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C-Phase gate

Identifier Operator Example statement
CZ \(CZ\) CZ q[0], q[1]

Description

The C-Phase, or CZ, gate is a two-qubit gate. It performs a Z gate (negates the relative phase) on the second qubit, conditional on the state of the first qubit. The first qubit is usually referred to as the control qubit and the second qubit as the target qubit.

In the standard computational basis for two qubits \(\{|00\rangle ,|01\rangle \,|10\rangle ,|11\rangle \}\), the CZ gate:

  • leaves the control qubit unchanged,
  • performs a Z gate on the target qubit, when the control qubit is in state \(|1\rangle\),
  • leaves the target qubit unchanged, when the control qubit is in state \(|0\rangle\).

Note

The notion of a control qubit and a target qubit (for any controlled operation) only holds for the standard computational basis. Generally, in another basis, both the 'control' and 'target' qubit change.

The CZ gate is a specific example of the controlled phase shift gate (CR gate): \(CZ = CR(\pi)\).

Aliases

Also known as C-Phase , controlled-Z , or controlled phase-flip gate.

Properties

  • Clifford gate;
  • Involutory operation (its own inverse);
  • Controlled gate;
  • Perfect Entangler (maximally entangles specific product states);
  • Ising gate.

Representation

\[\begin{align} CZ &= \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}\right) \end{align}\]

which is equal to:

\[CZ = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes Z.\]

Operation examples

Standard basis

\[\begin{align} CZ\,|00\rangle &= |00\rangle \\ \\ CZ\,|01\rangle &= |01\rangle \\ \\ CZ\,|10\rangle &= |10\rangle \\ \\ CZ\,|11\rangle &= -|11\rangle \\ \end{align}\]

Qubit state ordering convention and matrix representation

Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,

\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]

Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).