QfTransferVertexOperator

struct QfTransferVertexOperator

A transfer-vertex operator.

Note

This is an opaque data structure to the C API whose internals are implemented entirely in Rust. The remainder of this page describes the design and related functions to work with this struct.

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Definition

This operator is defined in terms of the transfer-vertex (\(T_{jk}\), \(V_j\)) operators:

\[\begin{align} V_j &= -i \gamma_{2j-1} \gamma_{2j} = 1 - 2 a^\dagger_j a_j \, , \nonumber \\ T_{jk} &= \frac{i}{2} V_j E_{jk} = \frac{i}{2} \gamma_{2j} \gamma_{2k-1} \, , \nonumber \\ T_{kj} &= \frac{i}{2} E_{jk} V_k = -\frac{i}{2} \gamma_{2j-1} \gamma_{2k} \nonumber \end{align}\]

where \(E_{jk}\) is an edge operator of the QfEdgeVertexOperator and these individual terms fulfill the following mixed fermionic-bosonic commutation relations for \(j \lt k \lt l \lt m\): [1]

\[\begin{align} \left\{ T_{jk}, V_k \right\} &= 0 \nonumber \\ \left\{ T_{jk}, T_{lk} \right\} &= 0 \nonumber \\ \left[ V_k, V_l \right] &= 0 \nonumber \\ \left[ T_{jk}, V_l \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{lm} \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{kj} \right] &= 0 \nonumber \\ \left[ T_{jk}, T_{km} \right] &= 0 \nonumber \, . \end{align}\]

Caution

Unlike the edge operators, where \(E_{kj} = -E_{jk}\) makes the two orientations two representations of a single operator, \(T_{jk}\) and \(T_{kj}\) are different operators. The order of the two index arrays is therefore significant, and there is no orientation convention to normalize — which is why qf_transfer_op_normal_ordered() takes no ascending parameter where qf_edge_op_normal_ordered() does.

We can abuse the notation a little bit and define \(V_j = T_{jj}\) which reflects how the internal data structure of this operator works. This makes the definition of the entire operator the following:

\[\text{\texttt{TransferVertexOperator}} = \sum_i c_i \bigotimes_{lr} T_{lr} \, ,\]

where \(lr\) indexes the involved operator terms and \(c_i\) is the (complex) coefficient making up the linear combination of products. The indices \(l\) and \(r\) can take any value between 0 and the number of fermionic modes acted upon by the operator minus 1.

We will refer to \(T_{lr}\) as generalized transfer operators.

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Implementation

This struct stores the terms and coefficients in multiple sparse vectors, akin to the compressed sparse row format commonly used for sparse matrices. More concretely, a single operator contains 4 arrays:

coeffs	A vector of complex coefficients consisting of two 64-bit floating point numbers.
left_indices	A vector of 32-bit integers storing the left fermionic mode indices (\(l\)).
right_indices	A vector of 32-bit integers storing the right fermionic mode indices (\(r\)).
boundaries	A vector of integers indicating the boundaries between terms.

Fermionic modes indexed by left_indices and right_indices are considered spinless. The two index arrays always have the same length, since every generator is identified by exactly one (left, right) pair; this is why the constructor takes a single num_indices length for both.

Note

You can access read-only borrows of these internal arrays via their respective functions:

• qf_transfer_op_get_coeffs()

• qf_transfer_op_get_left_indices()

• qf_transfer_op_get_right_indices()

• qf_transfer_op_get_boundaries()

The returned pointers stay valid only until the operator is modified or freed, and must not be freed by the caller.

This data structure allows for very efficient construction and manipulation of operators. However, it implies that duplicate terms might be contained in an operator at any moment. These must be resolved manually through the use of qf_transfer_op_simplify().

Construction

A new operator can be constructed directly by specifying the corresponding arrays outlined above. Alternatively, an empty QfTransferVertexOperator can be initialized with qf_transfer_op_zero() and terms can be added iteratively via qf_transfer_op_add_term().

qf_transfer_op_new()	Constructs a new operator from the provided arrays.
qf_transfer_op_zero()	Constructs the additive identity operator.
qf_transfer_op_one()	Constructs the multiplicative identity operator.
qf_transfer_op_add_term()	Adds a term to an existing QfTransferVertexOperator.

Note

A QfTransferVertexOperator can be freed with qf_transfer_op_free().

