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 non-Clifford gates and logical qubits for a quantum phase estimation algorithm in first quantization using a plane-wave 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 1-Norm of the Hamiltonian >>> algo.gates, # estimated number of non-Clifford 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, ϵ, is distributed among four different sources of error using Eq. (131)
ϵ2≥ϵ2qpe+(ϵM+ϵR+ϵT)2,where ϵqpe is the quantum phase estimation error and ϵM, ϵR, and ϵT are defined in Eqs. (132-134).
Here, we fix ϵM=ϵR=ϵT=αϵ with a default value of α=0.01 and obtain
ϵqpe=√ϵ2[1−(3α)2].Note that the user only needs to define the target algorithm error ϵ. 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 parameter-shift method.
Integer hash that uniquely represents the operator.
Dictionary of non-trainable 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 ⟨ψ|U(p)†ˆOU(p)|ψ⟩.
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 parameter-shift method.
This is a tuple with one nested list per operation parameter. For parameter ϕk, the nested list contains elements of the form [ci,ai,si] where i is the index of the term, resulting in a gradient recipe of
∂∂ϕkf=∑icif(aiϕk+si).If
None
, the default gradient recipe containing the two terms [c0,a0,s0]=[1/2,1,π/2] and [c1,a1,s1]=[−1/2,1,−π/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¶
- has_plxpr_decomposition = False¶
- has_sparse_matrix = False¶
- hash¶
Integer hash that uniquely represents the operator.
- Type
int
- hyperparameters¶
Dictionary of non-trainable 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 ⟨ψ|U(p)†ˆOU(p)|ψ⟩.
These frequencies encode the behaviour of the operator U(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 non-negative 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_qfunc_decomposition
(*args, ...)Experimental method to compute the dynamic decomposition of the operator with program capture enabled.
compute_sparse_matrix
(*params[, format])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.
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 single-parameter-form.
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 1-norm of a first-quantized Hamiltonian in the plane-wave 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 single-qubit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order, format])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=O1O2…On.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) – non-trainable 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=UΣU† where Σ is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary U†.
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) – non-trainable 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†, the operator can be reconstructed asO=UΣU†,where Σ 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) – non-trainable 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) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
matrix representation
- Return type
tensor_like
- static compute_qfunc_decomposition(*args, **hyperparameters)¶
Experimental method to compute the dynamic decomposition of the operator with program capture enabled.
When the program capture feature is enabled with
qml.capture.enable()
, the decomposition of the operator is computed with this method if it is defined. Otherwise, thecompute_decomposition()
method is used.The exception to this rule is when the operator is returned from the
compute_decomposition()
method of another operator, in which case the decomposition is performed withcompute_decomposition()
(even if this method is defined), and not with this method.When
compute_qfunc_decomposition
is defined for an operator, the control flow operations within the method (specifying the decomposition of the operator) are recorded in the JAX representation.Note
This method is experimental and subject to change.
See also
- Parameters
*args (list) – positional arguments passed to the operator, including trainable parameters and wires
**hyperparameters (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- static compute_sparse_matrix(*params, format='csr', **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
attributeformat (str) – format of the returned scipy sparse matrix, for example ‘csr’
**hyperparams (dict) – non-trainable 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=O1O2…OnA
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=UΣU† where Σ is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary U†.
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†, the operator can be reconstructed asO=UΣU†,where Σ 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
- 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 single-parameter-form.
For example, for operator
U(ϕ)=eiϕ(0.5Y+Z⊗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
LinearCombination
andSparseHamiltonian
respectively).
- 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 non-parameter 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 matrix-valued 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 1-norm of a first-quantized Hamiltonian in the plane-wave 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
1-norm of a first-quantized Hamiltonian in the plane-wave 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 1‖ 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^{\mu-2} / \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 single-qubit 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, format='csr')¶
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
format (str) – format of the returned scipy sparse matrix, for example ‘csr’
- 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_iA
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