The Fredkin gate (also controlled-SWAP gate and conservative logic gate) is a computational circuit suitable for reversible computing, invented by Edward Fredkin. It is universal, which means that any logical or arithmetic operation can be constructed entirely of Fredkin gates. The Fredkin gate is a circuit or device with three inputs and three outputs that transmits the first bit unchanged and swaps the last two bits if, and only if, the first bit is 1.
The Fredkin gate, [1] conceptualized by Edward Fredkin and Tommaso Toffoli at the MIT Laboratory for Computer Science, represents a pivotal advancement in the field of reversible computing and conservative logic. Developed within the framework of conservative logic, this gate is designed to align computing processes with fundamental physical principles such as the reversibility of dynamical laws and the conservation of energy. The technical rationale behind the Fredkin gate is rooted in addressing the inefficiencies of traditional computing, where irreversible operations typically result in significant energy dissipation.
In contrast to conventional logic gates, which often erase information and thus dissipate heat as per Landauer's principle, [2] the Fredkin gate maintains reversibility — a property that ensures no information is lost during the computation process. Each output state of the gate uniquely determines its input state, which not only preserves information but also aligns with energy conservation principles. This characteristic is particularly crucial as the demand for computational power grows, making energy efficiency a key consideration.
The invention of the Fredkin gate was motivated by the quest to minimize the energy footprint of computational operations. It allows for the construction of computing systems that are not only efficient in terms of processing speed and power consumption but also environmentally sustainable. By embodying principles of reversible computing, the Fredkin gate offers a practical solution to reducing the energy costs associated with digital computations, marking a significant shift towards more sustainable computing technologies.
The basic Fredkin gate [3] is a controlled swap gate (CSWAP gate) that maps three inputs (C, I1, I2) onto three outputs (C, O1, O2). The C input is mapped directly to the C output. If C = 0, no swap is performed; I1 maps to O1, and I2 maps to O2. Otherwise, the two outputs are swapped so that I1 maps to O2, and I2 maps to O1. It is easy to see that this circuit is reversible, i.e., "undoes" itself when run backwards. A generalized n × n Fredkin gate passes its first n − 2 inputs unchanged to the corresponding outputs and swaps its last two outputs if and only if the first n − 2 inputs are all 1.
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The Fredkin gate can be defined using truth functions with AND, OR, XOR, and NOT, as follows:
where S = (I1 XOR I2) AND C.
Alternatively:
One way to see that the Fredkin gate is universal is to observe that it can be used to implement AND, NOT and OR:
We can encode the truth table in a hardware description language such as Verilog:
modulefredkin_gate(inputu,inputx1,inputx2,outputv,outputy1,outputy2);always@(*)beginv=u;y1=(~u&x1)|(u&x2);y2=(u&x1)|(~u&x2);endendmodule
Three-bit full adder (add with carry) using five Fredkin gates. The "garbage" output bit g is (p NOR q) if r = 0, and (p NAND q) if r = 1.
Inputs on the left, including two constants, go through three gates to quickly determine the parity. The 0 and 1 bits swap places for each input bit that is set, resulting in parity bit on the 4th row and inverse of parity on 5th row.
Then the carry row and the inverse parity row swap if the parity bit is set and swap again if one of the p or q input bits are set (it doesn't matter which is used) and the resulting carry output appears on the 3rd row.
The p and q inputs are only used as gate controls so they appear unchanged in the output.
Recent research has demonstrated the Fredkin gate on programmable silicon photonic chips. These chips use a network of Mach-Zehnder interferometers to route photons efficiently, creating a versatile and scalable platform that can handle multiple quantum gates. This approach allows for integrating Fredkin gates into large-scale quantum processors, paving the way for future quantum computing advancements. [4]
In a photonic setup, the Fredkin gate serves as an effective controlled-SWAP mechanism, enabling the conditional swap of target qubits. This is particularly valuable in generating high-fidelity Greenberger-Horne-Zeilinger (GHZ) states, which are crucial for quantum communication and other protocols. The gate thus provides a powerful tool for quantum protocols that require efficient conditional operations. [5]
The Fredkin gate's controlled operations allow for estimating the overlap between quantum states without requiring resource-intensive quantum state tomography. This makes it particularly useful for quantum communication, measurement, and cryptography, where efficiency and accuracy are paramount. [5]
On March 25, 2016, researchers from Griffith University and the University of Queensland announced they had built a quantum Fredkin gate that uses the quantum entanglement of particles of light to swap qubits. The availability of quantum Fredkin gates may facilitate the construction of quantum computers. [5] [6]
In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state.
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ. With multiple inputs, XOR is true if and only if the number of true inputs is odd.
In logic circuits, the Toffoli gate, also known as the CCNOT gate (“controlled-controlled-not”), invented by Tommaso Toffoli, is a CNOT gate with two control qubits and one target qubit. That is, the target qubit will be inverted if the first and second qubits are both 1. It is a universal reversible logic gate, which means that any classical reversible circuit can be constructed from Toffoli gates.
An adder, or summer, is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used in the arithmetic logic units (ALUs). They are also used in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators and similar operations.
In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly other actions. The minimum set of actions that a circuit needs to be able to perform on the qubits to enable quantum computation is known as DiVincenzo's criteria.
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.
Reversible computing is any model of computation where the computational process, to some extent, is time-reversible. In a model of computation that uses deterministic transitions from one state of the abstract machine to another, a necessary condition for reversibility is that the relation of the mapping from states to their successors must be one-to-one. Reversible computing is a form of unconventional computing.
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XOR gate is a digital logic gate that gives a true output when the number of true inputs is odd. An XOR gate implements an exclusive or from mathematical logic; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "must have one or the other but not both".
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Norman H. Margolus is a Canadian-American physicist and computer scientist, known for his work on cellular automata and reversible computing. He is a research affiliate with the Computer Science and Artificial Intelligence Laboratory at the Massachusetts Institute of Technology.
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