Arithmetics

The following functions provide arithmetic manipulation:

qf_transfer_op_add()	Adds two operators together.
qf_transfer_op_mul()	Multiplies an operator by a scalar.
qf_transfer_op_compose()	Composes two operators with each other.
qf_transfer_op_adjoint()	Returns the Hermitian conjugate operator.

Manipulation

The following functions provide operator manipulation logic:

qf_transfer_op_ichop()	Removes terms with small coefficient magnitudes.
qf_transfer_op_simplify()	Returns an equivalent but simplified operator.
qf_transfer_op_normal_ordered()	Returns an equivalent operator with normal ordered terms.
qf_transfer_op_relabel_modes()	Relabels the modes of an operator.

Properties

The following functions exist to check certain properties of an operator.

qf_transfer_op_is_hermitian()	Returns whether an operator is Hermitian.
qf_transfer_op_len()	Returns the number of terms in this operator.

Mapping

The following functions map this operator into another representation:

qf_transfer_vertex_to_fermion()	Maps to a QfFermionOperator.
qf_transfer_vertex_to_majorana()	Maps to a QfMajoranaOperator.
qf_transfer_vertex_to_edge_vertex()	Maps to a QfEdgeVertexOperator.

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[1]

https://arxiv.org/abs/2512.11418

Members

QfTransferVertexOperator *qf_transfer_op_new(uint64_t num_terms, uint64_t num_indices, const QkComplex64 *coeffs, const uint32_t *left_indices, const uint32_t *right_indices, const uint32_t *boundaries)

Constructs a new operator.

Any of the pointer arguments can be NULL if and only if their corresponding length is zero.

A generator with equal indices is a vertex operator, \(V_j\); one with differing indices is a transfer operator, \(T_{jk}\).

Caution

Unlike the edge operators, \(T_{jk}\) and \(T_{kj}\) are different operators, not two representations of one. The order of the two index arrays is therefore significant.

Example

// this builds `1.0 + (-1.0) * V(0) T(0,1)`
uint64_t num_terms = 2;
uint64_t num_indices = 2;
uint32_t left_indices[2] = {0, 0};
uint32_t right_indices[2] = {0, 1};
QkComplex64 coeffs[2] = {{1.0, 0.0}, {-1.0, 0.0}};
uint32_t boundaries[3] = {0, 0, 2};
QfTransferVertexOperator *op = qf_transfer_op_new(num_terms, num_indices, coeffs,
                                                  left_indices, right_indices, boundaries);

Parameters:

• num_terms – The number of terms in the operator.

• num_indices – The number of generators summed over all terms. Both index arrays have this same length, since every generator is identified by exactly one (left, right) pair.

• coeffs – A pointer to an array of term coefficients. The length of this array should be num_terms.

• left_indices – A pointer to an array of left-hand mode indices over all terms. The length of this array should be num_indices.

• right_indices – A pointer to an array of right-hand mode indices over all terms. The length of this array should be num_indices.

• boundaries – A pointer to an array of the boundaries between terms. The length of this array should be num_terms + 1.

Returns:

A pointer to the created operator.

void qf_transfer_op_free(QfTransferVertexOperator *op)

Frees an existing operator.

Example

QfTransferVertexOperator *op = qf_transfer_op_one();
qf_transfer_op_free(op);

Parameters:

• op – A pointer to the transfer-vertex operator to be freed.

void qf_transfer_op_get_coeffs(const QfTransferVertexOperator *op, QkComplex64 **coeffs_out, uint64_t *coeffs_len)

Provides read-only access to the operator’s coefficients.

See also

The explanation of the internal data structure, here.

Example

uint64_t num_terms = 2;
QkComplex64 coeffs[2] = {{1.0, 0.0}, {0.0, -1.0}};
uint32_t boundaries[3] = {0, 0, 0};
QfTransferVertexOperator *op =
    qf_transfer_op_new(num_terms, 0, coeffs, NULL, NULL, boundaries);

QkComplex64 *coeffs_out;
uint64_t coeffs_len;

qf_transfer_op_get_coeffs(op, &coeffs_out, &coeffs_len);

assert(coeffs_len == 2);

Note

This function borrows the operator’s internal buffer rather than copying it. The returned pointer stays valid only until the operator is modified or freed; do not free it yourself.

Parameters:

• op – A pointer to the transfer-vertex operator whose coefficients to access.

• coeffs_out – A pointer to the array of complex values into which to write the coefficients.

