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Quantum Walks and Quantum Bioinformatics
Talk delivered at
US Naval Research Laboratory NCARAI 2007-2008 Seminar Series
May 27th 2008
Salvador Elías Venegas-Andraca
Quantum Information Processing Group
Tecnológico de Monterrey, Campus Estado de México
http://mindsofmexico.org/sva
[email protected] , salvador.venegas-andraca@ keble.oxon.org,
[email protected]
Motivation
Quantum computation is a scientific field that uses the quantum
mechanical properties of (very small) systems in order to build
computers and algorithms that have a better (faster) performance
than current computer technology.
Bioinformatics is a scientific field that employs algorithms for both
modeling biological systems (like genomes or proteins) and
simulating the behavior of complex biological processes (like
protein folding).
Thus, employing quantum algorithms for solving bioinformatics
challenges may lead to expedite development of new drugs,
vaccines, deeper understanding of genome structure and several
other product and technologies relevant to biodefense.
Outline
1. Concise introduction to quantum computation.
2.Quantum walks.
3.Protein Folding and quantum adiabatic algorithms.
4.Future research directions and Conclusions.
Part I
Very quick introduction to quantum computation
What is quantum computation?
It is a multidisciplinary field in which physicists, computer
scientists, mathematicians and engineers work towards the
development of hardware and software based on the rules of
quantum mechanics.
- Computer scientists working in this field usually think of how to
harness the laws of nature in order to create faster algorithms.
- Theoretical physicists may also think of this field as a test bed
for new discoveries about the fundamental properties of nature.
- Finally, society may think of it as a powerful tool that may be
used to answer some of the most challenging scientific problems
we face now.
Scientific and technological reasons for thinking of
quantum computers
1.Simulating quantum systems using other quantum systems
(R.P. Feynman [1].)
2. Miniaturization of transistor technology is pushing the physical
limits towards quantum regimes.
3. It has been shown that some algorithms based on the rules of
quantum mechanics lead to algorithmic speed up.
For example: Grover’s search [2], Shor’s factorization [3] and
Childs et al’s continuous quantum walk [4].
4. Quantum algorithms are an exciting new platform for proposing
solutions to challenging problems from several scientific fields.
Concise introduction to Quantum Mechanics (1/6)
Primus inter pares: the qubit
In classical computer science, information is stored and processed in
bits. The relationship between a logical bit and the storage of binary
information in a physical system is simple:
Choose a physical system with a degree of
freedom with two mutually exclusive
measurement values.
For example, we may take two different voltage
values, measured between terminals E and C.
Concise introduction to Quantum Mechanics (2/6)
These and “bras” and “kets”: they’re just vectors!
Marvin Mermin, quoting a ‘newly enlightened computer scientist’
We shall introduce four postulates of quantum mechanics in a form
suitable for algorithm development.
Postulate 1. Description of a quantum system.
To each isolated physical system we associate a Hilbert space H,
hereinafter known as the state space of the system.
The physical system is completely described by its state vector
which is a unit vector of H.

Note: A linear combination of state vectors is also a state vector. This
is known as the superposition principle and it is an important
feature for quantum algorithm development.
Concise introduction to Quantum Mechanics (3/6)
Primus inter pares: the qubit
The quantum counterpart of a bit is a qubit. A qubit is defined as a
mathematical representation of a physical quantum system with two
distinguishable states.
Postulate 1 states that an isolated quantum system can be described
by a vector state. Thus, a qubit may be written as
  0   1
Example of a ket.
Interchangeable by vector columns
Where  , 
are complex numbers and     1
2
2
Concise introduction to Quantum Mechanics (4/6)
Postulate 2. Evolution of a quantum system (behavior over time).
The evolution of a closed quantum system with state vector
can be written in two different and equivalent ways:

