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
→
ℝ
=
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
Comments 0
Log in to post a comment