Diapositiva 1

1
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//mindsofm
exico.org/sva sva_at_mindsofmexico.org ,
salvador.venegas-andraca_at_ keble.oxon.org,
svenegas_at_itesm.mx
2
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.
3
Outline

• Concise introduction to quantum computation.
• Quantum walks.
• Protein Folding and quantum adiabatic algorithms.
• Future research directions and Conclusions.

4
Part I Very quick introduction to quantum
computation
5
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.

6
Scientific and technological reasons for
thinking of quantum computers

• 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 Grovers search 2, Shors
  factorization 3 and Childs et als continuous
  quantum walk 4.
• 4. Quantum algorithms are an exciting new
  platform for proposing solutions to challenging
  problems from several scientific fields.

7
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.
8
Concise introduction to Quantum Mechanics
(2/6) These and bras and kets theyre 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.
9
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
Example of a ket. Interchangeable by vector
columns
Where are complex numbers and
10
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.
is a unitary operator.
Hamiltonian evolution (Schrödinger Equation)
Leads to a continuous model of computation. Time
is a real variable.
, the Hamiltonian of the system, is a
Hermitian operator.
11
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.

12
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
are used to build a composite
quantum system , then
. Mathematical operators, examples and details
can be found in handouts.
13
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 Church-Turing
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.
14
Part II Quantum Walks
15
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
Binomial distribution
16
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
17
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 Bx1, 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. F1?

Example. Instance of 3-SAT
18
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)
19
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
20

• Why are Quantum Walks relevant in
• Quantum Computation?
• Quantum walks produce position probability
  distributions that are very different from those
  probability distributions obtained by the
  computation of classical random walks. We expect
  to use those new probability distributions
  produced by quantum walks for the development of
  new and faster algorithms.
• Quantum walks can be a good test for the
  quantumness of physical realizations of quantum
  computers.

21
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.
22
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.
Conditional Shift Operator. As with the previous
operator, the only requirement is that of
unitarity. A suitable conditional shift operator
is
So, the total evolution operator is given by
23
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
• 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

24
Results for a Quantum Walk of an Infinite Line
(1/2)
Figure 1. Unbalanced coin
Total initial state
Probability
Evolution operator
Number of steps
t 100
Position
cf.
25
Results for a Discrete Quantum Walk of an
Infinite Line (2/2)
Figure 2. Balanced coin
Total initial state
Probability
Evolution operator
Number of steps
t 100
Position
cf.
26
Relevant Properties of Discrete Quantum Walks on
an Infinite Line

• 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).

27
Continuous quantum walks (1/3)
Let G(V,E) be a graph with Vn ? 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
28
Continuous quantum walks (2/3)
Using matrix Mab, we define a Hamiltonian
with matrix elements given by
The unitary operator
Defines a continuous quantum walk on graph G.
29
Continuous quantum walks (3/3)

• Childs et als main results 4
• No classical algorithm may go from Entrance to
  Exit in polynomial time.
• 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)
30
Part III Protein Folding and Quantum Adiabatic
Algorithms
31
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.
32
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.
33
Globular proteins

• The native state of a typical globular protein
  has, amongst others, the following properties
• It is tightly packed as a small-molecule crystal
  but, in general, a globular protein does not have
  the spatial regularity of a crystal (bad news for
  complexity in simulation algorithms).
• 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).
34
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.
35
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 proteins free energy. This is known as the
Thermodynamic Hypothesis which, as stated in C.
Anfinsens 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 GU pV - TS, where U internal energy
(kinetic energy due to molecule motion and energy
of chemical bonds), p pressure, Vvolume of the
molecule, T temperature and Sentropy.
36
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.


37
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
• Unzip DNA chains
• Create RNAm from unzipped DNA chains.
• 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.

38
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 Levinthals 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.


39
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
• Amino acids can be divided into two sets
  hydrophobic (H) and polar (P) i.e. keen on
  interacting with water. Hence the acronym HP.
• 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.
• HP amino acid interaction with its milieu is a
  driving force of the folding process.
• The only interaction among amino acids is the
  favorable contact between two non-adjacent (in
  the amino acid sequence) H amino acids.

40
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.
41
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)

42
Quantum adiabatic algorithms (2/3)

• Characteristics of the quantum adiabatic model of
  computation
• It is a continuous model of computation, i.e. it
  is based on the Schrödinger equation.
• 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.
• It has been proved 18 that quantum adiabatic
  algorithms can provide quadratic algorithmic
  speed up for certain problems.
• It remains an open question whether quantum
  adiabatic algorithms may provide exponential
  algorithmic speed up.

