Toffoli Gate (CCNOT) Simulator

Qubit Register |q₀ q₁ q₂⟩

|000⟩

Apply Gate

Toffoli flips q₂ only when q₀ = 1 AND q₁ = 1.

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State index

0 / 8

Gate applications

Would flip q₂?

Reversible?

Yes (T² = I)

How it works

The Toffoli gate (CCNOT, "doubly controlled NOT") acts on 3 qubits: two controls q₀, q₁ and one target q₂. It flips q₂ if and only if both controls are |1⟩ — otherwise it does nothing:

Toffoli|a,b,c⟩ = |a, b, c ⊕ (a·b)⟩

Truth table (only rows where a=b=1 change c):
000→000  001→001  010→010  011→011
100→100  101→101  110→111  111→110

As an 8×8 permutation matrix on the basis states |000⟩…|111⟩, Toffoli is unitary and its own inverse: applying it twice returns the exact starting state (T² = I). That reversibility is what makes it a valid quantum gate, even though its action on classical bit strings is identical to a purely classical reversible gate.

Universality: Toffoli alone is universal for classical reversible computation — any classical logic circuit (AND, OR, NOT, FANOUT) can be built from Toffoli gates with ancilla bits, with zero information loss.
Quantum role: combined with Hadamard, {H, T, Toffoli} forms a universal gate set for quantum computation — it is the standard way to embed classical arithmetic (e.g. inside Shor's algorithm's modular exponentiation) into a quantum circuit.
Cube view: the 8 corners of the cube are the 8 basis states |000⟩…|111⟩. Applying Toffoli swaps the two corners that differ only where both controls are 1 — visually, the highlighted vertex jumps to its Toffoli-paired corner and back.

Current state |q₀q₁q₂⟩ Other basis states

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