• coeffs_len – A pointer to the integer into which to write the length of the output array.

void qf_transfer_op_get_left_indices(const QfTransferVertexOperator *op, uint32_t **left_indices_out, uint64_t *left_indices_len)

Provides read-only access to the operator’s left-hand mode indices.

See also

The explanation of the internal data structure, here.

Example

uint64_t num_terms = 1;
uint64_t num_indices = 2;
uint32_t left_indices[2] = {0, 0};
uint32_t right_indices[2] = {0, 1};
QkComplex64 coeffs[1] = {{1.0, 0.0}};
uint32_t boundaries[2] = {0, 2};
QfTransferVertexOperator *op = qf_transfer_op_new(num_terms, num_indices, coeffs,
                                                  left_indices, right_indices, boundaries);

uint32_t *left_out;
uint64_t left_len;

qf_transfer_op_get_left_indices(op, &left_out, &left_len);

assert(left_len == 2);

Note

This function borrows the operator’s internal buffer rather than copying it. The returned pointer stays valid only until the operator is modified or freed; do not free it yourself.

Parameters:

• op – A pointer to the transfer-vertex operator whose left indices to access.

• left_indices_out – A pointer to the array of integers into which to write the indices.

• left_indices_len – A pointer to the integer into which to write the length of the output array.

void qf_transfer_op_get_right_indices(const QfTransferVertexOperator *op, uint32_t **right_indices_out, uint64_t *right_indices_len)

Provides read-only access to the operator’s right-hand mode indices.

See also

The explanation of the internal data structure, here.

Example

uint64_t num_terms = 1;
uint64_t num_indices = 2;
uint32_t left_indices[2] = {0, 0};
uint32_t right_indices[2] = {0, 1};
QkComplex64 coeffs[1] = {{1.0, 0.0}};
uint32_t boundaries[2] = {0, 2};
QfTransferVertexOperator *op = qf_transfer_op_new(num_terms, num_indices, coeffs,
                                                  left_indices, right_indices, boundaries);

uint32_t *right_out;
uint64_t right_len;

qf_transfer_op_get_right_indices(op, &right_out, &right_len);

assert(right_len == 2);

Note

This function borrows the operator’s internal buffer rather than copying it. The returned pointer stays valid only until the operator is modified or freed; do not free it yourself.

Parameters:

• op – A pointer to the transfer-vertex operator whose right indices to access.

• right_indices_out – A pointer to the array of integers into which to write the indices.

• right_indices_len – A pointer to the integer into which to write the length of the output array.

void qf_transfer_op_get_boundaries(const QfTransferVertexOperator *op, size_t **boundaries_out, uint64_t *boundaries_len)

Provides read-only access to the indices indicating the boundaries between operator terms.

See also

The explanation of the internal data structure, here.

Example

uint64_t num_terms = 1;
uint64_t num_indices = 2;
uint32_t left_indices[2] = {0, 0};
uint32_t right_indices[2] = {0, 1};
QkComplex64 coeffs[1] = {{1.0, 0.0}};
uint32_t boundaries[2] = {0, 2};
QfTransferVertexOperator *op = qf_transfer_op_new(num_terms, num_indices, coeffs,
                                                  left_indices, right_indices, boundaries);

size_t *boundaries_out;
uint64_t boundaries_len;

qf_transfer_op_get_boundaries(op, &boundaries_out, &boundaries_len);

assert(boundaries_len == 2);

Note

This function borrows the operator’s internal buffer rather than copying it. The returned pointer stays valid only until the operator is modified or freed; do not free it yourself.

Parameters:

• op – A pointer to the transfer-vertex operator whose boundaries to access.

• boundaries_out – A pointer to the array of integers into which to write the boundaries.

• boundaries_len – A pointer to the integer into which to write the length of the output array.

uint32_t qf_transfer_op_num_support(const QfTransferVertexOperator *op)

Gets the size of the support of an operator.

Use this to size the output buffer of qf_transfer_op_get_support(), which does not report a length of its own.

Example

uint32_t left_indices[3] = {2, 0, 2};
uint32_t right_indices[3] = {2, 1, 2};
QkComplex64 coeffs[2] = {{1.0, 0.0}, {1.0, 0.0}};
uint32_t boundaries[3] = {0, 2, 3};
QfTransferVertexOperator *op =
    qf_transfer_op_new(2, 3, coeffs, left_indices, right_indices, boundaries);

uint32_t num_support = qf_transfer_op_num_support(op);

assert(num_support == 3);

Parameters:

• op – A pointer to the transfer-vertex operator whose support size to get.

