSC

best
-
known examples are:

Shor’s

algorithm for integer factorization. Solves the problem in
polynomial time (exponential speedup).

current quantum computers can factor all integers up to
21
.

Grover's
algorithm is a quantum algorithm for searching an unsorted
database with

entries
in


1
/
2

importance is huge (cryptanalysis, etc.).

Motivation

i
n
what ways quantum computers are
more efficient than
classical computers?

w
hat
problems could be solved more
efficiently on a q
uantum
computer
?

Itay Hen

June 6
th
, 2012

UCSC

people are considering “hard” satisfiability (SAT) and other

optimization problems which are at least NP
-
complete.

t
hese
are hard to solve classically;
time needed is
exponential in the input (exponential complexity).

c
ould
a quantum computer solve
these problems in an
efficient manner? perhaps even in polynomial
time
?

Motivation

Computing

is a general
approach to solve a broad range
of hard
optimization problems using a
quantum
computer
[
Farhi

et al
.,2001]

w
hat other problems can
quantum computers solve?

Itay Hen

June 6
th
, 2012

UCSC

quantum computation

Itay Hen

June 6
th
, 2012

UCSC

is different that circuit
-
based

computation (there are still other models of computation).

computation is
analog

in nature (as

opposed to 0/1 “digital” circuits).

it is based on the fact that the
state of a system will tend

to reach a minimum configuration
.
t
his is the
principle of least

action
.

The
nature of physical systems

Itay Hen

June 6
th
, 2012

UCSC

Analog classical optimization

we are given a landscape f(x) for which we need to find the

minimum configuration, i.e., the stable state of the ball.

if energy landscape is “simple”:

the
ball will
eventually end up

at the bottom of the hill.

x

f

Itay Hen

June 6
th
, 2012

UCSC

Analog classical optimization

we are given a landscape f(x) for which we need to find the

minimum configuration, i.e.,
the stable
state of the ball.

if energy landscape is
“jagged”:

ball will eventually end up in a

minimum but is likely to end up

in a local minimum.

this is no good.

x

f

Itay Hen

June 6
th
, 2012

UCSC

Analog classical optimization

we are given a landscape f(x) for which we need to find the

minimum configuration, i.e.,
the stable
state of the ball.

if energy landscape is
“jagged”:

ball will eventually end up in a

minimum but is likely to end up

in a local minimum.

this is no good.

probability of jumping over the

barrier is exponentially small in

the height of the barrier.

ball is unlikely to end up in the true minimum.

x

f

Itay Hen

June 6
th
, 2012

UCSC

Analog quantum optimization

we are given a landscape f(x) for which we need to find the

minimum configuration, i.e.,
the stable
state of the ball.

if energy landscape is
“jagged”:

ball will eventually end up in a

minimum but is likely to end up

in a local minimum.

this is no good.

quantum mechanically, the

ball can “tunnel through” thin but

high barriers!

sometimes more likely to end up in the true minimum.

tunneling is a key feature of quantum mechanics.

x

f

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

of
QM

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

of
QM

??

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

rephrasing:

if the ball is at
the
minimum (ground state)

of some function
(Hamiltonian)
, and

the function is changes
slowly enough
,

ball
will stay close to the
instantaneous

global minimum throughout
the evolution.

here,
initial function is easy to find the

minimum of
.
final function is much harder
.

to solve the problem!

of
QM

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

real QM example: change the strength of a
h
armonic potential

of a system in the ground state:

of
QM

𝜓
(
𝑥
,

)
2

harmonic

potential

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

real QM example: change the strength of a
h
armonic potential

of a system in the ground state:

an abrupt change

(a
diabatic

process):

of
QM

𝜓
(
𝑥
,

)
2

harmonic

potential

Itay Hen

June 6
th
, 2012

UCSC

of QM tells
us that
a
physical system

remains
in its instantaneous eigenstate if a given perturbation

is
acting on it slowly enough and if there is a gap between

the
eigenvalue and the rest of the Hamiltonian's spectrum
.

real QM example: change the strength of a
h
armonic potential

of a system in the ground state:

(an
process):

wave function can “keep up”

with the change.

of
QM

harmonic

potential

𝜓
(
𝑥
,

)
2

Itay Hen

June 6
th
, 2012

UCSC

The quantum

1. take a difficult (classical) optimization problem.

4. vary the Hamiltonian
slowly

and smoothly from

𝐻

𝑑

to
𝐻

𝑝

until
ground state of
𝐻

𝑝

is reached
.

the mechanism proposed by
Farhi

et al
., the QAA:

2. encode its solution in the ground state of a quantum

“problem” Hamiltonian
𝐻

𝑝
.

3. prepare the system in the ground state of another

easily solvable “driver” Hamiltonian”
𝐻

𝑑
.
𝐻

𝑝
,
𝐻

𝑑

0
.

