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.

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.

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`.

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.

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.

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.

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_group_weights(const QfTransferVertexOperator *op, double *weights_out)¶

Gets the mean absolute coefficient magnitude of each group.

The i-th entry is the sum of abs(coeff) over the terms in group i, divided by the number of terms in that group. This is the sampling weight of a randomized product formula (e.g. qDRIFT) that draws whole groups rather than individual terms, and is computed in a single pass over the operator rather than by reducing qf_transfer_op_get_coeffs() and qf_transfer_op_get_groups() (one value per ungrouped term each) on the caller’s side.

Example¶

QfTransferVertexOperator *op = ...;

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

double weights[2];
qf_transfer_op_group_weights(op, weights);

Note

A group index that no term carries weighs 0.0, which keeps it out of the sample.

Parameters:

op – A pointer to the transfer-vertex operator whose group weights to compute.
weights_out – A pointer to the array of doubles into which to write the weights. Must be sized to :c:func:qf_transfer_op_num_groups.

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.

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.

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

Sets the groups attribute of the provided operator.

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.

void qf_transfer_op_del_groups(QfTransferVertexOperator *op)¶

Deletes the groups attribute from the provided operator.

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.

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_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));

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_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.