Unitary evolution
- Leads to gate-oriented model of
computation, very natural to computer
scientists.
- Time evolve in discrete steps.
t1
is a unitary operator.
d | ( t )  ^
i
 H ( t )| ( t ) 
dt
Hamiltonian evolution
(Schrödinger Equation)
Leads to a continuous model
computation. Time is a real variable.
Û
t2
 Uˆ 

of
^
H, the Hamiltonian of the system, is a
Hermitian operator.
Concise introduction to Quantum Mechanics (5/6)
Postulate 3. Quantum measurement.
Measurement in quantum mechanics is a highly counter-intuitive
and non-trivial process, as opposed to common wisdom in classical
computer science:
-Measurement outcomes are inherently probabilistic.
-Once a measurement has been performed, a quantum system is
unavoidably altered.
Mathematical operators, examples and details can be found in
handouts.
Concise introduction to Quantum Mechanics (6/6)
Postulate 4. Composite quantum systems.
The state space of a composite quantum system, i.e. a system
made up of several qubits, is the tensor product of the component
system state spaces.
Main property to be remembered: the tensor product allows for an
exponential increase in the dimension of the total Hilbert space.
For example: if three qubits
composite quantum system
 1,  2 ,  3 2
 T , then 
are used to build a
23
T
 .
Mathematical operators, examples and details can be found in
handouts.
Theoretical and universality aspects of quantum computers
1. D. Deutsch proposed in [6] a quantum universal Turing
machine, as well as a physics-oriented version of the ChurchTuring thesis:
Every finitely realizable physical system can be perfectly simulated
by a universal model computing machine operating by finite means.
2. In classical computer science, a set of gates is universal if it is
possible to build a circuit with such gates in order to compute any
computable function.
The same holds for quantum computation: the set made by a
Hadamard gate (one-qubit) and a C-NOT gate (two qubits) is also
universal. In fact, there a many universal gate sets in quantum
computation [7,8].
3. Excellent introductions to QC for non-physicists: [9,10].
Part II
Quantum Walks
Quick reminder
Classical Random Walk on the Line
Froggy jumps either forward or
backwards, depending on the outcome
of corresponding coin toss, heads or
tails respectively.
Let us suppose that Mr. Money has a
coin with probability p of getting heads
and probability q of getting tails.
If Froggy begins its journey in position
zero, what is the probability of finding
our dear frog at position k after n
steps?
Answer:
n

 1
1
( k n)
( nk )
n
 p2
P0 k   1
q2
 ( k  n) 
2

[Binomial distribution]
Why are Classical Random Walks important in Computer Science?
Random Walks are used to develop algorithms that may
outperform
their deterministic counterparts for the solution of
certain problems.
An example follows
K-SAT, a Fundamental NP Problem
The K-SAT problem plays a most important role in the Theory of
Computation (NP complete problem). The setup of the K-SAT
problem is as follows
• Let B={x1, x2, …, xn} be a set of Boolean variables.
• Let Ci be a disjunction of k elements of B
• Finally, let F be a conjunction of m clauses Ci.
Question: Is there an assignment of Boolean variables in F
that satisfies all clauses simultaneously, i.e. F=1?
Example. Instance of 3-SAT
Solving the 3SAT Problem
One of the best algorithms for solving the 3SAT
problem is based on a classical random walk: T.
Hofmeister, U. Schöning, R. Schuler and O.
Watanabe, “A Probabilistic 3-SAT algorithm Further
Improved”, Symposium on Theoretical Aspects of
Computer Science, pp. 192-202 (2002)
Quantum Walks
What is a Quantum Walk?
A Quantum Walk is a generalization of classical random walks
in the quantum world. In this new paradigm, the constituent
elements of the walk are quantum particles.
Are there different kinds of Quantum Walks?
Yes, there are two kinds of quantum walks: discrete and
continuous.
What about Discrete Quantum Walks?
It is possible to define discrete quantum walks on the line (both
unrestricted and restricted) and in general graphs.
The basic model: Quantum Walk on an infinite line
Why are Quantum Walks relevant in
Quantum Computation?
1. Quantum walks produce position
distributions that are very different
probability
distributions
obtained
computation of classical random walks.
use those new probability distributions
quantum walks for the development
faster algorithms.
probability
from those
by
the
We expect to
produced by
of new and
2. Quantum walks can be a good test for the
“quantumness” of physical realizations of quantum
computers.
Elements of Discrete Quantum Walks on an Infinite Line (1/3)
1. Walker. A quantum system living in a Hilbert space of
infinite but countable dimension Hp. The walker is usually
initialized at the origin.
2. Coin. A quantum system living in a 2-dimensional Hilbert
space Hc. The initial coin state depends on the symmetry we
want to imprint on the position probability distribution of the
walker.
Total and initial states of the Quantum Walk. The total state
of the quantum walk resides in Ht= Hp  Hc. The initial state
that has been used so far is simply the product state of
corresponding walker and coin initial states.
Elements of Quantum Walks on an Infinite Line (2/3)
3. Evolution Operators.
Coin Evolution Operator. Any 2-dimensional unitary
operator can be a coin evolution operator. The Hadamard
operator is customary.

