qml.resource.DoubleFactorization¶

class
DoubleFactorization
(one_electron, two_electron, error=0.0016, rank_r=None, rank_m=None, rank_max=None, tol_factor=1e05, tol_eigval=1e05, br=7, alpha=10, beta=20, chemist_notation=False)[source]¶ Bases:
pennylane.operation.Operation
Estimate the number of nonClifford gates and logical qubits for a quantum phase estimation algorithm in second quantization with a doublefactorized Hamiltonian.
Atomic units are used throughout the class.
 Parameters
one_electron (array[array[float]]) – oneelectron integrals
two_electron (tensor_like) – twoelectron integrals
error (float) – target error in the algorithm
rank_r (int) – rank of the first factorization of the twoelectron integral tensor
rank_m (int) – average rank of the second factorization of the twoelectron integral tensor
tol_factor (float) – threshold error value for discarding the negligible factors
tol_eigval (float) – threshold error value for discarding the negligible factor eigenvalues
br (int) – number of bits for ancilla qubit rotation
alpha (int) – number of bits for the keep register
beta (int) – number of bits for the rotation angles
chemist_notation (bool) – if True, the twoelectron integrals need to be in chemist notation
Example
>>> symbols = ['O', 'H', 'H'] >>> geometry = np.array([[0.00000000, 0.00000000, 0.28377432], >>> [0.00000000, 1.45278171, 1.00662237], >>> [0.00000000, 1.45278171, 1.00662237]], requires_grad = False) >>> mol = qml.qchem.Molecule(symbols, geometry, basis_name='sto3g') >>> core, one, two = qml.qchem.electron_integrals(mol)() >>> algo = DoubleFactorization(one, two) >>> print(algo.lamb, # the 1Norm of the Hamiltonian >>> algo.gates, # estimated number of nonClifford gates >>> algo.qubits # estimated number of logical qubits >>> ) 53.62085493277858 103969925 290
Theory
To estimate the gate and qubit costs for implementing this method, the Hamiltonian needs to be factorized using the
factorize()
function following [PRX Quantum 2, 030305 (2021)]. The objective of the factorization is to find a set of symmetric matrices, \(L^{(r)}\), such that the twoelectron integral tensor in chemist notation, \(V\), can be computed as\[V_{ijkl} = \sum_r^R L_{ij}^{(r)} L_{kl}^{(r) T},\]with the rank \(R \leq n^2\), where \(n\) is the number of molecular orbitals. The matrices \(L^{(r)}\) are diagonalized and for each matrix the eigenvalues that are smaller than a given threshold (and their corresponding eigenvectors) are discarded. The average number of the retained eigenvalues, \(M\), determines the rank of the second factorization step. The 1norm of the Hamiltonian can then be computed using the
norm()
function from the electron integrals and the eigenvalues of the matrices \(L^{(r)}\).The total number of gates and qubits for implementing the quantum phase estimation algorithm for the given Hamiltonian can then be computed using the functions
gate_cost()
andqubit_cost()
with a target error that has the default value of 0.0016 Ha (chemical accuracy). The costs are computed using Eqs. (C39C40) of [PRX Quantum 2, 030305 (2021)].Attributes
Arithmetic depth of the operator.
The basis of an operation, or for controlled gates, of the target operation.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
Gradient recipe for the parametershift method.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an operator is hermitian.
String for the name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends on.
Number of wires the operator acts on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.Wires that the operator acts on.

arithmetic_depth
¶ Arithmetic depth of the operator.

