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We introduce a new model of interacting spin 1/2. It describes interactions of three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin gates. The Fredkin gate (also known as the controlled swap gate) is a computational circuit suitable for reversible computing. Our construction generalizes the model presented by Peter Shor and Ramis Movassagh to half-integer spins. Our model can be solved by means of Catalan combinatorics in the form of random walks on the upper half plane of a square lattice (Dyck walks). Each Dyck path can be mapped on a wave function of spins. The ground state is an equally weighted superposition of Dyck walks (instead of Motzkin walks). We can also express it as a matrix product state. We further construct a model of interacting spins 3/2 and greater half-integer spins. The models with higher spins require coloring of Dyck walks. We construct a $SU(k)$ symmetric model (where $k$ is the number of colors). The leading term of the entanglement entropy is then proportional to the square root of the length of the lattice (like in the Shor–Movassagh model). The gap closes as a high power of the length of the lattice [5, 11].

An alternative scheme is proposed in this paper to realize an n-qubit Fredkin gate with superconducting quantum-interference device (SQUID) in cavity QED. The multiqubit logical gate is essentially accomplished by a sequence of operations on only two SQUIDs in a large-detuned cavity. In the scheme, the two logical states of a qubit are presented by the two lowest states of a SQUID and the cavity is always in vacuum state so that the limit on the cavity is not serious.

Barrel shifter is an integral component of processor datapaths in computing systems since it can shift and rotate multiple bits in a single cycle. Furthermore, reversible logic has applications in emerging computing paradigms such as quantum computing, quantum dot cellular automata, optical computing, etc. In this work, we propose efficient methodologies for designing reversible barrel shifters. The proposed methodologies are designed using Fredkin gate and Feynman gate (FG). The Fredkin gate is used because it can implement a 2:1 MUX with minimum quantum cost, minimum number of ancilla inputs and garbage outputs, and the Feynman gate is used to avoid a fanout since a fanout is not allowed in reversible logic. In the existing literature, design methodologies are limited to the design of a ($n,k$) reversible left rotator that can only perform the left rotate operation. This work explores the other primary functionalities of a reversible barrel shifter such as the design of a reversible: (i) logical right shifter, (ii) universal right shifter that supports logical right shifter, arithmetic right shifter and right rotate operation, (iii) bidirectional logical shifter and (iv) universal bidirectional shifter that supports bidirectional logical and arithmetic shifter and rotate operations. The other types of reversible barrel shifters can also be easily designed by making minor modifications in the proposed methodologies. The proposed design methodologies are generic in nature and can be implemented using any barrel shifter of ($n,k$) size, where n and k are the number of data bits and shift value, respectively. In order to minimize the number of ancilla inputs and garbage outputs, strategies such as the implementation of an n number of 2:1 MUXes as a chain of n Fredkin gates and the mapping of the two different 2:1 MUXes that are controlled by a common control signal but having the swapped controlled signals on a single Fredkin gate, are utilized. The design methodologies are evaluated in terms of the number of garbage outputs, the number of ancilla inputs and quantum cost. For a ($n,k$) reversible barrel shifter, the relations between the varying values of n and k and their impact on the number of garbage outputs, the number of ancilla inputs and quantum cost are also established to help the designers in choosing an efficient barrel shifter according to their design needs.

A thin-layer Belousov–Zhabotinsky (BZ) medium is a powerful computing device capable for implementing logical circuits, memory, image processors, robot controllers, and neuromorphic architectures. We design the reversible logical gates — Fredkin gate and Toffoli gate — in a BZ medium network of excitable channels with subexcitable junctions. Local control of the BZ medium excitability is an important feature of the gates’ design. An excitable thin-layer BZ medium responds to a localized perturbation with omnidirectional target or spiral excitation waves. A subexcitable BZ medium responds to an asymmetric perturbation by producing traveling localized excitation wave-fragments similar to dissipative solitons. We employ interactions between excitation wave-fragments to perform the computation. We interpret the wave-fragments as values of Boolean variables. The presence of a wave-fragment at a given site of a circuit represents the logical truth, absence of the wave-fragment — logically false. Fredkin gate consists of ten excitable channels intersecting at 11 junctions, eight of which are subexcitable. Toffoli gate consists of six excitable channels intersecting at six junctions, four of which are subexcitable. The designs of the gates are verified using numerical integration of two-variable Oregonator equations.

A scheme is proposed to directly implement the optical swap gate and quantum Fredkin gate based on the Mach–Zehnder interferometer (MZI). The distinct advantage of the present scheme is that ancilla single-photons are not needed. The optical swap gate is deterministic and does not need the photon number resolving detectors. The total success probability of the present Fredkin gate can reach 1/16 by using basic linear-optics elements.

In the framework of quantum computation with mixed states, fuzzy representations based on continuous t-norms for Toffoli and Fredkin quantum gates are introduced. A comparison between both gates is also studied.