1
H
 0 0  0 1  1 0  1 1
2
Conditional Shift Operator. As with the previous operator,
the only requirement is that of unitarity. A suitable
conditional shift operator is

S 0
c
0   i 1
i
p
i 1
c
1   i 1
p
i
i
^
So, the total evolution operator is given by
^
^
^
U  (H  I ) S
Elements of Discrete Quantum Walks on an Infinite Line (3/3)
The operational idea of a quantum walk does resemble that
of a classical random walk
i) Toss the coin (apply coin evolution operator)
ii) Move the walker according to the coin outcome (apply
conditional shift operator)
iii) Do steps i) and ii) a total number of t times, and finish the
walk by measuring the position of the walker
Results for a Quantum Walk of an Infinite Line (1/2)
Figure 1. Unbalanced coin
Total initial state

P
r
o
b
a
b
i
l
i
t
y
0
 0 c 0
p
Evolution operator
^
^
^
^
U  (H  I ) S
Number of steps
t = 100
Position
cf.
Results for a Discrete Quantum Walk of an Infinite Line (2/2)
Figure 2. Balanced coin
Total initial state
1
 0
 0  i 1 c  0
2
P
r
o
b
a
b
i
l
i
t
y
Evolution operator
^
^
^
^
U  (H  I ) S
Number of steps
t = 100
Position
cf.
p
Relevant Properties of Discrete Quantum Walks on an Infinite Line
1. The standard deviation of a quantum walk is n. In
contrast, the standard deviation of a classical random walk
is of order n. Therefore, the quantum walk propagates
quadratically faster (A. Nayak and A. Vishwanath, quant-ph/
0010117)
2. The position probability distribution of a quantum walk does
depend on the initial quantum state, as opposed to a
classical random walk.
3. Discrete quantum walks in algorithmic development may
provide polynomial speed-up (more on this shortly).
Continuous quantum walks (1/3)
Let G=(V,E) be a graph with |V|=n  a continuous time
random walk on G can be described by the infinitesimal
generator matrix M given by
Using Mab, it is possible to prove that the probability of being at
vertex a at time t is given by
Continuous quantum walks (2/3)
Using matrix Mab, we define a Hamiltonian
elements given by
The unitary operator
^
^
U  exp( i H t )
Defines a continuous quantum walk on graph G.
with matrix
Continuous quantum walks (3/3)
Childs et al’s main results [4]:
1. No classical algorithm may go from
Entrance to Exit in polynomial time.
2. A continuous quantum walk may
probabilistically go from Entrance to Exit
in polynomial time.
Full introduction to the field:
Quantum walks for computer scientists
S.E. Venegas Andraca
Morgan and Claypool (second half 2008)
Part III
Protein Folding
and
Quantum Adiabatic Algorithms
Let us start with a fresh topic: Proteins
A protein is a polymer molecule, a chain of tens to thousands of
monomer units. The monomers are the 20 naturally occurring
amino acids.
Proteins play a key role in thousands of chemical and physical
processes essential to keep an organism alive. Example: digestion
enzymes and hemoglobin.
Therefore, scientific research on proteins is a fundamental task
for understanding life. Also, protein research would lead
mankind to new forms of medicine practice and personalized
drug design.
Different kinds of Proteins
Proteins may be classified into three types: fibrous, membrane
and globular.
We shall focus on globular proteins as they are an essential
element in the chemistry of life.
The most important state of a globular protein, known as its
native or folded state, is extremely compact and is unique, i.e. a
given protein folds to only one native state.
Globular proteins
The native state of a typical globular
protein has, amongst others, the
following properties:
1. It is tightly packed as a smallmolecule crystal but, in general, a
globular protein does not have
the spatial regularity of a crystal
(bad news for complexity in
simulation algorithms).
2. Complex globular proteins have
domains, i.e. subsets of amino
acids that are often independently
stable and folded (helpful for
mathematical and computational
analysis).
Schematic diagram of a globular protein.
Introduction to Protein Structure.
C. Branden and J. Tooze.
Taylor and Francis (1999).
The Protein Folding Problem
Given the amino acid sequence of a protein, predict its
compact three-dimensional native state
The Protein Folding problem is a key challenge in modern science,
for both its intrinsic importance in the foundations of biological
science and its applications in medicine, agriculture, and many
other areas.
Some physics and mathematics of the Protein Folding
Problem (1/4)
The native fold of a globular protein is usually assumed to
correspond to the global minimum of the protein’s free energy.
This is known as the Thermodynamic Hypothesis which, as stated
in C. Anfinsen’s Nobel lecture [11], reads :
“The three-dimensional structure of a native protein, in its normal
physiological milieu, is the one in which the Gibbs free energy2 of
the whole system is lowest; that is, that the native conformation is
determined by the totality of interatomic interactions and hence by