basis
¶ The basis of an operation, or for controlled gates, of the target operation. If not
None
, should take a value of"X"
,"Y"
, or"Z"
.For example,
X
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "Z"
. Type
str or None

batch_size
¶ Batch size of the operator if it is used with broadcasted parameters.
The
batch_size
is determined based onndim_params
and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size isNone
. Returns
Size of the parameter broadcasting dimension if present, else
None
. Return type
int or None

control_wires
¶ Control wires of the operator.
For operations that are not controlled, this is an empty
Wires
object of length0
. Returns
The control wires of the operation.
 Return type

grad_method
= None¶

grad_recipe
= None¶ Gradient recipe for the parametershift method.
This is a tuple with one nested list per operation parameter. For parameter \(\phi_k\), the nested list contains elements of the form \([c_i, a_i, s_i]\) where \(i\) is the index of the term, resulting in a gradient recipe of
\[\frac{\partial}{\partial\phi_k}f = \sum_{i} c_i f(a_i \phi_k + s_i).\]If
None
, the default gradient recipe containing the two terms \([c_0, a_0, s_0]=[1/2, 1, \pi/2]\) and \([c_1, a_1, s_1]=[1/2, 1, \pi/2]\) is assumed for every parameter. Type
tuple(Union(list[list[float]], None)) or None

has_adjoint
= False¶

has_decomposition
= False¶

has_diagonalizing_gates
= False¶

has_generator
= False¶

has_matrix
= False¶

hash
¶ Integer hash that uniquely represents the operator.
 Type
int

hyperparameters
¶ Dictionary of nontrainable variables that this operation depends on.
 Type
dict

id
¶ Custom string to label a specific operator instance.

is_hermitian
¶ This property determines if an operator is hermitian.

name
¶ String for the name of the operator.

ndim_params
¶ Number of dimensions per trainable parameter of the operator.
By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.
 Returns
Number of dimensions for each trainable parameter.
 Return type
tuple

num_params
¶ Number of trainable parameters that the operator depends on.
By default, this property returns as many parameters as were used for the operator creation. If the number of parameters for an operator subclass is fixed, this property can be overwritten to return the fixed value.
 Returns
number of parameters
 Return type
int

num_wires
= 1¶ Number of wires the operator acts on.

parameter_frequencies
¶ Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
These frequencies encode the behaviour of the operator \(U(\mathbf{p})\) on the value of the expectation value as the parameters are modified. For more details, please see the
pennylane.fourier
module. Returns
Tuple of frequencies for each parameter. Note that only nonnegative frequency values are returned.
 Return type
list[tuple[int or float]]
Example
>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1]) >>> op.parameter_frequencies [(0.5, 1), (0.5, 1), (0.5, 1)]
For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:
>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1]) >>> op.parameter_frequencies [(1,)] >>> gen = qml.generator(op, format="observable") >>> gen_eigvals = qml.eigvals(gen) >>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals)) (1.0,)
For more details on this relationship, see
eigvals_to_frequencies()
.

parameters
¶ Trainable parameters that the operator depends on.

pauli_rep
¶ A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(*params[, wires])Representation of the operator as a product of other operators (static method).
compute_diagonalizing_gates
(*params, wires, …)Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals
(*params, **hyperparams)Eigenvalues of the operator in the computational basis (static method).
compute_matrix
(*params, **hyperparams)Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix
(*params, **hyperparams)Representation of the operator as a sparse matrix in the computational basis (static method).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
estimation_cost
(lamb, error)Return the number of calls to the unitary needed to achieve the desired error in quantum phase estimation.
expand
()Returns a tape that contains the decomposition of the operator.
gate_cost
(n, lamb, error, rank_r, rank_m, …)Return the total number of Toffoli gates needed to implement the double factorization algorithm.
Generator of an operator that is in singleparameterform.
label
([decimals, base_label, cache])A customizable string representation of the operator.
map_wires
(wire_map)Returns a copy of the current operator with its wires changed according to the given wire map.
matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
norm
(one, two, eigvals)Return the 1norm of a molecular Hamiltonian from the one and twoelectron integrals and eigenvalues of the factorized twoelectron integral tensor.
pow
(z)A list of new operators equal to this one raised to the given power.
qubit_cost
(n, lamb, error, rank_r, rank_m, …)Return the number of logical qubits needed to implement the double factorization method.
queue
([context])Append the operator to the Operator queue.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order])Representation of the operator as a sparse matrix in the computational basis.
terms
()Representation of the operator as a linear combination of other operators.
unitary_cost
(n, rank_r, rank_m, rank_max[, …])Return the number of Toffoli gates needed to implement the qubitization unitary operator for the double factorization algorithm.

