Aug 18, 2014

COMPUTER ENGINEERING STUDIES

A seminar report on :Quantum Structural Complexity

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Durga Datta Kandel U07CO264 B.TECH. IV (Computer Engineering) 7th Semester 2010-2011 Dr. Devesh Jinwala

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DEPARTMENT OF COMPUTER

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SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY, SURAT - 395 007 (GUJARAT, INDIA)

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IndexTitle Certificate Abstract List of Figures Notation Used Acknowledgement 1. Quantum Computation: Historical Development 1 2. Elements of Quantum Computing3 2.1 2.2 2.3 2.4 3.1 3.2 Qubit.3 Quantum Gate ..5 2.2.1 The CNOT gate6 Quantum Circuits.6 Quantum Algorithm.7 Complexity Models..9 Computational Complexity..10

3. Complexity Theory...9

4. Quantum Complexity Theory...11 4.1 4.2 4.3 4.4 Definition of the subject and Importance & Importance.11 Basic Complexity Class: Definition.12 Polynomial Time Quantum Computation15 4.3. 1 Polynomial Circuit Families and BQP...15 Quantum Proof.17 4.4.1 Definition of QMA..17

5. Conclusion19 Acknowledgement20 Reference...................................................................................................................... 21

List of Figures Fig. 2.1 The Bloch Sphere ..4 Fig. 2.2: The CNOT Gate5 Fig. 2.3: The Hadamard Gate..8 Fig 3.1: An example of a quantum circuit10 Fig. 4.1: Relation among Complexity Class...14 Fig. 4.2: Suspected relation among P, NP , BQP and QMA.......18

Notations : Pr[A] [ .] i AB AB AB |x| X A probability of event A reference index Quantum Stare Vector (-1) alphabet {0,1} All the language over A is subset of B Set A intersection Set B Set A union Set B modulus ( Length) X belongs to A

Abstract

Quantum Computing generalizes and extend the notion of conventional computation by directly using the quantum mechanical phenomena such as entanglement and superposition to perform operation (quantum rule) on data encoded in physical system [1]. With the discovery of Shors Factorization Algorithm [2] and Grover's Search Algorithm [3], significant interest has been drawn in the field of Quantum Computing. Though practical quantum computing is still in its infancy, both practical and theoretical continues. It has become an attractive interdisciplinary research area in Physics, Mathematics and Computer Science with profound implication to all of these. Quantum effects like interference and entanglement play no direct role in conventional information processing, but they canin principle now, but probably eventually in practicebe harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations [5]. Following the sequences of results [2, 6, 7] suggesting that quantum computers are more powerful than classical probabilistic computers, a great deal of attention has focused on quantum computing. Several outstanding problems in Theoretical Computer Science can be tackled in a new approach. Several Important results have been found in Quantum Computational Complexity which can potentially shake the foundations of Theoretical Computer Science. Here, I give a brief introduction to quantum computing and track through the developments in Quantum Computational Complexity along with its implication to Computer Science.

1Quantum Computation: Historical DevelopmentA nice historical perspective of evolution of computational model as a physical system is given at [2]. The first person to look at the interaction between computation and quantum mechanics appears to have been Benioff [9]. Although he did not ask whether quantum mechanics conferred extra power to computation, he showed that reversible unitary evolution was sufficient to realize the computational power of a Turing machine, thus showing that quantum mechanics is at least as powerful computationally as a classical computer. This work was fundamental in making later investigation of quantum computers possible. Feynman [8] seems to have been the first to suggest that quantum mechanics might be more powerful computationally than a Turing machine. He gave arguments as to why quantum mechanics might be intrinsically expensive computationally to simulate on a classical computer. He also raised the possibility of using a computer based on quantum mechanical principles to avoid this problem, thus implicitly asking the converse question: by using quantum mechanics in a computer can you compute more efficiently than on a classical computer? Deutsch [10] was the first to ask this question explicitly. In order to study this question, he defined both quantum Turing machines and quantum circuits and investigated some of their properties. The question of whether using quantum mechanics in a computer allows one to obtain more computational power was addressed by Deutsch and Jozsa [11] and Berthiaume and Brassard [12]. These papers showed that there are problems which quantum computers can quickly solve exactly, but that classical computers can only solve quickly with high probability and the aid of a random number generator. However, these papers did not show how to solve any problem in quantum polynomial time that was not already known to be solvable in polynomial time with the aid of a random number generator, allowing a small probability of error; this isthe characterization of the complexity class BPP (defined later), which is widely viewed as the class of efficiently solvable problems. 1

Further work on this problem was stimulated by Bernstein and Variani [13]. One of the results contained in their paper was an oracle problem (that is, a problem involving a black box subroutine that the computer is allowed to perform, but for which no code is accessible) which can be done in polynomial time on a quantum Turing machine but which requires superpolynomial time on a classical computer. This result was improved by Simon [14], who gave a much simpler construction of an oracle problem which takes polynomial time on a quantum computer but requires exponential time on a classical computer. Indeed, while Bernstein and Vaziarnis problem appears contrived, Simons problem looks quite natural. Simons algorithm inspired the work presented in this paper. Two number theory problems which have been studied extensively but for which no polynomial-time algorithms have yet been discovered are finding discrete logarithms and factoring integers [2]. Its been sown that these problems can be solved in polynomial time on a quantum computer with a small probability of error.[2].Currently, nobody knows how to build a quantum computer, although it seems as though it might be possible within the laws of quantum mechanics. Some suggestions have been made as to possible designs for such computers [15], but there will be substantial difficulty in building any of these [16, 17]. The most difficult obstacles appear to involve the decoherence of quantum superposition through the interaction of the computer with the environment, and the implementation of quantum state transformations with enough precision to give accurate results after many computation steps. Both of these obstacles become more difficult as the size of the computer grows, so it may turn out to be possible to build small quantum computers, while scaling up to machines large enough to do interesting computations may present fundamental difficulties. Even if no useful quantum computer is ever built, this research does illuminate the problem of simulating quantum mechanics on a classical computer. Any method of doing this for an arbitrary Hamiltonian would necessarily be able to simulate a quantum computer. Thus, any general method for simulating quantum mechanics with at most a polynomial slowdown would lead to a polynomial-time algorithm for factoring.

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Elements of Quantum Computing2.1 The QubitThe qubit is the quantum analogue of the bit, the classical fundamental unit of information. It is a mathematical object with specific properties that can be realized physically in many different ways as an actual physical system. Just as the classical bit has a state (either 0 or 1), a qubit also has a state. Yet contrary to the classical bit, 0 and 1 are but two possible states of the qubit, and any linear combination (superposition) thereof is also physically possible. In general, thus, the physical state of a qubit is the superposition = 0 + 1 (where and are complex numbers). The state of a qubit can be described as a vector in a two-dimensional Hilbert space, a complex vector space . The special states 0 and 1 are known as the computational basis states, and form an orthonormal basis for this vector space. According to quantum theory, when we try to measure the qubit in this basis in order to determine its state, we get either 0 with probability or 1 with probability . Since + = 1 (i.e., the qubit is a unit vector in the aforementioned two-dimensional Hilbert state), we may (ignoring the overall phase factor) effectively write its state as = cos() 0 + eisin() 1 , where the numbers and define a point on the unit three-dimensional sphere, as shown here. This sphere is often called the Bloch sphere, and it provides a useful means to visualize the state of a single qubit.

Theoretically, a single qubit can store an infinite amount of information, yet when measured it yields only the classical result (0 or 1) with certain probabilities that are specified by the quantum state. In other words, the measurement changes the state of the qubit, collapsing it from the superposition to one of its terms. The crucial p

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