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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

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Quantum walks and quantum bioinformatics

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]


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

Outline

  • Concise introduction to quantum computation.

  • Quantum walks.

  • Protein Folding and quantum adiabatic algorithms.

  • Future research directions and Conclusions.


Quantum walks and quantum bioinformatics

Part I

Very quick introduction to quantum computation


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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: 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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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 anexponential 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.


Quantum walks and quantum bioinformatics

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].


Quantum walks and quantum bioinformatics

Part II

Quantum Walks


Quantum walks and quantum bioinformatics

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]


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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 and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

  • 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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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).


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

Continuous quantum walks (3/3)

  • Childs et al’s 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)


Quantum walks and quantum bioinformatics

Part III

Protein Folding

and

Quantum Adiabatic Algorithms


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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).


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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].


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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 walks and quantum bioinformatics

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 walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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…)


Quantum walks and quantum bioinformatics

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 walks and quantum bioinformatics

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).


Quantum walks and quantum bioinformatics

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


Quantum walks and quantum bioinformatics

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 < 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

for all the computing time t  [0,T]


Quantum walks and quantum bioinformatics

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!


Quantum walks and quantum bioinformatics

Quantum mechanical algorithms for Protein Folding (5/9)

For how long should we run the quantum adiabatic computer?

where


Quantum walks and quantum bioinformatics

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…


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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).


Quantum walks and quantum bioinformatics

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.


Quantum walks and quantum bioinformatics

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?


Quantum walks and quantum bioinformatics

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)


Quantum walks and quantum bioinformatics

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 200:1012-1018 (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. ArXiv:quant-ph/0001106

[17]A quantum adiabaticevolutionalgorithmappliedto 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] Howpowerful is adiabatic quantum computation?Win Van Dam, Michele Mosca, and UmeshVazirani. 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.


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