Returns:

The number of distinct mode indices acted upon by the operator.

void qf_transfer_op_get_support(const QfTransferVertexOperator *op, uint32_t *support_out)

Gets the support of an operator, i.e. the mode indices acted upon by the operator.

The indices are written in ascending order, and every index appears exactly once no matter how many terms act upon it.

Example

uint32_t left_indices[3] = {2, 0, 2};
uint32_t right_indices[3] = {2, 1, 2};
QkComplex64 coeffs[2] = {{1.0, 0.0}, {1.0, 0.0}};
uint32_t boundaries[3] = {0, 2, 3};
QfTransferVertexOperator *op =
    qf_transfer_op_new(2, 3, coeffs, left_indices, right_indices, boundaries);

uint32_t num_support = qf_transfer_op_num_support(op);
uint32_t *support_out = malloc(num_support * sizeof(uint32_t));

qf_transfer_op_get_support(op, support_out);

assert(support_out[0] == 0);
assert(support_out[1] == 1);
assert(support_out[2] == 2);

free(support_out);

Note

Unlike the other getters, this does not hand out a pointer into the operator: the support is computed on demand rather than stored, so the caller provides the output buffer. Query its required length with qf_transfer_op_num_support() first.

Note

Both the left and the right indices of every generalized transfer operator contribute to the support.

Parameters:

• op – A pointer to the transfer-vertex operator whose support to get.

• support_out – A pointer to the integer array into which to write the support. It must be sized to at least :c:func:qf_transfer_op_num_support elements.

QfTransferVertexOperator *qf_transfer_op_zero(void)

Constructs the additive identity operator.

Adding the operator that is constructed by this method to another one has no effect.

Example

QfTransferVertexOperator *zero = qf_transfer_op_zero();

QfTransferVertexOperator *op_plus_zero = qf_transfer_op_add(op, zero);

assert(qf_transfer_op_equal(op, op_plus_zero));

Returns:

A pointer to the created operator.

QfTransferVertexOperator *qf_transfer_op_one(void)

Constructs the multiplicative identity operator.

Composing the operator that is constructed by this method with another one has no effect.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();

QfTransferVertexOperator *op_times_one = qf_transfer_op_compose(op, one);

assert(qf_transfer_op_equal(op, op_times_one));

Returns:

A pointer to the created operator.

bool qf_transfer_op_has_groups(const QfTransferVertexOperator *op)

Checks whether an operator has its groups attribute set.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;

bool has_groups = qf_transfer_op_has_groups(op);

Parameters:

• op – A pointer to the transfer-vertex operator to check.

Returns:

Whether the operator tracks group indices.

uint32_t qf_transfer_op_num_groups(const QfTransferVertexOperator *op)

Gets the number of groups from an operator.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;

uint32_t num_groups = qf_transfer_op_num_groups(op);

Note

The number of groups is evaluated lazily as the largest occurring group index plus 1.

Parameters:

• op – A pointer to the transfer-vertex operator whose number of groups to get.

Returns:

The number of group indices from the operator’s groups attribute.

void qf_transfer_op_get_groups(const QfTransferVertexOperator *op, uint32_t **groups_out, uint64_t *groups_len)

Gets the group indices for all operator terms.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;
uint32_t *groups_out;
uint64_t groups_len;

qf_transfer_op_get_groups(op, &groups_out, &groups_len);

Note

This function borrows the operator’s internal buffer rather than copying it. The returned pointer stays valid only until the operator is modified or freed; do not free it yourself.

Parameters:

• op – A pointer to the transfer-vertex operator whose group indices to get.

• groups_out – A pointer to the integer array into which to write the group indices.

• groups_len – A pointer to the integer into which to write the length of the output array.

QfExitCode qf_transfer_op_set_groups(QfTransferVertexOperator *op, const uint32_t *groups_in, uint64_t groups_len)

Sets the groups attribute of the provided operator.

The length is validated here because nothing downstream re-checks it: too few indices would silently drop the trailing terms wherever terms are iterated together with their groups, and too many would make qf_transfer_op_num_groups() report groups that no term carries.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;

uint32_t num_terms = 4;
uint32_t groups_in[4] = {0, 1, 0, 1};
qf_transfer_op_set_groups(op, groups_in, num_terms);

Parameters:

• op – A pointer to the transfer-vertex operator whose groups attribute to write.

• groups_in – A pointer to the groups integer array to write into the operator.

• groups_len – The number of terms in the groups_in array.