adjoint
()¶ Create an operation that is the adjoint of this one.
Adjointed operations are the conjugated and transposed version of the original operation. Adjointed ops are equivalent to the inverted operation for unitary gates.
 Returns
The adjointed operation.

static
compute_decomposition
(*params, wires=None, **hyperparameters)¶ Representation of the operator as a product of other operators (static method).
\[O = O_1 O_2 \dots O_n.\]Note
Operations making up the decomposition should be queued within the
compute_decomposition
method.See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
decomposition of the operator
 Return type
list[Operator]

static
compute_diagonalizing_gates
(*params, wires, **hyperparams)¶ Sequence of gates that diagonalize the operator in the computational basis (static method).
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
list of diagonalizing gates
 Return type
list[Operator]

static
compute_eigvals
(*params, **hyperparams)¶ Eigenvalues of the operator in the computational basis (static method).
If
diagonalizing_gates
are specified and implement a unitary \(U^{\dagger}\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
eigenvalues
 Return type
tensor_like

static
compute_matrix
(*params, **hyperparams)¶ Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
matrix representation
 Return type
tensor_like

static
compute_sparse_matrix
(*params, **hyperparams)¶ Representation of the operator as a sparse matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

decomposition
()¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n\]A
DecompositionUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
decomposition of the operator
 Return type
list[Operator]

diagonalizing_gates
()¶ Sequence of gates that diagonalize the operator in the computational basis.
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
A
DiagGatesUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
a list of operators
 Return type
list[Operator] or None

eigvals
()¶ Eigenvalues of the operator in the computational basis.
If
diagonalizing_gates
are specified and implement a unitary \(U^{\dagger}\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
Note
When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.
A
EigvalsUndefinedError
is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.See also
 Returns
eigenvalues
 Return type
tensor_like

static
estimation_cost
(lamb, error)[source]¶ Return the number of calls to the unitary needed to achieve the desired error in quantum phase estimation.
The expression for computing the cost is taken from Eq. (45) of [PRX Quantum 2, 030305 (2021)].
 Parameters
lamb (float) – 1norm of a secondquantized Hamiltonian
error (float) – target error in the algorithm
 Returns
number of calls to unitary
 Return type
int
Example
>>> lamb = 72.49779513025341 >>> error = 0.001 >>> estimation_cost(lamb, error) 113880

expand
()¶ Returns a tape that contains the decomposition of the operator.
 Returns
quantum tape
 Return type

static
gate_cost
(n, lamb, error, rank_r, rank_m, rank_max, br=7, alpha=10, beta=20)[source]¶ Return the total number of Toffoli gates needed to implement the double factorization algorithm.
The expression for computing the cost is taken from Eqs. (45) and (C39) of [PRX Quantum 2, 030305 (2021)].
 Parameters
n (int) – number of molecular spinorbitals
lamb (float) – 1norm of a secondquantized Hamiltonian
error (float) – target error in the algorithm
rank_r (int) – rank of the first factorization of the twoelectron integral tensor
rank_m (float) – average rank of the second factorization of the twoelectron tensor
rank_max (int) – maximum rank of the second factorization of the twoelectron tensor
br (int) – number of bits for ancilla qubit rotation
alpha (int) – number of bits for the keep register
beta (int) – number of bits for the rotation angles
 Returns
the number of Toffoli gates for the double factorization method
 Return type
int
Example
>>> n = 14 >>> lamb = 52.98761457453095 >>> error = 0.001 >>> rank_r = 26 >>> rank_m = 5.5 >>> rank_max = 7 >>> br = 7 >>> alpha = 10 >>> beta = 20 >>> gate_cost(n, lamb, error, rank_r, rank_m, rank_max, br, alpha, beta) 167048631

generator
()¶ Generator of an operator that is in singleparameterform.
For example, for operator
\[U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}\]we get the generator
>>> U.generator() 0.5 * Y(0) + Z(0) @ X(1)
The generator may also be provided in the form of a dense or sparse Hamiltonian (using
Hermitian
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.

