The general computational process

A suitably programmed quantum computer should act on a number x to produce another number f(x) for some specified function f. Appropriately interpreted, with an accuracy that increases with increasing k, we can treat such numbers as non-negative integers less than 2k. Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits.

If we specify the numbers x as n-bit integers and the numbers f(x) as m-bit integers, then we shall need at least n + m Qbits: a set of n-Qbits, called the input register, to represent x, and another set of m-Qbits, called the output register, to represent f(x). Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f(x) assigns the same value to different values of x, as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f(x). Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall find that this dual-register architecture can also be usefully exploited by a quantum computer in some strikingly nonclassical ways.

Period finding, factoring, and cryptography

Simon's problem (Section 2.5) starts with a subroutine that calculates a function f(x), which satisfies f(x) = f(y) for distinct x and y if and only if y = x ⊕ a, where ⊕ denotes the bitwise modulo-2 sum of the n-bit integers a and x. The number of times a classical computer must invoke the subroutine to determine a grows exponentially with n, but with a quantum computer it grows only linearly.

This is a rather artificial example, of interest primarily because it gives a simple demonstration of the remarkable computational power a quantum computer can possess. It amounts to finding the unknown period a of a function on n-bit integers that is “periodic” under bitwise modulo-2 addition. A more difficult, but much more natural problem is to find the period r of a function f on the integers that is periodic under ordinary addition, satisfying f(x) = f(y) for distinct x and y if and only if x and y differ by an integral multiple of r. Finding the period of such a periodic function turns out to be the key to factoring products of large prime numbers, a mathematically natural problem with quite practical applications.

We illustrate here the mathematics of the final (post-quantum-computational) stage of Shor's period-finding procedure. The final measurement produces (with high probability) an integer y that is within ½ of an integral multiple of 2n/r, where n is the number of Qbits in the input register, satisfying 2n > N2 > r2. Deducing the period r of the function f from such an integer y makes use of the theorem that if x is an estimate for the fraction j/r that differs from it by less than ½r2, then j/r will appear as one of the partial sums in the continued-fraction expansion of x. In the case of Shor's period finding algorithm x = y/2n. If j and r happen to have no factors in common, r is given by the denominator of the partial sum with the largest denominator less than N. Otherwise the continued-fraction expansion of x gives r0: r divided by whatever factor it has in common with the random integer j. If several small multiples of r0 fail to be a period of f, one repeats the whole procedure, getting a different submultiple r1 of r. There is a good chance that r will be the least common multiple of r0 and r1, or a not terribly large multiple of it. If not, one repeats the whole procedure a few more times until one succeeds in finding a period of f.

INTRODUCTION

We know from Section 2.2.2 that configuration spaces of general linkages that are permitted to self-intersect, even in ℝ2, can have exponentially many connected components. On the other hand, we know from Section 5.1.1.1 (p. 59) and Section 5.1.2 (p. 66) that configuration spaces of open and closed 3D chains that are permitted to self-intersect have just one connected component. We also know from Section 5.3 (p. 70) that configuration spaces of planar chains have just one connected component when permitted to move into ℝ3 but forbidden to self-cross. We have until now avoided the most natural questions, which concern chains embedded in ℝd, with motion confined to the same space ℝd, without self-crossings. These questions avoid the generality of linkages on the one hand, and the special assumptions of planar embeddings or projections on the other hand.

The main question addressed in this context is which types of linkages always have connected configuration spaces. A linkage with a connected configuration space is unlocked: no two configurations are prevented from reaching each other. If a linkage in 3D or higher dimensions has a disconnected configuration space, it is locked. But for connectivity of the configuration space to be possible in 2D, we need to place an additional constraint, because planar closed chains cannot be turned “inside-out” as they could in Section 5.1.2 when we permitted the chain to self-intersect.