Quantum circuit

Builds on Quantum logic gate Qubit Tensor product of Hilbert spaces Boolean circuit

Definition

Let $n \ge 1$ and let $\mathcal{G}$ be a set of quantum logic gates, called the gate set. A quantum circuit on $n$ qubits over $\mathcal{G}$ is a finite sequence $((G_1, t_1), \dots, (G_m, t_m))$ with the following properties.

Each $G_j$ is a $k_j$ -qubit gate in $\mathcal{G}$ , with $k_j \le n$ .
Each $t_j$ is a $k_j$ -tuple of distinct indices in $\set{1, \dots, n}$ .

Let $\widetilde{G}_j$ be the operator on $(\C^2)^{\otimes n}$ obtained by applying $G_j$ to the qubits $t_j$ . The circuit implements the unitary operator $U = \widetilde{G}_m \cdots \widetilde{G}_2 \widetilde{G}_1$ . The number $m$ is the size of the circuit.

Remarks

Example: let $n = 2$ and the circuit be $((H, (1)), (\mathrm{CNOT}, (1, 2)))$ , with $H$ the Hadamard gate. Then $\lvert 00 \rangle \mapsto \frac{1}{\sqrt2}(\lvert 00 \rangle + \lvert 10 \rangle) \mapsto \frac{1}{\sqrt2}(\lvert 00 \rangle + \lvert 11 \rangle)$ , a Bell state.

Non-example: the sequence $((\mathrm{CNOT}, (1, 1)))$ is not a circuit, because the indices in $t_1$ are not distinct; a gate cannot act twice on the same qubit at once.

The depth is the least number of layers into which the sequence can be cut, in order, such that the gates in one layer act on disjoint qubits. Many authors also allow measurements and classical control inside a circuit; by the principle of deferred measurement, these can be moved to the end at small cost.

Reading path

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Boolean circuit concept
Quantum logic gate concept
Qubit concept
Tensor product of Hilbert spaces concept
Quantum circuit definition

Tag	`CP0JWNNWXG`
URI	`m2://CP0JWNNWXG`
Kind	definition
Writing	W0 stub
MSC	81P68, 68Q12, 81P65
Defines	Quantum circuit
Uses	Quantum logic gate, Qubit, Tensor product of Hilbert spaces, Boolean circuit
Changed	3 November 2026

Quantum circuit

Definition

Remarks

Reading path

Further reading

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