the amino acid sequence, in a given environment.”
The protein folding problem can be thus analyzed as a global
optimization problem.
2 G=U
+ pV - TS, where U =internal energy (kinetic energy due to molecule motion and energy
of chemical bonds), p =pressure, V=volume of the molecule, T= temperature and S=entropy.
Some physics and mathematics of the Protein Folding
Problem (2/4)
Ideally we should compute, for every possible 3D conformation of
the chain, the sum of free energies of the atomic interactions in the
protein.
However, as you all can easily infer, such an exhaustive procedure
becomes unfeasible for even a small number of amino acids, as the
number of conformations obeys an exponential relation:
N  n
where n is the number of amino acids and   {2,3,4,5,6} [12].
Some physics and mathematics of the Protein Folding
Problem (3/4)
The exhaustive approach is not reasonable not only from a
computational point of view, but also with respect to what Nature
can do, as our bodies can do the following in milliseconds:
1. Unzip DNA chains
2. Create RNAm from unzipped DNA chains.
3. Assemble proteins, using RNAm, in our protein manufacturing
plant: ribosome.
For example, the human body produces thousands of enzymes (a
kind of protein) at least three times every day: when we digest our
meals.
Some physics and mathematics of the Protein Folding
Problem (4/4)
The computational complexity of exhaustive search and the (very) quick
procedure performed by Nature leads to Levinthal’s paradox [13]:
How does a protein find the global optimum (its native state) without a
global search?
What vast parts of conformational space does the protein avoid?
It is indeed evident that clever and faster approaches, based on the laws
of physics and chemistry, should be developed in order to efficiently
solve the protein folding problem.
The HP model (1/2)
A simplified but very useful model for studying protein folding is
known as the HP model [14], which is based on the following
assumptions:
1. Amino acids can be divided into two sets: hydrophobic (H) and
polar (P) [i.e. keen on interacting with water]. Hence the
acronym HP.
2. The interaction of proteins with its surroundings makes polar
amino acids be on the surface of the protein, while
hydrophobic amino acids tend to stay at the core of the
globular structure.
3. HP amino acid interaction with its milieu is a driving force of
the folding process.
4. The only interaction among amino acids is the favorable
contact between two non-adjacent (in the amino acid sequence)
H amino acids.
The HP model (2/2)
As stated, in the HP model the only
interaction among amino acids is the
favorable contact between two non-adjacent
(in the amino acid sequence) H amino acids.
For example, in this figure, there is a total
number of 12 H-H favorable interactions.
The protein folding problem, under the HP model, is
NP-complete [15].
Quantum adiabatic algorithms (1/3)
One of the novel approaches towards simulation of protein
folding is the employment of quantum adiabatic algorithms
based on the HP model.
The quantum adiabatic model of computation was originally
proposed by E. Farhi, J. Goldstone, S. Gutmann and M. Sipser
[16,17], and has the following characteristics (next slide):
Quantum adiabatic algorithms (2/3)
Characteristics of the quantum adiabatic model of computation
1. It is a continuous model of computation, i.e. it is based on
the Schrödinger equation.
2. It has been proved that it is a universal model of quantum
computation [18], i.e. a quantum adiabatic computer can do
anything a quantum Turing machine can do.
3. It has been proved [18] that quantum adiabatic algorithms
can provide quadratic algorithmic speed up for certain
problems.
4. It remains an open question whether quantum adiabatic
algorithms may provide exponential algorithmic speed up.
Quantum adiabatic algorithms (3/3)
The main idea behind quantum adiabatic algorithms is to
employ the Schrödinger equation:
d | ( t )  ^
i
 H ( t )| ( t ) 
dt
with a specific structure for the Hamiltonian of the system
(more on this very shortly, but before that…)
A connection between quantum walks and quantum
adiabatic algorithms
1. Universal quantum walks and adiabatic algorithms by 1D Hamiltonians.
B.D. Chase and A. J. Landahl [5].
A proposal for building Hamiltonians that enable universal computation.
Hamiltonians in this family are achieved by either a continuous quantum
walk or by executing an adiabatic algorithm.
Quantum mechanical algorithms for Protein Folding (1/9)
Very quick reminder of the algebraic properties of Hamiltonians
1. As any other linear operators, Hamiltonians have eigenvectors
and eigenvalues.
2. For historical reasons, the eigenvector(s) corresponding to the
smallest eigenvalue is(are) known as ground state(s).
Quantum mechanical algorithms for Protein Folding (2/9)
d | ( t )  ^
i
 H ( t )| ( t ) 
dt
where
t
t

