qml.resource.FirstQuantization

class FirstQuantization(n, eta, omega=None, error=0.0016, charge=0, br=7, vectors=None)

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() and qubit_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 = qml.resource.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, \(\epsilon\), 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. (132-134).

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	Arithmetic depth of the operator.
basis	The basis of an operation, or for controlled gates, of the target operation.
batch_size	Batch size of the operator if it is used with broadcasted parameters.
control_wires	Control wires of the operator.
gates	Return the total number of Toffoli gates needed to implement the first quantization algorithm.
grad_method
grad_recipe	Gradient recipe for the parameter-shift method.
has_adjoint
has_decomposition
has_diagonalizing_gates
has_generator
has_matrix
has_qfunc_decomposition
has_sparse_matrix
hash	Integer hash that uniquely represents the operator.
hyperparameters	Dictionary of non-trainable variables that this operation depends on.
id	Custom string to label a specific operator instance.
is_hermitian	This property determines if an operator is hermitian.
lamb	Return the 1-norm of a first-quantized Hamiltonian in the plane-wave basis.
name	String for the name of the operator.
ndim_params	Number of dimensions per trainable parameter of the operator.
num_params	Number of trainable parameters that the operator depends on.
num_wires	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 \(\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle\).
parameters	Trainable parameters that the operator depends on.
pauli_rep	A PauliSentence representation of the Operator, or None if it doesn't have one.
qubits	Return the number of logical qubits needed to implement the first quantization algorithm.
resource_keys
resource_params	A dictionary containing the minimal information needed to compute a resource estimate of the operator's decomposition.
wires	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 and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "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 on ndim_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 is None.

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 length 0.

Returns

The control wires of the operation.

Return type

Wires

gates

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)].

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 \(\phi_k\), 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

has_qfunc_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.

lamb

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.

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 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, or None if it doesn’t have one.

qubits

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].

resource_keys = {}

resource_params

A dictionary containing the minimal information needed to compute a resource estimate of the operator’s decomposition.

The keys of this dictionary should match the resource_keys attribute of the operator class. Two instances of the same operator type should have identical resource_params iff their decompositions exhibit the same counts for each gate type, even if the individual gate parameters differ.

Examples

The MultiRZ has non-empty resource_keys:

>>> qml.MultiRZ.resource_keys
{"num_wires"}

The resource_params of an instance of MultiRZ will contain the number of wires:

>>> op = qml.MultiRZ(0.5, wires=[0, 1])
>>> op.resource_params
{"num_wires": 2}

Note that another MultiRZ may have different parameters but the same resource_params:

>>> op2 = qml.MultiRZ(0.7, wires=[1, 2])
>>> op2.resource_params
{"num_wires": 2}

wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

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).
decomposition()	Representation of the operator as a product of other operators.
diagonalizing_gates()	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()	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.
single_qubit_rot_angles()	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).

\[O = O_1 O_2 \dots O_n.\]

Note

Operations making up the decomposition should be queued within the compute_decomposition method.

See also

decomposition().

Parameters

• *params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (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 \(O = U \Sigma U^{\dagger}\) 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

diagonalizing_gates().

Parameters

• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (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 \(U^{\dagger}\), 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

Operator.eigvals() and qml.eigvals()

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

Operator.matrix() and qml.matrix()

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, the compute_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 with compute_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

compute_decomposition().

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

sparse_matrix()

Parameters

• *params (list) – trainable parameters of the operator, as stored in the parameters attribute

• format (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.

\[O = O_1 O_2 \dots O_n\]

A DecompositionUndefinedError is raised if no representation by decomposition is defined.

See also

compute_decomposition().

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

compute_diagonalizing_gates().

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

compute_eigvals()

Returns

eigenvalues

Return type

tensor_like

static estimation_cost(n, eta, omega, error, br=7, charge=0, cubic=True, vectors=None)

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)

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

\[U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}\]

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 LinearCombination and SparseHamiltonian 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

Operator

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

compute_matrix()

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)

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 \(\frac{1}{\left \| \nu \right \|^k}\) 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)

\[p_{\nu} = \sum_{\mu = 2}^{n_p + 1} \sum_{\nu \in B_{\mu}} \frac{\left \lceil M(2^{\mu-2} / \left \| \nu \right \|)^2 \right \rceil}{M 2^{2\mu} 2^{n_{\mu} + 1}},\]

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)

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

Operator

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

compute_sparse_matrix()

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)

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)

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

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