43
Quantum adiabatic algorithms (3/3)
The main idea behind quantum adiabatic algorithms
is to employ the Schrödinger equation

with a specific structure for the Hamiltonian of
the system (more on this very shortly, but
before that)
44
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.
45
Quantum mechanical algorithms for Protein Folding
(1/9) Very quick reminder of the algebraic
properties of Hamiltonians

• As any other linear operators, Hamiltonians have
  eigenvectors and eigenvalues.
• For historical reasons, the eigenvector(s)
  corresponding to the smallest eigenvalue is(are)
  known as ground state(s).

46
Quantum mechanical algorithms for Protein Folding
(2/9)

where
This is the final Hamiltonian, which encodes in
its ground state the solution to the problem
under study.
This is the initial Hamiltonian, which has a
unique and easy to prepare ground state.
T is the total running time of the quantum
algorithm
47
Quantum mechanical algorithms for Protein Folding
(3/9)

• The rationale behind an adiabatic quantum
  algorithm is
• Start with an initial Hamiltonian
  that has
• i) an easy to prepare and unique ground state
• ii) Different eigenvalues Eo
• 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

for all the computing time t ? 0,T
48
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 We shall be very close to
the ground state of the Hamiltonian

which, by definition, has the solution to the
problem encoded in its ground state!
49
Quantum mechanical algorithms for Protein Folding
(5/9)
For how long should we run the quantum adiabatic
computer?
where
50
Quantum mechanical algorithms for Protein Folding
(6/9)
Hamiltonian for protein folding under the HP
model for an adiabatic quantum computer 20

It counts all non-adjacent H-H interactions. Each
favorable interaction is one energy unit.
One amino acid for each graph site
The folding follows the original amino acid
sequence HHPHPHHPPPHHP
51
Quantum mechanical algorithms for Protein Folding
(7/9)
Matrix elements for

It costs energy to put two amino acids on the
same site.
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.
52
Quantum mechanical algorithms for Protein Folding
(8/9)
Matrix elements for

It costs energy not to follow the primary
sequence.
where
is a distance function between amino acids P,Q.
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.
53
Quantum mechanical algorithms for Protein Folding
(9/9)
Matrix elements for

Each interaction counts for a -1 energy unit.
where
counts the number of non-adjacent H-H
interactions above, below, left and right of site
ij.
are the number of amino acids (frequently 100s),
and D is the dimension of the HP model (2 or 3).
54
Future research directions

• 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 and
  Aspuru-Guzik (incoming paper) are studying new
  quantum adiabatic Hamiltonians for the 3SAT
  problem.
• Both protein folding and 3SAT are NP-complete
  problems. Thus, faster algorithms for 3SAT may
  lead to faster algorithms for protein folding.

55
Conclusions

• 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.
• By combining the strengths of quantum walks and
  quantum adiabatic algorithms it may be possible
  to build families of Hamiltonians for universal
  computation.
• 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?

56
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.
212219 (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. 124134, 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)
57
References
11 Christian B. Anfinsen. Studies on the
principles that govern the folding of protein
chains. Nobel Lecture, December 11, 1972. 12
The protein folding problem. H. S. Chan y K.A.
Dill. Physics Today, pp. 24-32 (1993). 13 Are
there pathways for protein folding?. C.
Levinthal, Journal de Chimie Physique et de
Physico-Chimie Biologique vol. 65, pp. 44-45
(1968). 14 The hydrophobic effect and the
organization of living matter. C. Tanford.
Science 2001012-1018 (1978). 15 On the
complexity of protein folding. P. Crescenzi, D.
Goldman, C. H. Papadimitriou, A. Piccolboni y M.
Yannakakis. Journal of Computational Biology,
vol. 5(3) pp. 423-466 (1998). 16 Quantum
Computation by Adiabatic Evolution. E. Farhi, J.
Goldstone, S. Gutmann, and M. Sipser.
ArXivquant-ph/0001106 17 A quantum adiabatic
evolution algorithm applied to random instances
of an NP-complete problem. E. Farhi, J.
Goldstone, S. Gutmann, Joshua Lapan, Andrew
Lundgen, and Daniel Preda. Science vol. 292 pp.
472-476 (2001). 18 How powerful is adiabatic
quantum computation? Win Van Dam, Michele Mosca,
and Umesh Vazirani. Proceedings of the 42nd
Symposium on the Foundations of Computer Science
pp. 279-287 (2001). 19 Quantum Mechanics. A.
Messiah. Dover (1999). 20 On the construction
of model Hamiltonians for adiabatic quantum
computing and its application to finding low
energy conformations of lattice protein models.
A. Perdomo, C. Truncik, I. Tubert-Brohman, G.
Rose, and A. Aspuru-Guzik. To appear in PRA.