Barenco Gate

A general two-qubit controlled rotation introduced in the proof that a fixed two-qubit gate plus single-qubit gates is universal.

Definition

The Barenco gate is a parametric controlled operation A(φ, α, θ) that applies a general single-qubit unitary, parameterized by three Euler-like angles, to the target when the control is set. It was introduced by Adriano Barenco as an explicit universal two-qubit gate: any quantum computation can be built from copies of a single Barenco gate acting on different qubit pairs.

Matrix

The three angles φ, α, θ let the lower-right block sweep across all single-qubit unitaries, so choosing them recovers CNOT, controlled-phase, controlled rotations and more from one family.

Universality

Barenco's result complements the Barenco-DiVincenzo-et-al theorem that CNOT plus single-qubit gates is universal. It showed something stronger: a generic two-qubit gate is universal on its own when applied to enough pairs. Almost every two-qubit unitary shares this property; the exceptions form a measure-zero set.

import numpy as np
def barenco(phi, alpha, theta):
    m = np.eye(4, dtype=complex)
    c = np.exp(1j*alpha)*np.cos(theta)
    a = -1j*np.exp(1j*(alpha-phi))*np.sin(theta)
    b = -1j*np.exp(1j*(alpha+phi))*np.sin(theta)
    m[2,2]=c; m[3,3]=c; m[2,3]=a; m[3,2]=b
    return m

Significance

The Barenco gate is mainly of theoretical importance: it anchors the modern understanding of gate universality and motivated the ABC decomposition used to compile arbitrary controlled-U gates. See controlled-U and ABC decomposition.