How to choose the Trotter steps for an MPF

This guide teaches you how to find a good set of Trotter steps for an MPF. We assume that you have already read and understood the Getting started tutorial.

Understanding some heuristics

In principle, any \(k_j\) values can be chosen but some choices of Trotter steps will lead to a larger noise amplification on real devices than others. Thus, it is important that one tries to find “good” values of \(k_j\). In the following we will discuss some heuristics to help guide you in doing so.

  1. In practice, our largest \(k_j\) (\(k_{\text{max}}\)) value is simply bound by the depth of quantum circuit we can reliably execute.

  2. Since the Trotter error is only well behaved for \(\text{d}t = t/k \lt 1\) (see Understanding the stability of MPFs), our smallest \(k_j\) (\(k_{\text{min}}\)) should satisfy: \(t/k_{\text{min}} \lt 1\) (Note: empirically \(\text{d}t \leq 1\) seems to work fine, too).

  3. None of our \(x_j\) values should be close to \(0\), as this would imply that we might as well exclude the corresponding \(k_j\) from the expansion.

  4. Similarly, we do not want the \(x_j\) for \(k_{\text{max}}\) to be the only dominant one, since that implies that we are essentially just recovering a single product formula with \(k=k_{\text{max}}\).

  5. Finally, we want the L1-norm of our coefficients, \(x_j\), to be small, as this indicates a well-conditioned MPF (see Carrera Vazquez et al., 2023). Furthermore, small coefficients minimize the variance amplification of the MPF.

Applying these heuristics

Reusing the example from the Getting started tutorial, we want to simulate up to time \(t=8.0\). Thus, \(k_{\text{min}}=8\).

For the sake of this example, let us assume that our deepest circuit is bound by \(k_{\text{max}}=20\).

In addition to the constraints above, we will follow the heuristic presented by Zhuk et al., 2023, where they minimize a re-weighted L1-norm of the coefficients:

\[\text{min}_{k_j} \sum_j \frac{|c_j|}{k_j^{2\chi}} \, ,\]

where \(\chi\) is the order of our individual product formulas.

Preparing our LSE

In a first step, we prepare the LSE, \(Ax=b\), leveraging a feature of the setup_lse function that allows us to use a cvxpy.Parameter object for our \(k_j\) values.

[30]:

order = 2
symmetric = True

min_k = 8
max_k = 20

max_l1_norm = 5.0
min_coeff = 0.01

[31]:

import cvxpy as cp
from qiskit_addon_mpf.static import setup_lse

ks = cp.Parameter(3, integer=True)
lse = setup_lse(ks, order=order, symmetric=symmetric)

Next, we loop over all possible 3-tuples of \(k_j\) values within our specified range. For each one, we compute the analytical coefficients, \(x_j\), and then apply our filter criteria. We keep track of those \(k_j\) tuples which satisfy our criteria.

[32]:

import numpy as np

possible_ks = {}

for k1 in range(min_k, max_k):
    for k2 in range(k1 + 1, max_k):
        for k3 in range(k2 + 1, max_k):
            ks.value = [k1, k2, k3]
            coeffs = lse.solve()

            if np.any(np.isclose(coeffs, 0.0, atol=min_coeff)):
                continue

            norm1 = np.linalg.norm(coeffs, ord=1)
            if norm1 > max_l1_norm:
                continue

            weighted = np.sum([np.abs(c) / k ** (2 * order) for c, k in zip(coeffs, ks.value)])
            possible_ks[tuple(int(k) for k in ks.value)] = {
                "coeffs": tuple(coeffs),
                "weighted": float(weighted),
                "norm1": float(norm1),
            }

Finally, we sort our \(k_j\) by the re-weighted L1-norm presented by Zhuk et al., 2023:

[33]:

for ks_val in sorted(possible_ks, key=lambda ks_val: possible_ks[ks_val].get("weighted")):
    print(ks_val, possible_ks[ks_val])

