Adiabatic quantum computation –
a tutorial for computer scientists
Itay Hen
Dept. of Physics, UCSC
Advanced Machine Learning class
UCSC
June 6th 2012
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Outline

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(?)]
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
What is a
Quantum Computer?
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
input state



computer
computation
011
010011
What is a quantum computer?
output state
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).
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
input state
input state

computer
computation
output state
computation
10010
11101
01010
a classical algorithm (circuit) looks like this:
01011

011
010011
What is a quantum computer?
output state
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).
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
input state

computer
computation
011
010011
What is a quantum computer?
output state
quantum computers on the other hand manipulate objects called
quantum bits or qubits for short.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
input state

computer
computation
011
010011
What is a quantum computer?
output state
quantum computers on the other hand manipulate objects called
quantum bits or qubits for short.
what are qubits?
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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
Itay Hen
or
I’m there
quantum world
I’m here
and
Advanced Machine Learning class– UCSC
I’m there
June 6th, 2012
What are qubits?


while a bit can be either in the a |0 state or a |1 state (this is
Dirac’s |𝑘𝑒𝑡 notation).
a qubit can be in a superposition of these two states, e.g.,
|𝜑 =


1
2
|0 + |1
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 nonintuitive. important: only 1 bit of information is stored in a qubit.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
What are qubits?

take for example a system of 3 bits. Classically, we have 23 = 8
possible “classical configurations”:
|000 , |001 , |010 , |011 ,
|100 , |101 , |110 , |111 ,

the corresponding quantum state of a 3-qubit system would be:
7
|𝜑 =
𝑐𝑖 |𝑖
𝑖=0
this is a (normalized) 8-component vector over the complex
numbers (8 complex coefficients). |𝑖 enumerates the 8 classical
configurations listed above.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012

Encoding optimization problems
suppose that we are given a function 𝑓: {0,1}3 → ℝ
and are asked to find its minimum
and the corresponding minimizing configuration.

this is the same as having the diagonal matrix:
𝐹=
and asking which is the smallest eigenvalue (minimum of f ) and
corresponding eigenvector (minimizing configuration).

here, the basis vectors 𝑒𝑖 correspond to |𝑖 and the elements on
the diagonal are the evaluation of f on them.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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.,
𝑡
min 𝜑|𝐹|𝜑 = min 𝜑 𝐹𝜑
|𝜑
𝜑
but problem now is much harder. (this is Dirac’s 𝑏𝑟𝑎| 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.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
computer
input state
output state
input state
Itay Hen
computation
Advanced Machine Learning class– UCSC
10010
11010
+
11001
+
11101
01010
+
01001
quantum computers manipulate quantum bits or qubits for short.
a quantum algorithm (circuit) may look like this:
01011

computation
011
010011
What is a quantum computer?
output state
June 6th, 2012
input state
computer
computation

space of possible quantum states is huge.

range of operations on qubits is huge.


011
010011
What is a quantum computer?
output state
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.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Motivation for adiabatic
quantum computing
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Motivation
clearly, a quantum computer is a
generalization of a classical computer
in what ways quantum computers are
more efficient than classical computers?
what problems could be solved more
efficiently on a quantum computer?
these are two of the most
basic and important questions in the
field of quantum information/computation.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Motivation
in what ways quantum computers are
more efficient than classical computers?
what problems could be solved more
efficiently on a quantum computer?

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 𝑁 entries in 𝑂 𝑁1/2 time (quadratic speedup).

importance is huge (cryptanalysis, etc.).
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Motivation
what other problems can
quantum computers solve?

people are considering “hard” satisfiability (SAT) and other
optimization problems which are at least NP-complete.
 these are hard to solve classically; time needed is
exponential in the input (exponential complexity).
 could a quantum computer solve these problems in an
efficient manner? perhaps even in polynomial time?
Adiabatic Quantum Computing
is a general approach to solve a broad range
of hard optimization problems using a
quantum computer [Farhi et al.,2001]
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Adiabatic
quantum computation
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The nature of physical systems



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. this is the principle of least
action.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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.
f→

x→
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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.
f→

x→
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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.
f→

probability of jumping over the
barrier is exponentially small in
the height of the barrier.
x→
ball is unlikely to end up in the true minimum.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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.
f→

quantum mechanically, the
ball can “tunnel through” thin but
high barriers!
x→

sometimes more likely to end up in the true minimum.

tunneling is a key feature of quantum mechanics.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The adiabatic theorem of QM

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.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The adiabatic theorem of QM

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.
??
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The adiabatic theorem of QM


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!
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The adiabatic theorem of QM


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 harmonic potential
of a system in the ground state:
harmonic
potential
𝜓(𝑥, 𝑡)
Itay Hen
Advanced Machine Learning class– UCSC
2
June 6th, 2012
The adiabatic theorem of QM