Returns:

An exit code.

• QfExitCode_Success upon success

• QfExitCode_ValueError if groups_len differs from the number of terms in the operator

void qf_transfer_op_del_groups(QfTransferVertexOperator *op)

Deletes the groups attribute from the provided operator.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;

qf_transfer_op_del_groups(op);

Parameters:

• op – A pointer to the transfer-vertex operator whose groups attribute to delete.

void qf_transfer_op_split_out_groups(const QfTransferVertexOperator *op, const uint32_t *group_indices, uint64_t num_indices, QfTransferVertexOperator **group_ops_out)

Splits this operator into a list of new operators based on its groups attribute.

A duplicate index in group_indices is written once per occurrence in group_ops_out. Requesting only a small number of groups out of a much larger total is significantly cheaper than requesting all of them, since terms belonging to a group that is not requested are skipped rather than appended anywhere.

See also

The explanation on Group operator terms: use the operator structure.

Example

QfTransferVertexOperator *op = ...;

uint32_t groups_in[4] = {0, 1, 0, 1};
qf_transfer_op_set_groups(op, groups_in, 4);

// build every group, in index order
QfTransferVertexOperator *group_ops[2];
qf_transfer_op_split_out_groups(op, NULL, 0, group_ops);

// build only group 1
uint32_t group_indices[1] = {1};
QfTransferVertexOperator *group_op[1];
qf_transfer_op_split_out_groups(op, group_indices, 1, group_op);

Parameters:

• op – A pointer to the transfer-vertex operator whose groups to split out.

• group_indices – A pointer to the array of group indices for which to build operators, in the desired output order. May be NULL, in which case every group is built, in index order (equivalent to passing every index from 0 to :c:func:qf_transfer_op_num_groups - 1).

• num_indices – The number of indices in the group_indices array. Ignored if group_indices is NULL.

• group_ops_out – A pointer to the array of :c:struct:QfTransferVertexOperator into which to write the operators for each requested group. Must be sized to num_indices when group_indices is non-NULL, or to :c:func:qf_transfer_op_num_groups when it is NULL.

void qf_transfer_op_add_term(QfTransferVertexOperator *op, uint64_t num_indices, const uint32_t *left_indices, const uint32_t *right_indices, const QkComplex64 *coeff)

Adds a term to an existing operator.

Any of the pointer arguments can be NULL if and only if their corresponding length is zero.

Caution

This function resets the operator’s groups attribute to NULL.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();

QfTransferVertexOperator *op = qf_transfer_op_zero();
QkComplex64 coeff = {1.0, 0.0};

qf_transfer_op_add_term(op, 0, NULL, NULL, &coeff);

assert(qf_transfer_op_equal(op, one));

Parameters:

• op – A pointer to the transfer-vertex operator to be modified.

• num_indices – The length of both index arrays.

• left_indices – A pointer to an array of left-hand mode indices. The length of this array should be num_indices.

• right_indices – A pointer to an array of right-hand mode indices. The length of this array should be num_indices.

• coeff – A pointer to the complex coefficient.

QfTransferVertexOperator *qf_transfer_op_add(const QfTransferVertexOperator *left, const QfTransferVertexOperator *right)

Adds two operators together.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();
QfTransferVertexOperator *zero = qf_transfer_op_zero();

QfTransferVertexOperator *result = qf_transfer_op_add(one, zero);

assert(qf_transfer_op_equal(result, one));

Parameters:

• left – A pointer to the left operator.

• right – A pointer to the right operator.

Returns:

A pointer to the resulting operator.

QfTransferVertexOperator *qf_transfer_op_scaled_add(const QfTransferVertexOperator *left, const QfTransferVertexOperator *right, const QkComplex64 *factor)

Adds two operators together, scaling the coefficients of the right one.

This fuses the scaling into the addition, which avoids building a fully scaled copy of right just to append it. Passing a factor of -1 therefore subtracts, which is why no separate subtraction function is provided.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();
QfTransferVertexOperator *other = qf_transfer_op_one();

// Subtracting via a factor of -1 leaves the two terms cancelling each other out.
QkComplex64 factor = {-1.0, 0.0};
QfTransferVertexOperator *result = qf_transfer_op_scaled_add(one, other, &factor);

QfTransferVertexOperator *simplified = qf_transfer_op_simplify(result, 1e-10);
assert(qf_transfer_op_len(simplified) == 0);

Note

Like qf_transfer_op_add(), this appends the terms of right without combining them with those of left, so the result tracks no group indices. Call qf_transfer_op_simplify() to collect equal terms afterwards.

