a tutorial for computer scientists

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Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC

Itay

Hen

Advanced Machine Learning class

UCSC

June 6
th

2012

Adiabatic quantum
computation


a tutorial for computer scientists


Dept. of Physics, UCSC

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



introduction I
: what is a quantum computer?




introduction II
: motivation for adiabatic


quantum computing



adiabatic quantum computing




simulating adiabatic quantum computers




future of adiabatic computing


[AQC and machine learning(?)]

Outline

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC

What is a

Quantum Computer?

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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
asking
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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


UCSC

Motivation for adiabatic

quantum computing

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


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

Advanced Machine Learning class


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

time (quadratic speedup).



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

Advanced Machine Learning class


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

Adiabatic Quantum
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

Advanced Machine Learning class


UCSC

Adiabatic

quantum computation

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



adiabatic quantum computation
is different that circuit
-
based


computation (there are still other models of computation).



adiabatic quantum
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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


UCSC



the adiabatic theorem
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
.

The adiabatic theorem
of
QM

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



the adiabatic theorem
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
.

The adiabatic theorem
of
QM

??

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



the adiabatic theorem
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
.



take advantage of adiabatic evolution
to solve the problem!

The adiabatic theorem
of
QM

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



the adiabatic theorem
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:

The adiabatic theorem
of
QM

𝜓
(
𝑥
,

)
2

harmonic

potential

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



the adiabatic theorem
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):

The adiabatic theorem
of
QM

𝜓
(
𝑥
,

)
2

harmonic

potential

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC



the adiabatic theorem
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:



a gradual slow change


(an
adiabatic
process):


wave function can “keep up”


with the change.

The adiabatic theorem
of
QM

harmonic

potential

𝜓
(
𝑥
,

)
2

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC

The quantum
adiabatic algorithm (QAA)

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

Advanced Machine Learning class


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
adiabatic algorithm (QAA)



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

Advanced Machine Learning class


UCSC



the
adiabatic theorem ensures that
if the change in



is


made slowly enough, the
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
adiabatic algorithm (QAA)


how fast can
the process be?

𝐻

(

)
=

(

)
𝐻

𝑝
+
1


(

)
𝐻

𝑑

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


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
adiabatic
parameter

.


Quantum Phase Transition

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


UCSC



most interesting unknown about QAA to date:




The quantum adiabatic algorithm

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

Advanced Machine Learning class


UCSC


Simulating Adiabatic


Quantum Computers


Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


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

Advanced Machine Learning class


UCSC

Future of adiabatic

Quantum Computing

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC

Future of adiabatic QC



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

Advanced Machine Learning class


UCSC

Adiabatic QC and Machine Learning



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
://googleresearch.blogspot.com/2009/12/machine
-
learning
-
with
-
quantum.html#!/
2009/12/machine
-
learning
-
with
-
quantum.html

Itay Hen

June 6
th
, 2012

Advanced Machine Learning class


UCSC

Itay Hen

Advanced Machine Learning class

UCSC

June 6
th

2012

Adiabatic quantum
computation


a tutorial for computer scientists


Dept. of Physics, UCSC