Itay Hen

June 6
th
, 2012

UCSC

𝐻

(

)
=

(

)
𝐻

𝑝
+
1


(

)
𝐻

𝑑

𝐻

𝑝

is the problem Hamiltonian whose
ground state encodes the solution of
the optimization problem

𝐻

𝑑

is an easily solvable

driver Hamiltonian, which

does not commute with
𝐻

𝑝

the interpolating Hamiltonian is this:

The quantum

the
parameter


obeys
0




1
, with


0
=
0

and


𝒯
=
1
. also:
𝐻

0
=
𝐻

𝑑

and
𝐻

𝒯
=
𝐻

𝑝
.

here,


stands for time and
𝒯

is the running
time, or

complexity,
of the
algorithm.

Itay Hen

June 6
th
, 2012

UCSC

the
if the change in



is

system will stay close to the

ground
state
of
the instantaneous Hamiltonian
throughout

the evolution.

one
finally obtains a state close to
the ground
state of
𝐻

𝑝
.

measuring
the
state will
give the solution of the
original

problem
with
high probability.

the interpolating Hamiltonian is this:

The quantum

how fast can
the process be?

𝐻

(

)
=

(

)
𝐻

𝑝
+
1


(

)
𝐻

𝑑

Itay Hen

June 6
th
, 2012

UCSC

Quantum phase transition

bottleneck is likely to be something called a

happens when the energy
differenece

between the ground

state and the first excited state (i.e., the gap) is small.

there, the
probability to “get off track” is maximal
.

0



1

QPT

𝐻

=
𝐻

𝑑

𝐻

=
𝐻

𝑝

gap

1

a schematic picture of
the gap to the first
excited state as a
function of
the
parameter

.

Quantum Phase Transition

Itay Hen

June 6
th
, 2012

UCSC

Quantum phase transition

𝒯

1
Δ

𝑖
2

Landau
-
Zener

theory
tells us that to stay in the ground

state the running time needed is
:

exponentially closing gap (as a function of problem size N)

exponentially long running time

exponential complexity
.

Itay Hen

June 6
th
, 2012

UCSC

Quantum phase transition

𝒯

1
Δ

𝑖
2

system remains in its
i
nstantaneous ground state

a two
-
level avoided crossing

Landau
-
Zener

theory
tells us that to stay in the ground

state the running time needed is
:

exponentially closing gap (as a function of problem size N)

exponentially long running time

exponential complexity
.

Itay Hen

June 6
th
, 2012

UCSC

most interesting unknown about QAA to date:

c
ould the QAA solve in polynomial time
“hard” (NP
-
complete) problems?

f
or which hard problems is
𝒯

polynomial in

?

or

Itay Hen

June 6
th
, 2012

UCSC

Quantum Computers

Itay Hen

June 6
th
, 2012

UCSC

quantum computers are currently unavailable.
h
ow does one

study quantum computers
?

quantum systems are huge matrices
. sizes of matrices for
n

qubits
:

=
2

.

one method:
diagonalization

of the matrices. however, takes a

lot of time and resources

to do so. can’t go beyond

~25
qubits
.

another method:
“Quantum Monte Carlo”.

Exact
diagonalization

Itay Hen

June 6
th
, 2012

UCSC

Quantum Monte Carlo methods

quantum Monte Carlo (QMC)
is a generic name for
classical

algorithms designed to study quantum systems
(in equilibrium)

on a classical computer
by simulating them.

in quantum Monte Carlo, one only
samples the exponential number

of configurations
, where configurations with less energy are given

more weight and are sampled more often.

this is usually a stochastic Markov process (a
Markovian

chain).

there are statistical errors.

not all quantum physical systems can be simulated (sign problem).

quantum systems have an additional “extra” dimension
(called

“imaginary time” and is periodic). for example a 3D classical

system is similar to 2D quantum systems (with notable exceptions).

Itay Hen

June 6
th
, 2012

UCSC

main goal:

study the
dependence of the typical minimum gap

on the
size (number of bits) of the problem
.

this is because:

polynomial dependence

polynomial complexity!

Method

determine the complexity of the

QAA for the various optimization problems

𝒯

1
Δ

𝑖
2

Δ

min
=
min

0
,
1
Δ

Itay Hen

June 6
th
, 2012

UCSC

Quantum Computing

Itay Hen

June 6
th
, 2012

UCSC

so far, there is
no clear
-
cut example for a problem that is

solved efficiently using adiabatic quantum computation
(still

looking though).

the first quantum adiabatic computer (quantum
annealer
)

has been built. ~128
qubits
.
built by D
-
Wave
(Vancouver).

one piece has been sold to USC / Lockheed
-
Martin (~1M\$).

a strong candidate for future quantum computers
.

there are however
a lot of technological challenges
.

people are looking for other possible uses for it
.

Itay Hen

June 6
th
, 2012

UCSC

one of many possible avenues of research is Machine

Learning. people are starting to look into it.

supervised and unsupervised machine learning.
could
work

if problem can be cast in the form of an optimization problem.

Hartmut

Neven’s

blog:

http
-
learning
-
with
-
quantum.html#!/
2009/12/machine
-
learning
-
with
-
quantum.html

Itay Hen

June 6
th
, 2012

UCSC

Itay Hen

UCSC

June 6
th

2012