Static MPFs

qiskit_addon_mpf.static

Static MPF coefficients.

This module provides the generator function for the linear system of equations (LSE) for computing static (that is, time-independent) MPF coefficients.

setup_static_lse

setup_static_lse(trotter_steps, *, order=1, symmetric=False)

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Return the linear system of equations for computing static MPF coefficients.

This function constructs the following linear system of equations:

A x = b,

with

\begin{split}A_{0,j} &= 1 \\ A_{i>0,j} &= k_{j}^{-(\chi + s(i-1))} \\ b_0 &= 1 \\ b_{i>0} &= 0\end{split}

where $\chi$ is the order, $s$ is $2$ if symmetric is True and $1$ oterhwise, $k_{j}$ are the trotter_steps, and $x$ are the variables to solve for. The indices $i$ and $j$ start at $0$.

Here is an example:

>>> from qiskit_addon_mpf.static import setup_static_lse
>>> lse = setup_static_lse([1,2,3], order=2, symmetric=True)
>>> print(lse.A)
[[1.         1.         1.        ]
 [1.         0.25       0.11111111]
 [1.         0.0625     0.01234568]]
>>> print(lse.b)
[1. 0. 0.]

Parameters

$trotter_steps (list [int] | Parameter) - the sequence of trotter steps from which to build $A$. Rather than a list of integers, this may also be a Parameter instance of the desired size. In this case, the constructed LSE is parameterized whose values must be assigned before it can be solved.$
$order (int) - the order of the individual product formulas making up the MPF.$
$symmetric (bool) -$ $whether the individual product formulas making up the MPF are symmetric. For example, the Lie-Trotter formula is not symmetric, while Suzuki-Trotter is.$
Note
$Making use of this value is equivalent to the static MPF coefficient description provided by [1]. In contrast, [2] disregards the symmetry of the individual product formulas, effectively always setting symmetric=False .$

Returns

The LSE to find the static MPF coefficients as described above.

Return type

LSE

References

[1]: A. Carrera Vazquez et al., Quantum 7, 1067 (2023).

https://quantum-journal.org/papers/q-2023-07-25-1067/

[2]: S. Zhuk et al., Phys. Rev. Research 6, 033309 (2024).

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.6.033309

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