qml.labs.trotter_error.bch_expansion¶

bch_expansion(product_formula, order)[source]¶

Compute the Baker-Campbell-Hausdorff expansion of a ProductFormula object.

Parameters:

product_formula (ProductFormula) – The ProductFormula object whose BCH expansion will be computed.
order (int) – The maximum order of the expansion to return.

Returns:

A list of dictionaries. The ith dictionary contains the ith order commutators and their coefficients.

Return type:

List[Dict[Tuple[Hashable], complex]]

Example

In this example we compute the BCH expansion of the second order Trotter-Suzuki formula. The output is a list of dictionaries where each dictionary is indexed by a tuple representing a right-nested commutator. For example, ('A', 'A', 'B') represents the commutator \([A, [A, B]]\).

>>> from pprint import pp
>>> 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)
>>> pp(bch_expansion(second_order, order=3))
[defaultdict(<class 'complex'>,
             {('A',): (1+0j),
              ('B',): (1+0j),
              ('C',): (1+0j)}),
 defaultdict(<class 'complex'>, {}),
 defaultdict(<class 'complex'>,
             {('A', 'A', 'B'): (-0.04166666666666667+0j),
              ('B', 'A', 'B'): (-0.08333333333333333+0j),
              ('C', 'A', 'B'): (-0.08333333333333334+0j),
              ('A', 'A', 'C'): (-0.04166666666666667+0j),
              ('B', 'A', 'C'): (-0.08333333333333334+0j),
              ('B', 'B', 'C'): (-0.04166666666666667+0j),
              ('C', 'A', 'C'): (-0.08333333333333333+0j),
              ('C', 'B', 'C'): (-0.08333333333333333+0j)})]

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qml.labs.trotter_error.bch_expansion¶

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