qml.templates.subroutines.QuantumMonteCarlo

class QuantumMonteCarlo(probs, func, target_wires, estimation_wires, do_queue=True)[source]

Bases: pennylane.operation.Operation

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:

\[\mu = \sum_{i \in X} p(i) f(i).\]
../../_images/qmc.svg

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.

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.

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.

inverse

Boolean determining if the inverse of the operation was requested.

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.

string_for_inverse

wires

Wires of this operator.

base_name

Get base name of the operator.

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

Gradient computation 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

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

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

Current parameter values.

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.

expand()

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.

inv()

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

queue()

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

The adjointed operation.

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

multiplier, shift

Return type

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()

Append the operator to the Operator queue.