qml.resource.FirstQuantization¶

class
FirstQuantization
(n, eta, omega=None, error=0.0016, charge=0, br=7, vectors=None)[source]¶ Bases:
pennylane.operation.Operation
Estimate the number of nonClifford gates and logical qubits for a quantum phase estimation algorithm in first quantization using a planewave basis.
To estimate the gate and qubit costs for implementing this method, the number of plane waves, the number of electrons and the lattice vectors need to be defined. The costs 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). Atomic units are used throughout the class. Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
charge (int) – total electric charge of the system
br (int) – number of bits for ancilla qubit rotation
vectors (array[float]) – lattice vectors
Example
>>> n = 100000 >>> eta = 156 >>> vectors = np.array([[10.46219511, 0.00000000, 0.00000000], ... [ 0.00000000, 10.46219511, 0.00000000], ... [ 0.00000000, 0.00000000, 10.46219511]]) >>> algo = FirstQuantization(n, eta, vectors=vectors) >>> print(algo.lamb, # the 1Norm of the Hamiltonian >>> algo.gates, # estimated number of nonClifford gates >>> algo.qubits # estimated number of logical qubits >>> ) 649912.4804278888 1.1e+13 4416
Theory
Following PRX Quantum 2, 040332 (2021) , the target algorithm error, \(\epsilon\), is distributed among four different sources of error using Eq. (131)
\[\epsilon^2 \geq \epsilon_{qpe}^2 + (\epsilon_{\mathcal{M}} + \epsilon_R + \epsilon_T)^2,\]where \(\epsilon_{qpe}\) is the quantum phase estimation error and \(\epsilon_{\mathcal{M}}\), \(\epsilon_R\), and \(\epsilon_T\) are defined in Eqs. (132134).
Here, we fix \(\epsilon_{\mathcal{M}} = \epsilon_R = \epsilon_T = \alpha \epsilon\) with a default value of \(\alpha = 0.01\) and obtain
\[\epsilon_{qpe} = \sqrt{\epsilon^2 [1  (3 \alpha)^2]}.\]Note that the user only needs to define the target algorithm error \(\epsilon\). The error distribution takes place inside the functions.
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
(n, eta, omega, error[, br, …])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, eta, omega, error[, br, …])Return the total number of Toffoli gates needed to implement the first quantization 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
(n, eta, omega, error[, br, charge, …])Return the 1norm of a firstquantized Hamiltonian in the planewave basis.
pow
(z)A list of new operators equal to this one raised to the given power.
qubit_cost
(n, eta, omega, error[, br, …])Return the number of logical qubits needed to implement the first quantization algorithm.
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.
success_prob
(n, br)Return the probability of success for state preparation.
terms
()Representation of the operator as a linear combination of other operators.
unitary_cost
(n, eta, omega, error[, br, charge])Return the number of Toffoli gates needed to implement the qubitization unitary operator.

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
(n, eta, omega, error, br=7, charge=0, cubic=True, vectors=None)[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. (125) of [PRX Quantum 2, 040332 (2021)].
 Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
br (int) – number of bits for ancilla qubit rotation
charge (int) – total electric charge of the system
cubic (bool) – True if the unit cell is cubic
vectors (array[float]) – lattice vectors
 Returns
number of calls to unitary
 Return type
int
Example
>>> n = 100000 >>> eta = 156 >>> omega = 1145.166 >>> error = 0.01 >>> estimation_cost(n, eta, omega, error) 102133985

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

static
gate_cost
(n, eta, omega, error, br=7, charge=0, cubic=True, vectors=None)[source]¶ Return the total number of Toffoli gates needed to implement the first quantization algorithm.
The expression for computing the cost is taken from Eq. (125) of [PRX Quantum 2, 040332 (2021)].
 Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
br (int) – number of bits for ancilla qubit rotation
charge (int) – total electric charge of the system
cubic (bool) – True if the unit cell is cubic
vectors (array[float]) – lattice vectors
 Returns
the number of Toffoli gates needed to implement the first quantization algorithm
 Return type
int
Example
>>> n = 100000 >>> eta = 156 >>> omega = 169.69608 >>> error = 0.01 >>> gate_cost(n, eta, omega, error) 3676557345574

