# qml.templates.subroutines.QuantumMonteCarlo¶

class QuantumMonteCarlo(probs, func, target_wires, estimation_wires, do_queue=True, id=None)[source]

Performs the quantum Monte Carlo estimation algorithm.

Given a probability distribution $$p(i)$$ of dimension $$M = 2^{m}$$ for some $$m \geq 1$$ and a function $$f: X \rightarrow [0, 1]$$ defined on the set of integers $$X = \{0, 1, \ldots, M - 1\}$$, this function implements the algorithm that allows the following expectation value to be estimated:

$\mu = \sum_{i \in X} p(i) f(i).$

The algorithm proceeds as follows:

1. The probability distribution $$p(i)$$ is encoded using a unitary $$\mathcal{A}$$ applied to the first $$m$$ qubits specified by target_wires.

2. The function $$f(i)$$ is encoded onto the last qubit of target_wires using a unitary $$\mathcal{R}$$.

3. The unitary $$\mathcal{Q}$$ is defined with eigenvalues $$e^{\pm 2 \pi i \theta}$$ such that the phase $$\theta$$ encodes the expectation value through the equation $$\mu = (1 + \cos (\pi \theta)) / 2$$. The circuit in steps 1 and 2 prepares an equal superposition over the two states corresponding to the eigenvalues $$e^{\pm 2 \pi i \theta}$$.

4. The QuantumPhaseEstimation() circuit is applied so that $$\pm\theta$$ can be estimated by finding the probabilities of the $$n$$ estimation wires. This in turn allows for the estimation of $$\mu$$.

Visit Rebentrost et al. (2018) for further details. In this algorithm, the number of applications $$N$$ of the $$\mathcal{Q}$$ unitary scales as $$2^{n}$$. However, due to the use of quantum phase estimation, the error $$\epsilon$$ scales as $$\mathcal{O}(2^{-n})$$. Hence,

$N = \mathcal{O}\left(\frac{1}{\epsilon}\right).$

This scaling can be compared to standard Monte Carlo estimation, where $$N$$ samples are generated from the probability distribution and the average over $$f$$ is taken. In that case,

$N = \mathcal{O}\left(\frac{1}{\epsilon^{2}}\right).$

Hence, the quantum Monte Carlo algorithm has a quadratically improved time complexity with $$N$$.

Parameters
• probs (array) – input probability distribution as a flat array

• func (callable) – input function $$f$$ defined on the set of integers $$X = \{0, 1, \ldots, M - 1\}$$ such that $$f(i)\in [0, 1]$$ for $$i \in X$$

• target_wires (Union[Wires, Sequence[int], or int]) – the target wires

• estimation_wires (Union[Wires, Sequence[int], or int]) – the estimation wires

Raises

ValueError – if probs is not flat or has a length that is not compatible with target_wires

Note

This template is only compatible with simulators because the algorithm is performed using unitary matrices. Additionally, this operation is not differentiable. To implement the quantum Monte Carlo algorithm on hardware requires breaking down the unitary matrices into hardware-compatible gates, check out the quantum_monte_carlo() transformation for more details.

Consider a standard normal distribution $$p(x)$$ and a function $$f(x) = \sin ^{2} (x)$$. The expectation value of $$f(x)$$ is $$\int_{-\infty}^{\infty}f(x)p(x) \approx 0.432332$$. This number can be approximated by discretizing the problem and using the quantum Monte Carlo algorithm.

First, the problem is discretized:

from scipy.stats import norm

m = 5
M = 2 ** m

xmax = np.pi  # bound to region [-pi, pi]
xs = np.linspace(-xmax, xmax, M)

probs = np.array([norm().pdf(x) for x in xs])
probs /= np.sum(probs)

func = lambda i: np.sin(xs[i]) ** 2


The QuantumMonteCarlo template can then be used:

n = 10
N = 2 ** n

target_wires = range(m + 1)
estimation_wires = range(m + 1, n + m + 1)

dev = qml.device("default.qubit", wires=(n + m + 1))

@qml.qnode(dev)
def circuit():
qml.templates.QuantumMonteCarlo(
probs,
func,
target_wires=target_wires,
estimation_wires=estimation_wires,
)
return qml.probs(estimation_wires)

phase_estimated = np.argmax(circuit()[:int(N / 2)]) / N


The estimated value can be retrieved using the formula $$\mu = (1-\cos(\pi \theta))/2$$