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 harmonic potential
of a system in the ground state:
an abrupt change
(a diabatic process):
harmonic
potential
𝜓(𝑥, 𝑡)
Itay Hen
Advanced Machine Learning class– UCSC
2
June 6th, 2012
The adiabatic theorem of QM



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 harmonic potential
of a system in the ground state:
a gradual slow change
(an adiabatic process):
wave function can “keep up”
with the change.
harmonic
potential
𝜓(𝑥, 𝑡)
Itay Hen
Advanced Machine Learning class– UCSC
2
June 6th, 2012
The quantum adiabatic algorithm (QAA)

the mechanism proposed by Farhi et al., the QAA:
1. take a difficult (classical) optimization problem.
2. encode its solution in the ground state of a quantum
“problem” Hamiltonian 𝐻𝑝 .
3. prepare the system in the ground state of another
easily solvable “driver” Hamiltonian” 𝐻𝑑 . 𝐻𝑝 , 𝐻𝑑 ≠ 0.
4. vary the Hamiltonian slowly and smoothly from
𝐻𝑑 to 𝐻𝑝 until ground state of 𝐻𝑝 is reached.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The quantum adiabatic algorithm (QAA)

the interpolating Hamiltonian is this:
𝐻(𝑡) = 𝑠(𝑡)𝐻𝑝 + 1 − 𝑠(𝑡) 𝐻𝑑
𝐻𝑝 is the problem Hamiltonian whose
ground state encodes the solution of
the optimization problem


𝐻𝑑 is an easily solvable
driver Hamiltonian, which
does not commute with 𝐻𝑝
the parameter 𝑠 obeys 0 ≤ 𝑠 𝑡 ≤ 1, with 𝑠 0 = 0 and
𝑠 𝒯 = 1. also: 𝐻 0 = 𝐻𝑑 and 𝐻 𝒯 = 𝐻𝑝 .
here, 𝑡 stands for time and 𝒯 is the running time, or
complexity, of the algorithm.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The quantum adiabatic algorithm (QAA)

the interpolating Hamiltonian is this:
𝐻(𝑡) = 𝑠(𝑡)𝐻𝑝 + 1 − 𝑠(𝑡) 𝐻𝑑



the adiabatic theorem ensures that if the change in 𝑠 𝑡 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 𝐻𝑝 .
measuring the state will give the solution of the original
problem with high probability.
how fast can the process be?
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Quantum phase transition

bottleneck is likely to be something called a
Quantum Phase Transition

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.
a schematic picture of
the gap to the first
excited state as a
function of the adiabatic
parameter 𝑠.
gap
∆𝐸1

Itay Hen
0
𝐻 = 𝐻𝑑
QPT
1 𝑠
𝐻 = 𝐻𝑝
Advanced Machine Learning class– UCSC
June 6th, 2012
Quantum phase transition


Landau-Zener theory tells us that to stay in the ground
state the running time needed is:
𝒯∝1
2
Δ𝐸𝑚𝑖𝑛
exponentially closing gap (as a function of problem size N) 
exponentially long running time  exponential complexity.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Quantum phase transition


Landau-Zener theory tells us that to stay in the ground
state the running time needed is:
𝒯∝1
2
Δ𝐸𝑚𝑖𝑛
exponentially closing gap (as a function of problem size N) 
exponentially long running time  exponential complexity.
a two-level avoided crossing
system remains in its instantaneous ground state
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
The quantum adiabatic algorithm

most interesting unknown about QAA to date:
could the QAA solve in polynomial time
“hard” (NP-complete) problems?
or
for which hard problems is 𝒯 polynomial in 𝑁?
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Simulating Adiabatic
Quantum Computers
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Exact diagonalization




quantum computers are currently unavailable. how does one
study quantum computers?
quantum systems are huge matrices. sizes of matrices for n
qubits: 𝑁 = 2𝑛 .
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”.
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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
Advanced Machine Learning class– UCSC
June 6th, 2012
Method

main goal:
determine the complexity of the
QAA for the various optimization problems

study the dependence of the typical minimum gap
Δ𝐸min = min Δ𝐸
𝑠∈ 0,1
on the size (number of bits) of the problem.

this is because:
𝒯∝1

2
Δ𝐸𝑚𝑖𝑛
polynomial dependence  polynomial complexity!
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Future of adiabatic
Quantum Computing
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
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
Advanced Machine Learning class– UCSC
June 6th, 2012
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/machinelearning-with-quantum.html#!/2009/12/machine-learning-withquantum.html

Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012
Adiabatic quantum computation –
a tutorial for computer scientists
Itay Hen
Dept. of Physics, UCSC
Advanced Machine Learning class
UCSC
June 6th 2012
Itay Hen
Advanced Machine Learning class– UCSC
June 6th, 2012