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:
\[\mu = \sum_{i \in X} p(i) f(i).\]The algorithm proceeds as follows:
The probability distribution \(p(i)\) is encoded using a unitary \(\mathcal{A}\) applied to the first \(m\) qubits specified by
target_wires
.The function \(f(i)\) is encoded onto the last qubit of
target_wires
using a unitary \(\mathcal{R}\).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 \(e^{\pm 2 \pi i \theta}\).
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 withtarget_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 hardwarecompatible gates.
Usage 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
Attributes
Get base name of the operator.
Eigenvalues of an instantiated operator.
Generator of the operation.
Gradient computation method.
Gradient recipe for the parametershift method.
Boolean determining if the inverse of the operation was requested.
Matrix representation of an instantiated operator in the computational basis.
Get and set the name of the operator.
Current parameter values.
Wires of this operator.

base_name
¶ Get base name of the operator.

eigvals
¶

generator
¶ Generator of the operation.
A length2 list
[generator, scaling_factor]
, wheregenerator
is an existing PennyLane operation class or \(2\times 2\) Hermitian array that acts as the generator of the current operationscaling_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 parametershift method.'F'
: finite difference numerical differentiation.None
: the operation may not be differentiated.
Default is
'F'
, orNone
if the Operation has zero parameters.

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

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'¶
Methods
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

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.
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