Reversible Boolean networks

Abstract

We continue our consideration of a class of models describing the reversible dynamics of N Boolean variables, each with K inputs. We investigate in detail the behavior of the Hamming distance as well as of the distribution of orbit lengths as N and K are varied. We present numerical evidence for a phase transition in the behavior of the Hamming distance at a critical value K c ≈ 1.65 and also an analytic theory that yields the exact bounds on 1.5 ≤ K c ≤ 2.We also discuss the large oscillations that we observe in the Hamming distance for K < K c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate K both greater than and less than K c . We propose that local structures, or subsets of spins whose dynamics are not fully coupled to the other spins in the system, play a crucial role in generating these oscillations. The simplest of these structures are linear chains, called linkages, and rings, called circuits. We discuss the properties of the linkages in some detail, and sketch the properties of circuits. We argue that the observed oscillation phenomena can be largely understood in terms of these local structures.

Scaling laws for Shor's algorithm with a banded quantum Fourier transform

“…Therefore, reaching higher N is facilitated by reducing the requirements to run Shor's algorithm on a quantum computer. One such optimization is the use of an approximate, banded quantum Fourier transform [14] instead of the the full quantum Fourier transform (11). Further optimization is possible by using a banded version of the semi-classical quantum Fourier transform [22] defined in the following section.…”

“…Figure 1 (a) illustrates this single-qubit realization of the quantum Fourier transform for the special case of five qubits (we classify the conditional rotation gates θ in measurement results are used to control the phase rotation gates θ. As first pointed out by Coppersmith [14], even this quantum circuit may still be optimized by working with an approximate, banded quantum Fourier transform as illustrated in Fig. 1 (b).…”

“…This section also serves to introduce the basic notation and explains the central position of the quantum Fourier transform in Shor's algorithm. While the original version of Shor's algorithm [11] is formulated with the help of a full implementation of the quantum Fourier transform, it turns out that a reduced, approximate version of the quantum Fourier transform, the banded quantum Fourier transform [14][15][16], yields surprisingly good results when used in conjunction with Shor's algorithm. The banded quantum Fourier transform is introduced and discussed in Sec.…”

Scale-free network with Boolean dynamics as a function of connectivity

“…The simplicity and richness of behaviors of this model created a huge field of research, with many types of different boolean networks [5]. Moreover, this model became a good candidate to explain biological problems such as cell differentiation, gene expression, protein interaction and genetic regulatory networks [6][7][8]. A boolean network is a complex dynamical system constituted of logical variables connected by logical functions.…”

Results on two-bit gate design for quantum computers

“…Thus, by the criterion of length of its two-gate implementation, the Toffoli gate is only slightly less complex that the general three-bit gate! Previously, the best known implementation of the Toffoli gate was a six-gate implementation of topology (121231) found by Coppersmith [16]. (We do not obtain the same matrices as Coppersmith when we study this topology numerically, which is indicative of a weakness of our approach: since there are many unneeded free parameters in the (121231) implementation, our procedure does not pick out the one with "simple" two-bit gates which Coppersmith found.)…”