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This book started with the world s 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 algorithms and it is an ef cient 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 computers computers based on quantum physics principles are radically different 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
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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 stored for instance, in the state of a hydrogen atom The single electron in this atom can either be in the ground state (the lowest energy con guration) or it can be in an excited state (a high energy con guration) We can use these two states to encode for bit values 0 and 1, respectively Let us now introduce some quantum physics notation We denote the ground state of our electron by 0 , since it encodes for bit value 0, and likewise the excited state by 1 These are the two possible states of the electron in classical physics Many of the most counterintuitive aspects of quantum physics arise from the superposition principle which states that if a
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This eld 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 book s end
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Figure 101 An electron can be in a ground state or in an excited state In the Dirac notation used in quantum physics, these are denoted 0 and 1 But the superposition principle says that, in fact, the electron is in a state that is a linear combination of these two: 0 0 + 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 numbers, as long as the squares of their norms add up to 1!
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quantum system can be in one of two states, then it can also be in any linear superposition 1 1 of those two states For instance, the state of the electron could well be 2 0 + 2 1 or
1 2
perfectly valid quantum state! Such a superposition, 0 0 + 1 1 , is the basic unit of encoded information in quantum computers (Figure 101) It is called a qubit (pronounced cubit ) The whole concept of a superposition suggests that the electron does not make up its mind about whether it is in the ground or excited state, and the amplitude 0 is a measure of its inclination toward the ground state Continuing along this line of thought, it is tempting to think of 0 as the probability that the electron is in the ground state But then how are we to make sense of the fact that 0 can be negative, or even worse, imaginary This is one of the most mysterious aspects of quantum physics, one that seems to extend beyond our intuitions about the physical world This linear superposition, however, is the private world of the electron For us to get a glimpse of the electron s state we must make a measurement, and when we do so, we get a single bit of information 0 or 1 If the state of the electron is 0 0 + 1 1 , then the outcome of the measurement is 0 with probability | 0 |2 and 1 with probability | 1 |2 (luckily we normalized so | 0 |2 + | 1 |2 = 1) Moreover, the act of measurement causes the system to change its state: if the outcome of the measurement is 0, then the new state of the system is 0 (the ground state), and if the outcome is 1, the new state is 1 (the excited state) This feature of quantum physics, that a measurement disturbs the system and forces it to choose (in this case ground or excited state), is another strange phenomenon with no classical analog The superposition principle holds not just for 2-level systems like the one we just described, but in general for k-level systems For example, in reality the electron in the hydrogen atom can be in one of many energy levels, starting with the ground state, the rst excited state, the 292
coef cient 0 is called the amplitude of state 0 , and similarly with 1 And if things aren t already strange enough the s can be complex numbers, as long as they are normalized so 1 2i that | 0 |2 + | 1 |2 = 1 For example, 5 0 + 5 1 (where i is the imaginary unit, 1) is a
1 2
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