Parameters:

• left – A pointer to the left operator.

• right – A pointer to the right operator.

• factor – A pointer to the factor to scale the right operator’s coefficients with.

Returns:

A pointer to the resulting operator, left + factor * right.

void qf_transfer_op_add_inplace(QfTransferVertexOperator *left, const QfTransferVertexOperator *right)

Adds an operator into another one, in place.

The in-place counterpart of qf_transfer_op_add(), which saves copying left into a freshly allocated result.

Caution

This function resets the operator’s groups attribute to NULL.

Example

QfTransferVertexOperator *left = qf_transfer_op_one();
QfTransferVertexOperator *right = qf_transfer_op_one();

qf_transfer_op_add_inplace(left, right);

assert(qf_transfer_op_len(left) == 2);

Parameters:

• left – A pointer to the operator to add into.

• right – A pointer to the operator to add.

void qf_transfer_op_scaled_add_inplace(QfTransferVertexOperator *left, const QfTransferVertexOperator *right, const QkComplex64 *factor)

Adds an operator into another one in place, scaling the coefficients of the right one.

The in-place counterpart of qf_transfer_op_scaled_add(), computing left + factor * right without allocating a result. As there, a factor of -1 subtracts.

Caution

This function resets the operator’s groups attribute to NULL.

Example

QfTransferVertexOperator *left = qf_transfer_op_one();
QfTransferVertexOperator *right = qf_transfer_op_one();

// Subtract, leaving two terms that cancel each other out.
QkComplex64 factor = {-1.0, 0.0};
qf_transfer_op_scaled_add_inplace(left, right, &factor);

QfTransferVertexOperator *simplified = qf_transfer_op_simplify(left, 1e-10);
assert(qf_transfer_op_len(simplified) == 0);

Parameters:

• left – A pointer to the operator to add into.

• right – A pointer to the operator to add.

• factor – A pointer to the factor to scale the right operator’s coefficients with.

void qf_transfer_op_mul_inplace(QfTransferVertexOperator *op, const QkComplex64 *scalar)

Multiplies an operator by a scalar, in place.

The in-place counterpart of qf_transfer_op_mul(), which saves copying the operator into a freshly allocated result.

Example

QfTransferVertexOperator *op = qf_transfer_op_one();
QkComplex64 scalar = {2.0, 0.0};

qf_transfer_op_mul_inplace(op, &scalar);

Note

Unlike the two in-place additions above, this preserves the groups attribute: scaling the coefficients leaves the number of terms, and hence the one-index-per-term invariant, untouched.

Parameters:

• op – A pointer to the operator to scale.

• scalar – A pointer to the scalar.

QfTransferVertexOperator *qf_transfer_op_mul(const QfTransferVertexOperator *op, const QkComplex64 *scalar)

Multiplies an operator by a scalar.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();
QkComplex64 coeff = {2.0, 0.0};
QfTransferVertexOperator *result = qf_transfer_op_mul(one, &coeff);

QfTransferVertexOperator *expected = qf_transfer_op_zero();
qf_transfer_op_add_term(expected, 0, NULL, NULL, &coeff);

assert(qf_transfer_op_equal(result, expected));

Parameters:

• op – A pointer to the operator.

• scalar – A pointer to the scalar.

Returns:

A pointer to the resulting operator.

QfTransferVertexOperator *qf_transfer_op_compose(const QfTransferVertexOperator *left, const QfTransferVertexOperator *right)

Composes two operators with each other.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();
QfTransferVertexOperator *zero = qf_transfer_op_zero();

QfTransferVertexOperator *result = qf_transfer_op_compose(one, zero);

assert(qf_transfer_op_equal(result, zero));

Note

The composition of two operators tracks no groups, even when both operands do.

Parameters:

• left – A pointer to the left operator.

• right – A pointer to the right operator.

Returns:

A pointer to left.compose(right), which equals the operator result = right @ left in terms of the matrix multiplication @. In other words, right is applied first. To obtain left @ right, swap the arguments.

QfTransferVertexOperator *qf_transfer_op_adjoint(const QfTransferVertexOperator *op)

Returns the Hermitian conjugate (or adjoint) of an operator.