(8, 14, 19) {'coeffs': (0.10447913478216586, -1.7638200183654775, 2.6593408835833117), 'weighted': 9.182736455463762e-05, 'norm1': 4.527640036730955}
(8, 13, 19) {'coeffs': (0.13134519801186473, -1.4167162698412699, 2.285371071829405), 'weighted': 9.920634920634922e-05, 'norm1': 3.8334325396825397}
(8, 12, 19) {'coeffs': (0.17239057239057254, -1.1944700460829496, 2.022079473692377), 'weighted': 0.00011520737327188946, 'norm1': 3.388940092165899}
(9, 13, 19) {'coeffs': (0.2662743506493499, -1.6904000946969673, 2.4241257440476174), 'weighted': 0.00011837121212121191, 'norm1': 4.380800189393934}
(8, 13, 18) {'coeffs': (0.15003663003663004, -1.7549001536098316, 2.6048635235732016), 'weighted': 0.00012288786482334872, 'norm1': 4.509800307219663}
(8, 12, 18) {'coeffs': (0.19692307692307653, -1.4399999999999984, 2.243076923076922), 'weighted': 0.0001388888888888887, 'norm1': 3.879999999999997}
(8, 11, 19) {'coeffs': (0.24195168054817162, -1.070248538011695, 1.8282968574635234), 'weighted': 0.00014619883040935662, 'norm1': 3.14049707602339}
(9, 12, 19) {'coeffs': (0.37193877551020393, -1.5167873601053319, 2.144848584595128), 'weighted': 0.00014629507717065315, 'norm1': 4.033574720210664}
(8, 12, 17) {'coeffs': (0.22755555555555573, -1.7875862068965531, 2.5600306513409974), 'weighted': 0.00017241379310344843, 'norm1': 4.575172413793107}
(8, 11, 18) {'coeffs': (0.27638326585694917, -1.265318468585254, 1.988935202728305), 'weighted': 0.00017284590787313073, 'norm1': 3.530636937170508}
(9, 12, 18) {'coeffs': (0.42857142857142816, -1.8285714285714276, 2.3999999999999995), 'weighted': 0.00017636684303350958, 'norm1': 4.657142857142855}
(9, 11, 19) {'coeffs': (0.5858035714285708, -1.5251041666666654, 1.9393005952380946), 'weighted': 0.00020833333333333313, 'norm1': 4.05020833333333}
(8, 11, 17) {'coeffs': (0.3193762183235871, -1.5289264828738525, 2.2095502645502654), 'weighted': 0.00020885547201336693, 'norm1': 4.0578529657477045}
(8, 10, 19) {'coeffs': (0.383090160867938, -1.0642826734780744, 1.6811925126101364), 'weighted': 0.00021285653469561486, 'norm1': 3.1285653469561487}
(9, 11, 18) {'coeffs': (0.6750000000000009, -1.8030788177339916, 2.1280788177339907), 'weighted': 0.0002463054187192121, 'norm1': 4.606157635467984}
(8, 10, 18) {'coeffs': (0.43760683760683694, -1.2400793650793638, 1.8024725274725268), 'weighted': 0.0002480158730158727, 'norm1': 3.4801587301587276}
(8, 11, 16) {'coeffs': (0.3742690058479534, -1.9026640675763484, 2.528395061728395), 'weighted': 0.0002599090318388565, 'norm1': 4.805328135152697}
(8, 10, 17) {'coeffs': (0.5056790123456789, -1.4697236919459142, 1.9640446796002353), 'weighted': 0.00029394473838918284, 'norm1': 3.9394473838918285}
(8, 10, 16) {'coeffs': (0.592592592592593, -1.7806267806267817, 2.1880341880341887), 'weighted': 0.0003561253561253563, 'norm1': 4.561253561253563}
(8, 9, 19) {'coeffs': (0.8112497524262232, -1.3783613445378153, 1.5671115921115921), 'weighted': 0.00042016806722689084, 'norm1': 3.756722689075631}
(8, 9, 18) {'coeffs': (0.9266968325791858, -1.5882352941176474, 1.6615384615384616), 'weighted': 0.00048414427499394834, 'norm1': 4.176470588235295}
(8, 9, 17) {'coeffs': (1.0708496732026151, -1.855486425339368, 1.7846367521367528), 'weighted': 0.0005656108597285072, 'norm1': 4.710972850678736}

Recalling the Getting started tutorial, from these possible values we picked ks=(8, 12, 19) since it strikes a good balance of minimizing the re-weighted and plain L1-norms.