Fredkin gate

A reversible three-bit controlled-swap gate that preserves the control bit and swaps the other two bits exactly when the control is active.

Version

v2 · 2026-09-19 · History

Domain-specific #

4606

Domain group

Applied Sciences & Engineering

Origin domain

Computer Science & Software Engineering

Subdomain

Reversible Computing → Computer Science & Software Engineering

Core Idea

The Fredkin gate maps (c,a,b) to (c,a,b) when c=0 and to (c,b,a) when c=1, is its own inverse, conserves Hamming weight, and is universal for conservative reversible logic with suitable resources. Control condition selects identity or transposition; the bijective mapping retains every input bit in recoverable output structure and enables circuits without logical information erasure. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Swap-If Switch

Imagine three cups, each holding a marble or nothing. If the first cup is empty, the other two stay put. If the first cup has a marble, the other two cups swap places. No marble ever disappears or appears, and doing it twice puts everything back.

Controlled Swap Gate

The Fredkin gate is a tiny computer part that works on three bits, each a 0 or a 1. The first bit is the control. If it is 0, the other two bits stay where they are. If it is 1, those two bits swap places. Nothing is ever thrown away: you can always work backward from the output to the input, and running the gate twice undoes it. It also never changes how many 1s there are, and with some extra helper bits you can build any logic of this careful, reversible kind out of these gates.

Conservative Reversible Controlled-SWAP

The Fredkin gate, also called a controlled swap, takes bits (c, a, b) and outputs (c, a, b) when c = 0 and (c, b, a) when c = 1. It is a bijection, so it is reversible: every output comes from exactly one input, and the gate is its own inverse. It is also conservative, meaning the number of 1s (the Hamming weight) never changes. Unlike an AND gate, which loses information about its inputs, it erases nothing logically. Given suitable extra resources such as constant helper bits, Fredkin gates are universal for conservative reversible logic. When describing one precisely, you should state the bit order and which control value triggers the swap.

 

The Fredkin gate is a three-bit reversible gate: it maps (c, a, b) to (c, a, b) if c = 0 and to (c, b, a) if c = 1, i.e. a controlled transposition of the two target bits. As a permutation of the eight input states it is bijective and self-inverse, so applying it twice gives the identity. It conserves Hamming weight, which makes it the canonical conservative gate, and every input bit remains recoverable from the output, so circuits built from it need perform no logical information erasure. With suitable resources (ancilla/constant inputs and possibly garbage outputs), it is universal for conservative reversible logic. It can be realized classically or as a quantum gate (controlled-SWAP). A precise specification states the bit order, control polarity, truth table, realization, reversibility and conservation properties, ancilla assumptions, and the exact scope of any universality claim.

Scope of Application

Fredkin gate belongs to reversible computing and is useful where the analyst can specify the typed reversible computing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim are explicit. The scope is broad within that domain but bounded by the need for bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Fredkin gate can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Fredkin gate. Fredkin gate compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed reversible computing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of reversible computing because they reuse the typed reversible computing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Control condition selects identity or transposition; the bijective mapping retains every input bit in recoverable output structure and enables circuits without logical information erasure., and type the carrier, state every parameter and convention in the definition, test that bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Fredkin gate Domain-specific

Parents (1) — outgoing typed relations

• Fredkin gate is a kind of Bijectivity Prime

  It remains its own entry because its identity is fixed by bit order, control polarity, truth table, classical or quantum realization, reversibility, conservative property, ancilla assumptions, and universality claim.

Hierarchy paths (3) — routes to 1 parentless root

• Fredkin gate → is a kind of Bijectivity → is a kind of Function (Mapping)

Show alternative paths (2)

Neighborhood in Abstraction Space

Fredkin gate sits in a crowded region of the domain-specific corpus (34^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Quantum Protocols & Circuit Models (20 abstractions)

Nearest neighbors

• Counterfactual quantum computation — 0.89
• Quantum circuit — 0.89
• Nondeterministic Turing machine — 0.88
• And-inverter graph — 0.88
• AC0 — 0.87

Computed from structural-signature embeddings · 2026-10-25