H (t )  1   H b  H p
T
 T
^
T is the total running time
of the quantum algorithm
^
This is the initial
Hamiltonian, which has a
unique and easy to
prepare ground state.
^
This is the final
Hamiltonian, which
encodes in its ground
state the solution to the
problem under study.
Quantum mechanical algorithms for Protein Folding (3/9)
The rationale behind an adiabatic quantum algorithm is:
^
1. Start with an initial Hamiltonian
Hb
that has
i) an easy to prepare and unique ground state | ( 0 ) 
ii) Different eigenvalues Eo < E1<…<En
2. Evolve the system slowly.
By doing this evolution sufficiently slowly, the quantum adiabatic
theorem [19] allows us to predict that the system will stay in the
ground state of its Hamiltonian
^
H (t )
for all the computing time t  [0,T]
Quantum mechanical algorithms for Protein Folding (4/9)
3. If we let the system run for sufficiently long time T, the quantum
adiabatic algorithm allows us to predict that, when measuring the state
of the system described by
^
H (t )
We shall be very close to the ground state of the Hamiltonian
^
Hp
which, by definition, has the solution to the problem encoded in its
ground state!
Quantum mechanical algorithms for Protein Folding (5/9)
For how long should we run the quantum adiabatic computer?
T 
where
g
2
min

2
g min
 min E1 (t / T )  E0 (t / T ) 
2
0t / T 1
Quantum mechanical algorithms for Protein Folding (6/9)
Hamiltonian for protein folding under the HP model for an adiabatic
quantum computer [20]
t ^
t ^

H (t )  1   H b  H p
T
 T
^
^
^
^
^
H p  H onsite  H psc  H pairwise
One amino acid
for each graph site
The folding follows the
original amino acid
sequence
HHPHPHHPPPHHP…
It counts all non-adjacent H-H
interactions. Each favorable
interaction is one energy unit.
Quantum mechanical algorithms for Protein Folding (7/9)
^
Matrix elements for
^
^
H onsite, H psc , H pairwise
N 1
H onsite( N , D)  0 
i 1
N
H
j i 1
ij
onsite
( N , D)
It costs energy to put two amino acids on the same site.
D
ij
H onsite
( N , D)  
k 1
log2 N
 (1  q
f (i ,k ) r
 q f ( j ,k ) r  2q f (i ,k ) r q f (i ,k ) r
r 1
f (i, k )  D(i 1) log 2 N  (k 1) log 2 N
N , D, qi
is a distance function.
are the number of amino acids (frequently 100s), D is the
dimension of the HP model (2 or 3), and a quantum operator
in matrix representation, respectively.
Quantum mechanical algorithms for Protein Folding (8/9)
^
Matrix elements for
^
^
H onsite, H psc , H pairwise
N 1