>>> (1 - np.cos(np.pi * phase_estimated)) / 2
0.4327096457464369

 base_name Get base name of the operator. basis The basis of an operation, or for controlled gates, of the target operation. control_wires For operations that are controlled, returns the set of control wires. eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method Gradient computation method. grad_recipe Gradient recipe for the parameter-shift method. hash returns an integer hash uniquely representing the operator id String for the ID of the operator. inverse Boolean determining if the inverse of the operation was requested. is_composable_rotation True if composing multiple copies of the operation results in an addition (or alternative accumulation) of parameters. is_self_inverse True if the operation is its own inverse. is_symmetric_over_all_wires True if the operation is the same if you exchange the order of wires. is_symmetric_over_control_wires True if the operation is the same if you exchange the order of all but the last wire. matrix Matrix representation of an instantiated operator in the computational basis. name Get and set the name of the operator. num_params num_wires par_domain parameters Current parameter values. single_qubit_rot_angles The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase. string_for_inverse wires Wires of this operator.
base_name

Get base name of the operator.

basis = None

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

control_wires

For operations that are controlled, returns the set of control wires.

Returns

The set of control wires of the operation.

Return type

Wires

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

• generator is an existing PennyLane operation class or $$2\times 2$$ Hermitian array that acts as the generator of the current operation

• scaling_factor represents a scaling factor applied to the generator operation

For example, if $$U(\theta)=e^{i0.7\theta \sigma_x}$$, then $$\sigma_x$$, with scaling factor $$s$$, is the generator of operator $$U(\theta)$$:

generator = [PauliX, 0.7]


Default is [None, 1], indicating the operation has no generator.

grad_method

• 'A': analytic differentiation using the parameter-shift method.

• 'F': finite difference numerical differentiation.

• None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

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

hash

returns an integer hash uniquely representing the operator

Type

int

id

String for the ID of the operator.

inverse

Boolean determining if the inverse of the operation was requested.

is_composable_rotation = None

True if composing multiple copies of the operation results in an addition (or alternative accumulation) of parameters.

For example, qml.RZ is a composable rotation. Applying qml.RZ(0.1, wires=0) followed by qml.RZ(0.2, wires=0) is equivalent to performing a single rotation qml.RZ(0.3, wires=0).

If set to None, the operation will be ignored during compilation transforms that merge adjacent rotations.

Type

bool or None

is_self_inverse = None

True if the operation is its own inverse.

If None, all instances of the given operation will be ignored during compilation transforms involving inverse cancellation.

Type

bool or None

is_symmetric_over_all_wires = None

True if the operation is the same if you exchange the order of wires.

For example, qml.CZ(wires=[0, 1]) has the same effect as qml.CZ(wires=[1, 0]) due to symmetry of the operation.

If None, all instances of the operation will be ignored during compilation transforms that check for wire symmetry.

Type

bool or None

is_symmetric_over_control_wires = None

True if the operation is the same if you exchange the order of all but the last wire.

For example, qml.Toffoli(wires=[0, 1, 2]) has the same effect as qml.Toffoli(wires=[1, 0, 2]), but neither are the same as qml.Toffoli(wires=[0, 2, 1]).

If None, all instances of the operation will be ignored during compilation transforms that check for control-wire symmetry.

Type

bool or None

matrix
name

Get and set the name of the operator.

num_params = 3
num_wires = -1
par_domain = 'A'
parameters

Current parameter values.

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]

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

 adjoint([do_queue]) Create an operation that is the adjoint of this one. decomposition(*params, wires) Returns a template decomposing the operation into other quantum operations. Returns a tape containing the decomposed operations, rather than a list. get_parameter_shift(idx[, shift]) Multiplier and shift for the given parameter, based on its gradient recipe. Inverts the operation, such that the inverse will be used for the computations by the specific device. queue([context]) Append the operator to the Operator queue.
adjoint(do_queue=False)

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.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

static decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

expand()[source]

Returns a tape containing the decomposed operations, rather than a list.

Returns

Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.

Return type

JacobianTape

get_parameter_shift(idx, shift=1.5707963267948966)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index

Returns

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, float, float]]

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.

Any subsequent call of this method will toggle between the original operation and the inverse of the operation.

Returns

operation to be inverted

Return type

Operator

queue(context=<class 'pennylane.queuing.QueuingContext'>)

Append the operator to the Operator queue.