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Unformatted text preview: Chapter 10 Quantum algorithms This book started with the worlds oldest and most widely used algorithms (the ones for adding and multiplying numbers) and an ancient hard problem ( FACTORING ). In this last chapter the tables are turned: we present one of the latest algorithmsand it is an efficient algorithm for FACTORING ! There is a catch, of course: this algorithm needs a quantum computer to execute. Quantum physics is a beautiful and mysterious theory that describes Nature in the small, at the level of elementary particles. One of the major discoveries of the nineties was that quantum computerscomputers based on quantum physics principlesare radically differ- ent from those that operate according to the more familiar principles of classical physics. Surprisingly, they can be exponentially more powerful: as we shall see, quantum computers can solve FACTORING in polynomial time! As a result, in a world with quantum computers, the systems that currently safeguard business transactions on the Internet (and are based on the RSA cryptosystem) will no longer be secure. 10.1 Qubits, superposition, and measurement In this section we introduce the basic features of quantum physics that are necessary for understanding how quantum computers work. 1 In ordinary computer chips, bits are physically represented by low and high voltages on wires. But there are many other ways a bit could be storedfor instance, in the state of a hydrogen atom. The single electron in this atom can either be in the ground state (the lowest energy configuration) or it can be in an excited state (a high energy configuration). We can use these two states to encode for bit values and 1 , respectively. Let us now introduce some quantum physics notation. We denote the ground state of our electron by , since it encodes for bit value , and likewise the excited state by 1 . These are 1 This field is so strange that the famous physicist Richard Feynman is quoted as having said, I think I can safely say that no one understands quantum physics. So there is little chance you will understand the theory in depth after reading this section! But if you are interested in learning more, see the recommended reading at the books end. 311 312 Algorithms Figure 10.1 An electron can be in a ground state or in an excited state. In the Dirac notation used in quantum physics, these are denoted and 1 . But the superposition principle says that, in fact, the electron is in a state that is a linear combination of these two: + 1 1 . This would make immediate sense if the s were probabilities, nonnegative real numbers adding to 1 . But the superposition principle insists that they can be arbitrary complex num- bers , as long as the squares of their norms add up to 1!...
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- Spring '08