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--- quantum-gate-three-qubit [August 26, 2026 at 16:53] – created - external edit 127.0.0.1
+++ quantum-gate-three-qubit [August 26, 2026 at 17:49] (current) – created - external edit 127.0.0.1
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 # Three-qubit gates
-**Three-qubit gates** act on three qubits, typically as controlled versions of single-qubit or two-qubit gates. The most common are Toffoli (CCX) and Fredkin (CSWAP), which are universal for classical reversible computation and form building blocks for larger controlled operations.
+**Three-qubit gates** are unitary operations that act on a triple of qubits. They are represented by $8 \times 8$ unitary matrices and form the Lie group $\mathrm{SU}(8)$. Beyond what two-qubit gates offer, three-qubit gates expose genuinely tripartite entanglement, correlation that can't be reduced to any pair of qubits, and several of them (CCX, CSWAP) are universal for classical reversible computation on their own.
-## CCX (Toffoli)
+A three-qubit gate is applied to a joint state $|\psi\rangle \in \mathbb{C}^8$ to produce $U|\psi\rangle$. Most three-qubit gates used in practice are doubly-controlled versions of single-qubit gates (CCX, CCY, CCZ, CCRx, CCRy, CCRz, CCP) or controlled two-qubit gates (CSWAP, CiSWAP): two control qubits or one control plus a two-qubit operation, rather than an arbitrary point in $\mathrm{SU}(8)$.
-**[[quantum-gate-ccx|CCX]]** (also Toffoli or Controlled-Controlled-X) is the three-qubit version of CX: it flips the target if both controls are $|1\rangle$. The quantum AND gate and universal for classical reversible computation, it can be decomposed into single-qubit and CX gates but requires many gates (~6 CXs); native implementations reduce this overhead.
+Nearly all of them decompose into CNOT and single-qubit gates and none are native on current hardware; they exist as circuit-level building blocks rather than physical operations.
-Action: $|a\rangle|b\rangle|c\rangle \to |a\rangle|b\rangle|c \oplus (a \wedge b)\rangle$
+## List of gates
-$$\text{Toffoli} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}$$
+  - [[quantum-gate-ccx|CCX (Toffoli)]]
+  - [[quantum-gate-ccy|CCY (Controlled-Controlled-Y)]]
+  - [[quantum-gate-ccz|CCZ (Controlled-Controlled-Z)]]
+  - [[quantum-gate-cswap|CSWAP (Fredkin)]]
+  - [[quantum-gate-ciswap|CiSWAP (Controlled-iSWAP)]]
+  - [[quantum-gate-ccp|CCP (Controlled-Controlled-Phase)]]
+  - [[quantum-gate-ccrx|CCRx (Controlled-Controlled-RX)]]
+  - [[quantum-gate-ccry|CCRy (Controlled-Controlled-RY)]]
+  - [[quantum-gate-ccrz|CCRz (Controlled-Controlled-RZ)]]
-Basis action: flips target qubit only for $|11c\rangle$ states. Basis mapping: $|110\rangle \to |111\rangle$ and $|111\rangle \to |110\rangle$.
+## Genuine tripartite entanglement
-## CCY (Controlled-Controlled-Y)
+Two-qubit entanglement is a single number: concurrence, from 0 to 1. Three qubits need more than a number, because entanglement itself splits into inequivalent classes under local operations and classical communication (LOCC). No sequence of single-qubit gates and classical coordination between the three parties converts one class into another.
-**[[quantum-gate-ccy|CCY]]** (Controlled-Controlled-Y) applies Y to the target qubit if both control qubits are $|1\rangle$. Like CCX, it is an asymmetric doubly-controlled gate but less commonly native. It can be decomposed into CCX plus phase gates on the target.
