Commutator Generators

The operator representations provide efficient functions for the computation of various commutators. Since these functions work with generic operator representations (a notion that does not exist in C) we explain them generically in the following three sections. The actual functions contained in the C API for the various operator representations are listed at the bottom of this page.

Commutator

OpType qf_op_type_commutator(const OpType *op_a, const OpType *op_b)

Caution

The function signature here is generic! A real one will replace OpType with an actual operator representations and op_type with its matching prefix (e.g. ferm_op for QfFermionOperator).

The commutator is defined as:

\[[A, B] = AB - BA\]
Parameters:
  • op_a – A pointer to the operator \(A\).

  • op_b – A pointer to the operator \(B\).

Returns:

A pointer to the constructed operator commutator.

Example

 1QfFermionOperator *op1 = qf_ferm_op_zero();
 2QkComplex64 coeff1 = {1.0, 0.0};
 3bool action1[2] = {true, false};
 4uint32_t modes1[2] = {0, 0};
 5qf_ferm_op_add_term(op1, 2, action1, modes1, &coeff1);
 6QfFermionOperator *op2 = qf_ferm_op_zero();
 7QkComplex64 coeff2 = {2.0, 0.0};
 8bool action2[2] = {false, true};
 9uint32_t modes2[2] = {0, 0};
10qf_ferm_op_add_term(op2, 2, action2, modes2, &coeff2);
11
12QfFermionOperator *comm = qf_ferm_op_commutator(op1, op2);

Anti-Commutator

OpType qf_op_type_anti_commutator(const OpType *op_a, const OpType *op_b)

Caution

The function signature here is generic! A real one will replace OpType with an actual operator representations and op_type with its matching prefix (e.g. ferm_op for QfFermionOperator).

The anti-commutator is defined as:

\[\{A, B\} = AB + BA\]
Parameters:
  • op_a – A pointer to the operator \(A\).

  • op_b – A pointer to the operator \(B\).

Returns:

A pointer to the constructed operator anti-commutator.

Example

 1QfFermionOperator *op1 = qf_ferm_op_zero();
 2QkComplex64 coeff1 = {1.0, 0.0};
 3bool action1[2] = {true, false};
 4uint32_t modes1[2] = {0, 0};
 5qf_ferm_op_add_term(op1, 2, action1, modes1, &coeff1);
 6QfFermionOperator *op2 = qf_ferm_op_zero();
 7QkComplex64 coeff2 = {2.0, 0.0};
 8bool action2[2] = {false, true};
 9uint32_t modes2[2] = {0, 0};
10qf_ferm_op_add_term(op2, 2, action2, modes2, &coeff2);
11
12QfFermionOperator *anti_comm = qf_ferm_op_anti_commutator(op1, op2);

Double-Commutator

OpType qf_op_type_double_commutator(const OpType *op_a, const OpType *op_b, const OpType *op_c, bool sign)

Caution

The function signature here is generic! A real one will replace OpType with an actual operator representations and op_type with its matching prefix (e.g. ferm_op for QfFermionOperator).

The double-commutator is defined as follows (see also Equation (13.6.18) in [1]):

If sign is false, it returns

\[[[A, B], C]/2 + [A, [B, C]]/2 = (2ABC + 2CBA - BAC - CAB - ACB - BCA)/2. \]

If sign is true, it returns

\[\{[A, B], C\}/2 + \{A, [B, C]\}/2 = (2ABC - 2CBA - BAC + CAB - ACB + BCA)/2. \]
Parameters:
  • op_a – A pointer to the operator \(A\).

  • op_b – A pointer to the operator \(B\).

  • op_c – A pointer to the operator \(C\).

  • sign – the nature of the outer (anti-)commutator as per the definition above.

Returns:

A pointer to the constructed operator anti-commutator.

Example

 1QfFermionOperator *op1 = qf_ferm_op_zero();
 2QkComplex64 coeff1 = {1.0, 0.0};
 3bool action1[2] = {true, false};
 4uint32_t modes1[2] = {0, 0};
 5qf_ferm_op_add_term(op1, 2, action1, modes1, &coeff1);
 6QfFermionOperator *op2 = qf_ferm_op_zero();
 7QkComplex64 coeff2 = {2.0, 0.0};
 8bool action2[2] = {false, true};
 9uint32_t modes2[2] = {0, 0};
10qf_ferm_op_add_term(op2, 2, action2, modes2, &coeff2);
11QfFermionOperator *op3 = qf_ferm_op_zero();
12qf_ferm_op_add_term(op3, 2, action1, modes1, &coeff1);
13QkComplex64 coeff3 = {2.0, 0.5};
14qf_ferm_op_add_term(op3, 2, action2, modes2, &coeff3);
15
16QfFermionOperator *double_comm = qf_ferm_op_double_commutator(op1, op2, op3, false);

Functions

QfFermionOperator *qf_ferm_op_commutator(const QfFermionOperator *op_a, const QfFermionOperator *op_b)
QfFermionOperator *qf_ferm_op_anti_commutator(const QfFermionOperator *op_a, const QfFermionOperator *op_b)
QfFermionOperator *qf_ferm_op_double_commutator(const QfFermionOperator *op_a, const QfFermionOperator *op_b, const QfFermionOperator *op_c, bool sign)
QfMajoranaOperator *qf_maj_op_commutator(const QfMajoranaOperator *op_a, const QfMajoranaOperator *op_b)
QfMajoranaOperator *qf_maj_op_anti_commutator(const QfMajoranaOperator *op_a, const QfMajoranaOperator *op_b)
QfMajoranaOperator *qf_maj_op_double_commutator(const QfMajoranaOperator *op_a, const QfMajoranaOperator *op_b, const QfMajoranaOperator *op_c, bool sign)