// 3-parameter 2-pulse single qubit gate
gate u3(theta,phi,lambda) q { 
    U(theta,phi,lambda) q; 
}

// 2-parameter 1-pulse single qubit gate
gate u2(phi,lambda) q { 
    U(pi/2,phi,lambda) q; 
}

// 1-parameter 0-pulse single qubit gate
gate u1(lambda) q { 
    U(0,0,lambda) q; 
}

// controlled-NOT
gate cx c,t { 
    CX c,t; 
}

// idle gate (identity)
gate id a { 
    U(0,0,0) a; 
}

// --- QE Standard Gates ---
// Pauli gate: bit-flip
gate x a { 
    u3(pi,0,pi) a; 
}

// Pauli gate: bit and phase flip
gate y a {
    u3(pi,pi/2,pi/2) a;
}

// Pauli gate: phase flip
gate z a { 
    u1(pi) a; 
}

// Clifford gate: Hadamard
gate h a { 
    u2(0,pi) a;
}

// Clifford gate: sqrt(Z) phase gate
gate s a { 
    u1(pi/2) a; 
}

// Clifford gate: conjugate of sqrt(Z)
gate sdg a { 
    u1(-pi/2) a; 
}

// C3 gate: sqrt(S) phase gate
gate t a { 
    u1(pi/4) a; 
}

// C3 gate: conjugate of sqrt(S)
gate tdg a { 
    u1(-pi/4) a; 
}

// --- Standard rotations ---
// Rotation around X-axis
gate rx(theta) a { 
    u3(theta,-pi/2,pi/2) a; 
}

// rotation around Y-axis
gate ry(theta) a { 
    u3(theta,0,0) a; 
}

// rotation around Z axis
gate rz(phi) a { 
    u1(phi) a; 
}

// --- QE Standard User-Defined Gates ---
// controlled-Phase
gate cz a,b {
    h b; 
    cx a,b;
    h b;
}

// controlled-Y
gate cy a,b { 
    sdg b; 
    cx a,b; 
    s b; 
}

// controlled-H
gate ch a,b {
    h b; 
    sdg b;
    cx a,b;
    h b; 
    t b;
    cx a,b;
    t b; 
    h b; 
    s b; 
    x b; 
    s a;
}

// C3 gate: Toffoli
gate ccx a,b,c {
    h c;
    cx b,c; 
    tdg c;
    cx a,c; 
    t c;
    cx b,c; 
    tdg c;
    cx a,c; 
    t b; 
    t c; 
    h c;
    cx a,b; 
    t a; 
    tdg b;
    cx a,b;
}

// controlled rz rotation
gate crz(lambda) a,b {
    u1(lambda/2) b;
    cx a,b;
    u1(-lambda/2) b;
    cx a,b;
}

// controlled phase rotation
gate cu1(lambda) a,b {
    u1(lambda/2) a;
    cx a,b;
    u1(-lambda/2) b;
    cx a,b;
    u1(lambda/2) b;
}

// controlled-U
gate cu3(theta,phi,lambda) c, t {
    // implements controlled-U(theta,phi,lambda) with target t and control c
    u1((lambda-phi)/2) t;
    cx c,t;
    u3(-theta/2,0,-(phi+lambda)/2) t;
    cx c,t;
    u3(theta/2,phi,0) t;
}