+$\mathrm{GHZ} = \frac{|000\rangle + |111\rangle}{\sqrt{2}}, \quad W = \frac{|001\rangle + |010\rangle + |100\rangle}{\sqrt{3}}$
-$$\text{CCY} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \end{pmatrix}$$
+**GHZ-class** states, produced by a Hadamard followed by CCZ-style correlation (e.g. $H \otimes I \otimes I$ then CCX-chained), have maximal three-way correlation but lose all entanglement if any one qubit is traced out. **W-class** states keep some pairwise entanglement between any two of the three qubits even after tracing out the third, but never reach GHZ's three-way correlation. Between these sit **biseparable** states (entangled across one bipartition, product across another) and fully separable states.
-Basis action: applies Y to target only for $|11c\rangle$ states. Basis mapping: $|110\rangle \to i|111\rangle$ and $|111\rangle \to -i|110\rangle$.
+The three-tangle extends concurrence to detect GHZ-type correlation specifically; it vanishes on W states despite their nonzero pairwise entanglement, which is precisely how the two classes are told apart.
-## CCZ (Controlled-Controlled-Z)
+## Matrix representations
-**[[quantum-gate-ccz|CCZ]]** (Controlled-Controlled-Z) applies a phase to the $|111\rangle$ state only. Symmetric in all three qubits (unlike CCX and CCY), it is related to CZ via a doubly-controlled construction. Diagonal, it applies phases without changing basis states.
+**CC-Pauli family** (controls on qubits 1,2, target on qubit 3), all identity except the $|11\rangle$-target block:
-$$\text{CCZ} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}$$
+$$\text{CCX} = \begin{pmatrix} I_6 & 0 \\ 0 & X \end{pmatrix} \quad \text{CCY} = \begin{pmatrix} I_6 & 0 \\ 0 & Y \end{pmatrix} \quad \text{CCZ} = \begin{pmatrix} I_6 & 0 \\ 0 & Z \end{pmatrix}$$
-Basis action: applies phase only to $|111\rangle$. Basis mapping: $|111\rangle \to -|111\rangle$; all other states unchanged.
+where $I_6$ is the $6\times6$ identity on the subspace where at least one control is $|0\rangle$, and the bottom-right $2\times2$ block is the target-qubit Pauli, applied only in the $|11\rangle$-control subspace.
-## CSWAP (Fredkin)
+**Controlled two-qubit gates** (control on qubit 1, SWAP/iSWAP on qubits 2,3):
-**[[quantum-gate-cswap|CSWAP]]** (also Fredkin or Controlled-SWAP) swaps two qubits if the control is $|1\rangle$. The quantum controlled-SWAP and also universal for classical reversible computation, it is useful for reversible algorithms that require conditional swaps. Self-inverse ($\text{CSWAP}^2 = I$) and symmetric in the two swapped qubits, it conserves Hamming weight (the number of 1s in the bitstring).
+$$\text{CSWAP} = \begin{pmatrix} I_4 & 0 \\ 0 & \mathrm{SWAP} \end{pmatrix} \quad \text{CiSWAP} = \begin{pmatrix} I_4 & 0 \\ 0 & \mathrm{iSWAP} \end{pmatrix}$$
-Action: $|c\rangle|a\rangle|b\rangle \to |c\rangle|a'\rangle|b'\rangle$ where $a' = c \cdot b + \bar{c} \cdot a$ and $b' = c \cdot a + \bar{c} \cdot b$.
+## Universality for reversible classical computation
-$$\text{Fredkin} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}$$
+CCX and CSWAP are each, on their own, universal for classical reversible computation: any classical Boolean circuit can be built from either gate alone plus ancilla bits fixed to $|0\rangle$/$|1\rangle$. CCX with one control fixed to $|1\rangle$ reduces to CNOT; with the target fixed to $|0\rangle$ it computes AND; combined with ancillas it reaches NAND, and NAND alone is classically universal.
-Basis action: swaps the second and third qubits when the first qubit is $|1\rangle$. Basis mapping: $|101\rangle \to |110\rangle$ and $|110\rangle \to |101\rangle$.
+$\text{CCX}(|1\rangle, a, b) = |1\rangle, a, (b \oplus a)$, so fixing one control recovers CNOT
-## General controlled gates
+No single- or two-qubit gate has this property, since a $2\times2$ or $4\times4$ unitary can't implement a genuinely irreversible-looking classical function like AND without an extra output qubit to preserve unitarity. This is why Toffoli and Fredkin appear throughout reversible and quantum arithmetic circuits: they let a quantum computer host ordinary classical logic without discarding information.