label
(decimals=None, base_label=None, cache=None)¶ A customizable string representation of the operator.
 Parameters
decimals=None (int) – If
None
, no parameters are included. Else, specifies how to round the parameters.base_label=None (str) – overwrite the nonparameter component of the label
cache=None (dict) – dictionary that carries information between label calls in the same drawing
 Returns
label to use in drawings
 Return type
str
Example:
>>> op = qml.RX(1.23456, wires=0) >>> op.label() "RX" >>> op.label(base_label="my_label") "my_label" >>> op = qml.RX(1.23456, wires=0, id="test_data") >>> op.label() "RX("test_data")" >>> op.label(decimals=2) "RX\n(1.23,"test_data")" >>> op.label(base_label="my_label") "my_label("test_data")" >>> op.label(decimals=2, base_label="my_label") "my_label\n(1.23,"test_data")"
If the operation has a matrixvalued parameter and a cache dictionary is provided, unique matrices will be cached in the
'matrices'
key list. The label will contain the index of the matrix in the'matrices'
list.>>> op2 = qml.QubitUnitary(np.eye(2), wires=0) >>> cache = {'matrices': []} >>> op2.label(cache=cache) 'U(M0)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True)] >>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1)) >>> op3.label(cache=cache) 'U(M1)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True), tensor([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], requires_grad=True)]

map_wires
(wire_map)¶ Returns a copy of the current operator with its wires changed according to the given wire map.
 Parameters
wire_map (dict) – dictionary containing the old wires as keys and the new wires as values
 Returns
new operator
 Return type

matrix
(wire_order=None)¶ Representation of the operator as a matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.If the matrix depends on trainable parameters, the result will be cast in the same autodifferentiation framework as the parameters.
A
MatrixUndefinedError
is raised if the matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
matrix representation
 Return type
tensor_like

static
norm
(one, two, eigvals)[source]¶ Return the 1norm of a molecular Hamiltonian from the one and twoelectron integrals and eigenvalues of the factorized twoelectron integral tensor.
The 1norm of a doublefactorized molecular Hamiltonian is computed using Eqs. (1517) of [Phys. Rev. Research 3, 033055 (2021)]
\[\lambda = T + \frac{1}{4} \sum_r L^{(r)}^2,\]where the Schatten 1norm for a given matrix \(T\) is defined as
\[T = \sum_k \text{eigvals}[T]_k.\]The matrices \(L^{(r)}\) are obtained from factorization of the twoelectron integral tensor \(V\) such that
\[V_{ijkl} = \sum_r L_{ij}^{(r)} L_{kl}^{(r) T}.\]The matrix \(T\) is constructed from the one and twoelectron integrals as
\[T = h_{ij}  \frac{1}{2} \sum_l V_{illj} + \sum_l V_{llij}.\]Note that the twoelectron integral tensor must be arranged in chemist notation.
 Parameters
one (array[array[float]]) – oneelectron integrals
two (array[array[float]]) – twoelectron integrals
eigvals (array[float]) – eigenvalues of the matrices obtained from factorizing the twoelectron integral tensor
 Returns
1norm of the Hamiltonian
 Return type
array[float]
Example
>>> symbols = ['H', 'H', 'O'] >>> geometry = np.array([[0.00000000, 0.00000000, 0.28377432], >>> [0.00000000, 1.45278171, 1.00662237], >>> [0.00000000, 1.45278171, 1.00662237]], requires_grad=False) >>> mol = qml.qchem.Molecule(symbols, geometry, basis_name='sto3g') >>> core, one, two = qml.qchem.electron_integrals(mol)() >>> two = np.swapaxes(two, 1, 3) # convert to the chemists notation >>> _, eigvals, _ = qml.qchem.factorize(two, 1e5) >>> print(norm(one, two, eigvals)) 52.98762043980203

