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  4. qml.labs.trotter_error.ProductFormula

qml.labs.trotter_error.ProductFormula

class ProductFormula(terms, coeffs=None, exponent=1.0, label=None)[source]

Bases: object

Class for representing product formulas.

For a set of Hermitian operators \(H_1,\dots,H_n\) a product formula is any function in the form \(U(t) = \prod_{k=1}^n e^{it\alpha_k H_k}\) with \(\alpha_k \in \mathbb{R}\).

Parameters:

  • terms (Sequence[Hashable] | Sequence[ProductFormula]) – Either a list of labels for the Hermitian operators or a list of ProductFormula objcts. When a list of labels is given, the product formula returned is the product of exponentials of the lables. When a list of product formulas is given the product formula returned is the product of the given product formulas.

  • coeffs – (Sequence[float]): A list of coefficients corresponding to the given terms. This argument is not needed when the terms are ProductFormula objects.

  • exponent (float) – Raises the product formula to the power of exponent. Defaults to 1.0.

  • label (str) – Optional parameter used for pretty printing.

Example

This example uses ProductFormula to build the fourth order Troter-Suzuki formula on three fragments. First we build the second order formula.

>>> from pennylane.labs.trotter_error import ProductFormula
>>>
>>> frag_labels = ["A", "B", "C", "B", "A"]
>>> frag_coeffs = [1/2, 1/2, 1, 1/2, 1/2]
>>> second_order = ProductFormula(frag_labels, frag_coeffs)

Now we build the fourth order formula out of the second order formula using arithmetic operations.

>>> u = 1 / (4 - 4**(1/3))
>>> v = 1 - 4*u
>>>
>>> fourth_order = second_order(u)**2 @ second_order(v) @ second_order(u)**2

ordered_fragments

Return the fragment ordering used by the product formula.

ordered_terms

Return the term ordering used by the product formula.
ordered_fragments

Return the fragment ordering used by the product formula.

Example

>>> from pennylane.labs.trotter_error import ProductFormula
>>>
>>> pf1 = ProductFormula(["A", "B", "C"], [1, 1, 1])
>>> pf2 = ProductFormula(["X", "Y", "Z"], [1, 1, 1])
>>>
>>> pf = pf1 @ pf2
>>>
>>> pf.ordered_fragments
{'A': 0, 'B': 1, 'C': 2, 'X': 3, 'Y': 4, 'Z': 5}
ordered_terms

Return the term ordering used by the product formula.

Example

>>> from pennylane.labs.trotter_error import ProductFormula, bch_expansion
>>>
>>> frag_labels = ["A", "B", "C", "B", "A"]
>>> frag_coeffs = [1/2, 1/2, 1, 1/2, 1/2]
>>> second_order = ProductFormula(frag_labels, frag_coeffs, label="U")
>>>
>>> u = 1 / (4 - 4**(1/3))
>>> v = 1 - 4*u
>>>
>>> fourth_order = second_order(u)**2 @ second_order(v) @ second_order(u)**2
>>> fourth_order.ordered_terms
{U(0.4144907717943757)@U(-0.6579630871775028): 0, U(0.4144907717943757)**2.0: 1}

to_matrix(fragments)

Returns a numpy representation of the product formula.
to_matrix(fragments)[source]

Returns a numpy representation of the product formula.

Parameters:

fragments (Dict[Hashable, Fragment]) – The matrix representations of the fragment labels.

Example

>>> import numpy as np
>>> from pennylane.labs.trotter_error import ProductFormula
>>>
>>> frag_labels = ["A", "B", "C", "B", "A"]
>>> frag_coeffs = [1/2, 1/2, 1, 1/2, 1/2]
>>> second_order = ProductFormula(frag_labels, frag_coeffs)
>>>
>>> np.random.seed(42)
>>> fragments = {
>>>     "A": np.random.random(size=(3, 3)),
>>>     "B": np.random.random(size=(3, 3)),
>>>     "C": np.random.random(size=(3, 3)),
>>> }
>>>
>>> second_order.to_matrix(fragments)
[[20.53683969 24.33566914 25.4931284 ]
 [12.50207018 15.44505726 15.01069493]
 [13.52951601 17.64888648 18.04980336]]
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