Adiabatic Quantum Computation – a Tutorial for Computer Scientists

Itay Hen June 6th, 2012 Advanced Machine Learning class UCSC

Itay Hen

Advanced Machine Learning class

UCSC

June 6th 2012

Adiabatic quantum computation

a tutorial for computer scientists

Dept. of Physics, 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


What is a

Quantum Computer?


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 (0s and 1s).

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 0s and 1s).

computer

010011

011

input state output state computation



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 0s and 1s).

computer

010011

011


0 1

0 1

1

0 1

0 1

0

1 1

1 0

1

1 0

0 1

0




quantum computers on the other hand manipulate objects called

quantum bits or qubits for short.

010011

011


computer



quantum computers on the other hand manipulate objects called

quantum bits or qubits for short.

010011

011


computer

what are qubits?


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

Im here Im there or

quantum world

Im here Im there and


What are qubits?

while a bit can be either in the a |0 state or a |1 state (this is

Diracs | 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.

| =1

2|0 + |1


What are qubits?

take for example a system of 3 bits. Classically, we have 23 = 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). | enumerates the 8 classical configurations listed above.

|000 , |001 , |010 , |011 , |100 , |101 , |110 , |111 ,

| =

7

=0

|


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 correspond to | and the elements on

the diagonal are the evaluation of f on them.

: {0,1}3

=


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 Diracs | 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|

|| = min



quantum computers manipulate quantum bits or qubits for short.

a quantum algorithm (circuit) may look like this:

010011

011


computer

0 1

0 1

1

0 1

0 1

0

+

0 1

0 0

1

1 0

0 1

0


1 1

0 1

0

+

1 1

0 0

1

+

1 1

1 0

1



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


computer


Motivation for adiabatic

quantum computing


Motivation

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.

clearly, a quantum computer is a generalization of a classical computer


best-known examples are:

Shors 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

in what ways quantum computers are more efficient than classical computers?

what problems could be solved more efficiently on a quantum computer?


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?

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]

what other problems can quantum computers solve?


Adiabatic

quantum computation


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.

The nature of physical systems


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




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






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


Analog quantum optimization




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


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




??



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!




real QM example: change the strength of a harmonic potential of a system in the ground state:


(, ) 2

harmonic potential




an abrupt change (a diabatic process):


(, ) 2

harmonic potential




a gradual slow change (an adiabatic process): wave function can keep up with the change.


harmonic potential

(, ) 2


The quantum adiabatic algorithm (QAA)

1. take a difficult (classical) optimization problem.

4. vary the Hamiltonian slowly and smoothly from

to until ground state of is reached.

the mechanism proposed by Farhi et al., the QAA:

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

problem Hamiltonian .

3. prepare the system in the ground state of another

easily solvable driver Hamiltonian . , 0.


() = () + 1 ()

is the problem Hamiltonian whose

ground state encodes the solution of the optimization problem

is an easily solvable driver Hamiltonian, which

does not commute with

the interpolating Hamiltonian is this:


the parameter obeys 0 1, with 0 = 0 and = 1. also: 0 = and = .

here, stands for time and is the running time, or complexity, of the algorithm.


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 .

measuring the state will give the solution of the original problem with high probability.

the interpolating Hamiltonian is this:


how fast can the process be?

() = () + 1 ()


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 = =

gap

1 a schematic picture of

the gap to the first excited state as a function of the adiabatic parameter .

Quantum Phase Transition


Quantum phase transition

1

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.


Quantum phase transition

1

2

system remains in its instantaneous 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.


most interesting unknown about QAA to date:

The quantum adiabatic algorithm

could the QAA solve in polynomial time hard (NP-complete) problems?

for which hard problems is polynomial in ? or


Simulating Adiabatic

Quantum Computers


quantum computers are currently unavailable. how 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. cant go beyond

~25 qubits.

another method: Quantum Monte Carlo.

Exact diagonalization


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).


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

1

2

min = min 0,1


Future of adiabatic

Quantum Computing


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.


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 Nevens blog: http://googleresearch.blogspot.com/2009/12/machine-

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

quantum.html


Itay Hen

Advanced Machine Learning class

UCSC

June 6th 2012

Adiabatic quantum computation

a tutorial for computer scientists

Dept. of Physics, UCSC

Documents

Adiabatic Quantum Computation – a Tutorial for Computer Scientists