This affects the terms and coefficients as follows:

• the generators in each term reverse their order

• the coefficients are complex conjugated

Example

QfTransferVertexOperator *op = qf_transfer_op_zero();
QkComplex64 coeff = {0.0, 1.0};
qf_transfer_op_add_term(op, 0, NULL, NULL, &coeff);

QfTransferVertexOperator *adjoint = qf_transfer_op_adjoint(op);

QfTransferVertexOperator *expected = qf_transfer_op_zero();
QkComplex64 coeff_adj = {0.0, -1.0};
qf_transfer_op_add_term(expected, 0, NULL, NULL, &coeff_adj);

assert(qf_transfer_op_equal(adjoint, expected));

Note

Reversing the operator string is essential: the transfer and vertex generators anticommute when they share an index, so BA != AB in general.

Parameters:

• op – A pointer to the operator.

Returns:

A pointer to the created operator.

void qf_transfer_op_ichop(QfTransferVertexOperator *op, double atol)

Removes terms whose coefficient magnitude lies below the provided threshold.

Caution

This functions truncates coefficients greedily! If the acted upon operator might contain separate coefficients for duplicate terms consider calling qf_transfer_op_simplify() instead!

Example

QfTransferVertexOperator *op = qf_transfer_op_zero();
QkComplex64 coeff = {1e-8, 0.0};
qf_transfer_op_add_term(op, 0, NULL, NULL, &coeff);

qf_transfer_op_ichop(op, 1e-6);

QfTransferVertexOperator *expected = qf_transfer_op_zero();

assert(qf_transfer_op_equal(op, expected));

Parameters:

• op – A pointer to the operator.

• atol – The absolute tolerance for coefficient truncation.

QfTransferVertexOperator *qf_transfer_op_simplify(const QfTransferVertexOperator *op, double atol)

Returns an equivalent but simplified operator.

The simplification process first sums all coefficients that belong to equal terms and then only retains those whose total coefficient exceeds the specified tolerance (just like qf_transfer_op_ichop()).

When an operator has been arithmetically manipulated or constructed in a way that does not guarantee unique terms, this method should be called before applying any method that filters numerically small coefficients to avoid loss of information.

Example

QfTransferVertexOperator *op = qf_transfer_op_zero();
QkComplex64 coeff = {1e-5, 0.0};
for (int i = 0; i < 100; i++) {
  qf_transfer_op_add_term(op, 0, NULL, NULL, &coeff);
}

QfTransferVertexOperator *canon = qf_transfer_op_simplify(op, 1e-4);

assert(qf_transfer_op_len(canon) == 1);

Note

This groups terms by their exact stored index arrays. Call qf_transfer_op_normal_ordered() first to bring mathematically equal terms into a common form.

Parameters:

• op – A pointer to the transfer-vertex operator to be simplified.

• atol – The absolute tolerance for coefficient truncation.

Returns:

An equivalent but simplified operator.

QfTransferVertexOperator *qf_transfer_op_normal_ordered(const QfTransferVertexOperator *op, bool reduce)

Returns an equivalent operator with normal ordered terms.

The reduce flag contracts adjacent generators. There are exactly two rules: \(V_j V_j = 1\) and \(T_{jk} T_{jk} = 1/4\) (two identical adjacent factors contract to a scalar), and \(T_{jk} T_{kj} = -\frac{1}{4} V_j V_k\) (two anti-parallel transfer operators become a pair of vertex operators).

Example

// `T(0,1) T(1,0)` reduces to `-1/4 V(0) V(1)`
QfTransferVertexOperator *op = qf_transfer_op_zero();
uint32_t left[2] = {0, 1};
uint32_t right[2] = {1, 0};
QkComplex64 coeff = {4.0, 0.0};
qf_transfer_op_add_term(op, 2, left, right, &coeff);

QfTransferVertexOperator *ordered = qf_transfer_op_normal_ordered(op, true);

QfTransferVertexOperator *expected = qf_transfer_op_zero();
uint32_t left_exp[2] = {0, 1};
uint32_t right_exp[2] = {0, 1};
QkComplex64 coeff_exp = {-1.0, 0.0};
qf_transfer_op_add_term(expected, 2, left_exp, right_exp, &coeff_exp);

assert(qf_transfer_op_equal(ordered, expected));

Note

Unlike qf_edge_op_normal_ordered(), this function takes no ascending parameter. There is no orientation convention to fix: \(T_{jk}\) and \(T_{kj}\) are different operators rather than two representations of one.

Note

There is no fusion rule. Two transfer operators sharing a single mode never collapse into one, so a product like \(T_{jk} T_{jl}\) cannot be shortened.