2
H psc ( N , D)  1  ( N  1)   d m,m1 
m 1


It costs energy not to follow the primary sequence.
D
where
d
2
P ,Q
( N , D)  
k 1



log2 N
2
r 1
r 1
(q f  P ,k  r

 q f Q ,k  r ) 

2
is a distance function between amino acids P,Q.
N , D, qi
are the number of amino acids (frequently 100s), D is the
dimension of the HP model (2 or 3), and a quantum operator
in matrix representation, respectively.
Quantum mechanical algorithms for Protein Folding (9/9)
^
Matrix elements for
^
^
H onsite, H psc , H pairwise
N 1
H pairwise( N , D)  
i 1
N
ij
G
H
 ij pairwise
j i 1
Each interaction counts for a -1 energy unit.
where
,2D
H ijpairwise
( N )  xij, 2 D ( N )  xij, 2 D ( N )  yij, 2 D ( N )  yij, 2 D ( N )
counts the number of non-adjacent H-H interactions above, below, left
and right of site ij.
N , D, qi
are the number of amino acids (frequently 100s), and D is
the dimension of the HP model (2 or 3).
Future research directions
1. We are currently trying to determine how fast a quantum
adiabatic algorithm can be when simulating protein folding.
We are working towards an analytical characterization of
minimum gaps for this fundamental biological process.
2. 3SAT is one of the most studied NP-complete problems. Farhi et
al [17], Van Dam et al [18] have studied it under the light of
quantum adiabatic algorithms.
Moreover, Permodo, Venegas-Andraca
(incoming paper) are studying new
Hamiltonians for the 3SAT problem.
and Aspuru-Guzik
quantum adiabatic
Both protein folding and 3SAT are NP-complete problems. Thus,
faster algorithms for 3SAT may lead to faster algorithms for
protein folding.
Conclusions
1. Quantum algorithms are a powerful tool not only for attacking
theoretical computer science problems and physics
phenomena. We may also efficiently simulate the behavior of
very complex biological processes.
2. By combining the strengths of quantum walks and quantum
adiabatic algorithms it may be possible to build families of
Hamiltonians for universal computation.
3. Moreover, quantum algorithms may also be employed to get a
deeper understanding on how to use the laws and products of
Nature for computational purposes.
How about, after learning more about protein folding, we use
proteins for very complex computations?
References
[1] Richard P. Feynman. Simulating Physics with Computers. International Journal of Theoretical Physics, 21
(6/7) pp. 467-488 (1982) y The Feynman Lectures on Computation. Penguin Books (1999).
[2] K. Grover. A fast quantum mechanical algorithm for database search. Proceedings 28th annual ACM
Symposium Theory of Computing, pp. 212–219 (1996).
[3] P. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Algorithms on a Quantum
Computer. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–
134, IEEE Computer Society Press (1994).
[4] A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. Speilman. Exponential algorithmic speedup
by quantum walk. Proceedings of the 35th ACM Symposium on the Theory of Computation (STOC ’03)
ACM, pp. 59.68 (2003).
[5] Universal quantum walks and adiabatic algorithms by 1D Hamiltonians. B.D. Chase and A.J. Landhal. Arxiv
quant-ph/0802.1207
[6] D. Deutsch. Quantum theory, the Church-Turing Principle and the Universal Quantum Computer.
Proceedings of the Royal Society of London, series A, 400(1818) pp. 97-117, (1985).
[7] M.I. Nielsen e I.L. Chuang. Quantum Computation and Quantum Information. CUP (2000).
[8] D. DiVicenzo. Two-qubit gates are universal for quantum computation. PRA 51, pp. 1015 - 1022 (1995)
[9] E. Reiffel and W. Polak. A n introduction to quantum computing for non-physicists. ACM Comput. Surv.
32(3) pp. 300-335 (2000).
[10} D. Mermin. From Cbits to Qbits: Teaching computer scientists quantum mechanics. American Journal of
Physics vol. 71(1)pp. 23-30 (2003)
References
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