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
(n, eta, omega, error, br=7, charge=0, cubic=True, vectors=None)[source]¶ Return the 1norm of a firstquantized Hamiltonian in the planewave basis.
The expressions needed for computing the norm are taken from [PRX Quantum 2, 040332 (2021)]. The norm is computed assuming that amplitude ampliﬁcation is performed.
 Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
br (int) – number of bits for ancilla qubit rotation
charge (int) – total electric charge of the system
cubic (bool) – True if the unit cell is cubic
vectors (array[float]) – lattice vectors
 Returns
1norm of a firstquantized Hamiltonian in the planewave basis
 Return type
float
Example
>>> n = 10000 >>> eta = 156 >>> omega = 1145.166 >>> error = 0.001 >>> norm(n, eta, omega, error) 281053.75612801575
Theory
To compute the norm, for numerical convenience, we use the following modified expressions to obtain parameters that contain a sum over \(\frac{1}{\left \ \nu \right \^k}\) where \(\nu\) denotes an element of the set of reciprocal lattice vectors, \(G_0\), and \(k \in \left \{ 1, 2 \right \}\).
For \(\lambda_{\nu}\) defined in Eq. (25) of PRX Quantum 2, 040332 (2021) as
\[\lambda_{\nu} = \sum_{\nu \in G_0} \frac{1}{\left \ \nu \right \^2},\]we follow Eq. (F6) of PRX 8, 011044 (2018) and use
\[\lambda_{\nu} = 4\pi \left ( \frac{\sqrt{3}}{2} N^{1/3}  1 \right) + 3  \frac{3}{N^{1/3}} + 3 \int_{x=1}^{N^{1/3}} \int_{y=1}^{N^{1/3}} \frac{1}{x^2 + y^2} dydx.\]We also need to compute \(\lambda^{\alpha}_{\nu}\) defined in Eq. (123) of PRX Quantum 2, 040332 (2021)
\[\lambda^{\alpha}_{\nu} = \alpha \sum_{\nu \in G_0} \frac{\left \lceil \mathcal{M}(2^{\mu  2}) / \left \ \nu \right \^2 \right \rceil}{\mathcal{M} 2^{2\mu  4}},\]which we compute here, for \(\alpha = 1\), as
\[\lambda^{1}_{\nu} = \lambda_{\nu} + \epsilon_l,\]where \(\epsilon_l\) is simply defined as the difference of \(\lambda^{1}_{\nu}\) and \(\lambda_{\nu}\). We follow Eq. (113) of PRX Quantum 2, 040332 (2021) to derive an upper bound for its absolute value:
\[\epsilon_l \le \frac{4}{2^{n_m}} (7 \times 2^{n_p + 1} + 9 n_p  11  3 \times 2^{n_p}),\]where \(\mathcal{M} = 2^{n_m}\) and \(n_m\) is defined in Eq. (132) of PRX Quantum 2, 040332 (2021). Finally, for \(p_{\nu}\) defined in Eq. (128) of PRX Quantum 2, 040332 (2021)
\[p_{\nu} = \sum_{\mu = 2}^{n_p + 1} \sum_{\nu \in B_{\mu}} \frac{\left \lceil M(2^{\mu2} / \left \ \nu \right \)^2 \right \rceil}{M 2^{2\mu} 2^{n_{\mu} + 1}},\]we use the upper bound from Eq. (29) in arXiv:1807.09802v2 which gives \(p_{\nu} = 0.2398\).

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, eta, omega, error, br=7, charge=0, cubic=True, vectors=None)[source]¶ Return the number of logical qubits needed to implement the first quantization algorithm.
The expression for computing the cost is taken from Eq. (101) of [arXiv:2204.11890v1].
 Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
br (int) – number of bits for ancilla qubit rotation
charge (int) – total electric charge of the system
cubic (bool) – True if the unit cell is cubic
vectors (array[float]) – lattice vectors
 Returns
number of logical qubits needed to implement the first quantization algorithm
 Return type
int
Example
>>> n = 100000 >>> eta = 156 >>> omega = 169.69608 >>> error = 0.01 >>> qubit_cost(n, eta, omega, error) 4377

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

static
success_prob
(n, br)[source]¶ Return the probability of success for state preparation.
The expression for computing the probability of success is taken from Eqs. (59, 60) of [PRX Quantum 2, 040332 (2021)].
 Parameters
n (int) – number of basis states to create an equal superposition for state preparation
br (int) – number of bits for ancilla qubit rotation
 Returns
probability of success for state preparation
 Return type
float
Example
>>> n = 3 >>> br = 8 >>> success_prob(n, br) 0.9999928850303523

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, eta, omega, error, br=7, charge=0)[source]¶ Return the number of Toffoli gates needed to implement the qubitization unitary operator.
The expression for computing the cost is taken from Eq. (125) of [PRX Quantum 2, 040332 (2021)].
 Parameters
n (int) – number of plane waves
eta (int) – number of electrons
omega (float) – unit cell volume
error (float) – target error in the algorithm
br (int) – number of bits for ancilla qubit rotation
charge (int) – total electric charge of the system
 Returns
the number of Toffoli gates needed to implement the qubitization unitary operator
 Return type
int
Example
>>> n = 100000 >>> eta = 156 >>> omega = 169.69608 >>> error = 0.01 >>> unitary_cost(n, eta, omega, error) 17033