Parameters:

• op – A pointer to the operator.

• reduce – Whether to contract adjacent generators via the algebra’s identities.

Returns:

A pointer to the created operator.

bool qf_transfer_op_is_hermitian(const QfTransferVertexOperator *op, double atol)

Checks whether an operator is Hermitian.

Example

// `V(0) T(0,1)` is *not* Hermitian: the two generators share the index 0 and therefore
// anticommute.
QfTransferVertexOperator *op = qf_transfer_op_zero();
uint32_t left[2] = {0, 0};
uint32_t right[2] = {0, 1};
QkComplex64 coeff = {1.0, 0.0};
qf_transfer_op_add_term(op, 2, left, right, &coeff);

assert(!qf_transfer_op_is_hermitian(op, 1e-10));

Note

This check is implemented using qf_transfer_op_equiv() on the qf_transfer_op_normal_ordered() difference of op and its qf_transfer_op_adjoint() and qf_transfer_op_zero().

Parameters:

• op – A pointer to the transfer-vertex operator to be checked.

• atol – The absolute tolerance upto which coefficients are considered equal.

Returns:

Whether the provided operator is Hermitian.

bool qf_transfer_op_equal(const QfTransferVertexOperator *left, const QfTransferVertexOperator *right)

Compare two operators for equality.

Equality in this context means an exact match of the internal data arrays.

Example

QfTransferVertexOperator *one = qf_transfer_op_one();
QfTransferVertexOperator *zero = qf_transfer_op_zero();

assert(qf_transfer_op_equal(one, one));
assert(!qf_transfer_op_equal(one, zero));

Parameters:

• left – A pointer to the left operator.

• right – A pointer to the right operator.

Returns:

Whether the two operators are equal.

bool qf_transfer_op_equiv(const QfTransferVertexOperator *left, const QfTransferVertexOperator *right, double atol)

Compare two operators for equivalence.

Equivalence in this context means approximate equality up to the specified absolute tolerance. To be more precise, this method returns True, when all the absolute values of the coefficients in the difference other - self are below the specified threshold atol.

Example

QfTransferVertexOperator *zero = qf_transfer_op_zero();

QfTransferVertexOperator *op = qf_transfer_op_zero();
QkComplex64 coeff = {1e-7, 0.0};
qf_transfer_op_add_term(op, 0, NULL, NULL, &coeff);

assert(qf_transfer_op_equiv(op, zero, 1e-6));
assert(!qf_transfer_op_equiv(op, zero, 1e-8));

Parameters:

• left – A pointer to the left operator.

• right – A pointer to the right operator.

• atol – The absolute tolerance for coefficient equivalence.

Returns:

Whether the two operators are equivalent.

size_t qf_transfer_op_len(const QfTransferVertexOperator *op)

Returns the length (or number of terms) of the provided operator.

Example

QfTransferVertexOperator *op = qf_transfer_op_zero();
uint32_t left[2] = {0, 1};
uint32_t right[2] = {1, 2};
QkComplex64 coeff = {1.0, 0.0};
qf_transfer_op_add_term(op, 2, left, right, &coeff);

assert(qf_transfer_op_len(op) == 1);

Parameters:

• op – A pointer to the transfer-vertex operator.

Returns:

The length (or number of terms) of the operator.

QfExitCode qf_transfer_op_relabel_modes(QfTransferVertexOperator *op, uint64_t num_modes, const uint32_t *permutation)

Relabels the modes of the provided operator.

Example

QfTransferVertexOperator *op = qf_transfer_op_zero();
uint32_t left[2] = {0, 2};
uint32_t right[2] = {1, 3};
QkComplex64 coeff = {1.0, 0.0};
qf_transfer_op_add_term(op, 2, left, right, &coeff);

uint32_t permutation[4] = {3, 2, 1, 0};

QfExitCode exit = qf_transfer_op_relabel_modes(op, 4, permutation);

assert(exit == QfExitCode_Success);

Note

Both index arrays are relabelled. Unlike most operations, this preserves the operator’s groups attribute, since relabelling permutes mode indices without reordering, splitting or merging terms.

Parameters:

• op – A pointer to the transfer-vertex operator.

• num_modes – The number of mode indices in the provided permutation list.

• permutation – The index permutation list.

Returns:

An exit code.

• QfExitCode_Success upon success

• QfExitCode_DuplicateIndexError if duplicate indices were found in the permutation

• QfExitCode_IndexError for any other index errors, such as invalid indices.