-Any unitary $U$ can be controlled: a controlled-$U$ gate applies $U$ to target qubits if all control qubits are $|1\rangle$. Construction typically decomposes $U = e^{i\alpha} A X B X C$ where $ABC = I$, then uses controlled single-qubit gates and CNOTs to build the full controlled operation.
+## Group structure
-## Scalability
+Three-qubit gates form $\mathrm{SU}(8)$, dimension 63. Unlike two-qubit gates, there is no clean KAK-style decomposition that isolates all entangling content into a handful of parameters; the local subgroup $\mathrm{SU}(2)^{\otimes 3}$ has dimension 9, leaving a 54-dimensional entangling quotient with no simple canonical form like the two-qubit Weyl chamber.
-Multi-qubit gates are expensive: implementing a controlled-$U$ on $n$ qubits requires $O(n)$ elementary gates and time. For large $n$, it is often more efficient to avoid native multi-qubit gates and instead use shallow circuits composed of two-qubit gates.
+Classification instead falls back on the coarser GHZ/W/biseparable/separable partition above, which describes the *type* of entanglement a state can have rather than giving a continuous parametrization of gates the way the two-qubit case does.
+## Decomposition and circuit cost
+CCX costs about 6 CNOTs plus single-qubit gates; CSWAP and CiSWAP are comparably expensive. This generalizes to arbitrary $n$-controlled gates via the ancilla-based construction (Barenco et al.): an $n$-control Toffoli decomposes into $O(n)$ ordinary Toffolis using one borrowed ancilla qubit, or $O(n^2)$ without ancilla. Three-qubit gates are therefore the first rung of a ladder: they're where multi-control decomposition techniques start, and every higher-arity controlled gate reduces to chains of them.
+## Uses
+- **Quantum arithmetic**: CCX is the carry gate in ripple-carry quantum adders
+- **Grover's algorithm**: multi-controlled phase oracles built from CCZ/CCP chains
+- **Error correction**: some stabilizer measurements and magic-state protocols use CCZ
+- **Entanglement benchmarking**: GHZ-state fidelity is a standard hardware benchmark, prepared via $H$ then CCX-chained gates
+- **Reversible computing**: CCX/CSWAP host classical reversible logic inside a quantum circuit
+## Implementation
+Three-qubit gates are essentially never physically native; every platform compiles them down to single- and two-qubit primitives.
+- **Superconducting**: CCX decomposed into ~6 CX gates and single-qubit rotations; CCRz can be nearly free if implemented as a virtual phase
+- **Trapped ions**: decomposed via chains of native $\text{XX}(\theta)$-type gates; some proposals for direct multi-ion Mølmer-Sørensen gates remain experimental
+- **Photonic**: three-qubit gates require post-selection or probabilistic schemes; costly compared to two-qubit CZ
+## Relations
+- [[quantum-gate-ccx|CCX]], [[quantum-gate-cswap|CSWAP]]: the two gates universal for classical reversible computation
+- [[quantum-gate-ccy|CCY]], [[quantum-gate-ccz|CCZ]]: doubly-controlled Pauli gates, interconvertible with CCX via basis change
+- [[quantum-gate-ciswap|CiSWAP]]: controlled iSWAP, native on some transmon platforms
+- [[quantum-gate-ccp|CCP]], [[quantum-gate-ccrx|CCRx]], [[quantum-gate-ccry|CCRy]], [[quantum-gate-ccrz|CCRz]]: parametric doubly-controlled gates for variational circuits
+- [[quantum-gate-two-qubit|Two-qubit gates]]: three-qubit gates decompose into chains of these
+- [[quantum-gate-multiqubit|Multi-qubit gates]]: general $n$-qubit generalization, built from three-qubit primitives via ancilla constructions
+- $\mathrm{SU}(8)$: group structure of three-qubit unitaries
+- GHZ state, W state: the two inequivalent classes of genuine tripartite entanglement