pow
(z)¶ A list of new operators equal to this one raised to the given power.
 Parameters
z (float) – exponent for the operator
 Returns
list[
Operator
]

static
qubit_cost
(n, lamb, error, rank_r, rank_m, rank_max, br=7, alpha=10, beta=20)[source]¶ Return the number of logical qubits needed to implement the double factorization method.
The expression for computing the cost is taken from Eq. (C40) of [PRX Quantum 2, 030305 (2021)].
 Parameters
n (int) – number of molecular spinorbitals
lamb (float) – 1norm of a secondquantized Hamiltonian
error (float) – target error in the algorithm
rank_r (int) – rank of the first factorization of the twoelectron integral tensor
rank_m (float) – average rank of the second factorization of the twoelectron tensor
rank_max (int) – maximum rank of the second factorization of the twoelectron tensor
br (int) – number of bits for ancilla qubit rotation
alpha (int) – number of bits for the keep register
beta (int) – number of bits for the rotation angles
 Returns
number of logical qubits for the double factorization method
 Return type
int
Example
>>> n = 14 >>> lamb = 52.98761457453095 >>> error = 0.001 >>> rank_r = 26 >>> rank_m = 5.5 >>> rank_max = 7 >>> br = 7 >>> alpha = 10 >>> beta = 20 >>> qubit_cost(n, lamb, error, rank_r, rank_m, rank_max, br, alpha, beta) 292

queue
(context=<class 'pennylane.queuing.QueuingManager'>)¶ Append the operator to the Operator queue.

simplify
()¶ Reduce the depth of nested operators to the minimum.
 Returns
simplified operator
 Return type

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase. Returns
A list of values \([\phi, \theta, \omega]\) such that \(RZ(\omega) RY(\theta) RZ(\phi)\) is equivalent to the original operation.
 Return type
tuple[float, float, float]

sparse_matrix
(wire_order=None)¶ Representation of the operator as a sparse matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.A
SparseMatrixUndefinedError
is raised if the sparse matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

terms
()¶ Representation of the operator as a linear combination of other operators.
\[O = \sum_i c_i O_i\]A
TermsUndefinedError
is raised if no representation by terms is defined. Returns
list of coefficients \(c_i\) and list of operations \(O_i\)
 Return type
tuple[list[tensor_like or float], list[Operation]]

static
unitary_cost
(n, rank_r, rank_m, rank_max, br=7, alpha=10, beta=20)[source]¶ Return the number of Toffoli gates needed to implement the qubitization unitary operator for the double factorization algorithm.
The expression for computing the cost is taken from Eq. (C39) of [PRX Quantum 2, 030305 (2021)].
 Parameters
n (int) – number of molecular spinorbitals
rank_r (int) – rank of the first factorization of the twoelectron integral tensor
rank_m (float) – average rank of the second factorization of the twoelectron tensor
rank_max (int) – maximum rank of the second factorization of the twoelectron tensor
br (int) – number of bits for ancilla qubit rotation
alpha (int) – number of bits for the keep register
beta (int) – number of bits for the rotation angles
 Returns
number of Toffoli gates to implement the qubitization unitary
 Return type
int
Example
>>> n = 14 >>> rank_r = 26 >>> rank_m = 5.5 >>> rank_max = 7 >>> br = 7 >>> alpha = 10 >>> beta = 20 >>> unitary_cost(n, rank_r, rank_m, rank_max, br, alpha, beta) 2007