# Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure

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## Abstract

**:**

## 1. Introduction

## 2. Trotterization—Theoretical Background

## 3. Molecular Hydrogen

#### Geometry Dependence

## 4. Generalized Ordering Strategies

#### 4.1. Methods

#### 4.2. Results—Magnitude Ordering

#### 4.3. Statistics of Commuting Hamiltonian Subsets

#### 4.4. Subset-Based Ordering Schemes

## 5. Error Operator Strategies

#### 5.1. Small Systems

#### 5.2. Term Insertion Error Operator

#### 5.3. Error Operator Based Ordering Schemes

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CI | configuration interaction |

FCI | full configuration interaction |

VQE | variational quantum eigensolver |

JW | Jordan–Wigner |

BK | Bravyi–Kitaev |

## Appendix A. Alternative Ordering Schemes

#### Appendix A.1. Commutator Ordering

#### Appendix A.2. Reverse Commutator Ordering

#### Appendix A.3. Performance of Commutator and ReverseCommutator Orderings

**Figure A2.**Trotter error of the commutator and reverseCommutator ordering relative to a magnitude ordering. Upper plots are linear within $\pm {10}^{-6}$. (

**a**): by number of qubits. (

**b**): by maximum nuclear charge. The reverseCommutator ordering is best for almost all systems including heavy atoms, however in almost all cases, a simple magnitude ordering outperforms all orderings considered. (

**c**): Frequency of “ordering orders”, being the sequence of ordering performance, ignoring the magnitude ordering. (

**d**): as lower left, but with the magnitude ordering.

## Appendix B. Systems in Dataset

System | Multiplicity | Charge | Basis | Qubits |
---|---|---|---|---|

B_{1} | 2 | 0 | STO-3G | 10 |

Be_{1} | 1 | 0 | STO-3G | 10 |

C_{1}O_{1} | 1 | 0 | STO-3G | 20 |

C_{1} | 3 | 0 | STO-3G | 10 |

Cl_{1} | 2 | 0 | STO-3G | 18 |

F_{2} | 1 | 0 | STO-3G | 20 |

H_{1}Cl_{1}^{1} | 1 | 0 | STO-3G | 20 |

H_{1}F_{1}^{1} | 1 | 0 | 3-21G | 22 |

H_{1}F_{1} | 1 | 0 | STO-3G | 12 |

H_{1}He_{1} | 1 | +1 | 3-21G | 8 |

H_{1}He_{1} | 1 | +1 | 6-311G | 12 |

H_{1}He_{1} | 1 | +1 | 6-31G | 8 |

H_{1}He_{1} | 1 | +1 | STO-3G | 4 |

H_{1}Li_{1}O_{1}^{1} | 1 | 0 | STO-3G | 22 |

H_{1}Li_{1} | 1 | 0 | STO-3G | 12 |

H_{1}Na_{1} | 1 | 0 | STO-3G | 20 |

H_{1}O_{1} | 1 | -1 | STO-3G | 12 |

H_{2}Be_{1} | 1 | 0 | STO-3G | 14 |

H_{2}C_{1}O_{1}^{1} | 1 | 0 | STO-3G | 24 |

H_{2}C_{1} | 3 | 0 | STO-3G | 14 |

H_{2}C_{1} | 3 | 0 | STO-3G | 14 |

H_{2}C_{2}^{1} | 1 | 0 | STO-3G | 24 |

H_{2}Mg_{1} | 1 | 0 | STO-3G | 22 |

H_{2}O_{1} | 1 | 0 | STO-3G | 14 |

H_{2}S_{1} | 1 | 0 | STO-3G | 22 |

H_{2} | 1 | 0 | 3-21G | 8 |

H_{2}^{1} | 1 | 0 | 6-311G** | 24 |

H_{2} | 1 | 0 | 6-311G | 12 |

H_{2} | 1 | 0 | 6-31G** | 20 |

H_{2} | 1 | 0 | 6-31G | 8 |

H_{2} | 1 | 0 | STO-3G | 4 |

H_{3}N_{1} | 1 | 0 | STO-3G | 16 |

H_{3} | 1 | +1 | 3-21G | 12 |

H_{3} | 1 | +1 | STO-3G | 6 |

H_{4}C_{1} | 1 | 0 | STO-3G | 18 |

H_{4}N_{1}^{1} | 1 | +1 | STO-3G | 18 |

Li_{1} | 2 | 0 | STO-3G | 10 |

Mg_{1} | 1 | 0 | STO-3G | 18 |

N_{2} | 1 | 0 | STO-3G | 20 |

Na_{1} | 2 | 0 | STO-3G | 18 |

O_{2} | 3 | 0 | STO-3G | 20 |

P_{1} | 4 | 0 | STO-3G | 18 |

S_{1} | 3 | 0 | STO-3G | 18 |

Si_{1} | 3 | 0 | STO-3G | 18 |

System | Multiplicity | Charge | Basis | Qubits |
---|---|---|---|---|

Ar_{1} | 1 | 0 | STO-3G | 18 |

C_{1}O_{2} | 1 | 0 | STO-3G | 30 |

Cl_{1} | 1 | -1 | STO-3G | 18 |

F_{1} | 2 | 0 | STO-3G | 10 |

H_{1}He_{1} | 1 | +1 | 6-31G** | 20 |

H_{1}Li_{1} | 1 | 0 | 3-21G | 22 |

H_{1} | 2 | 0 | STO-3G | 2 |

H_{2}C_{1} | 3 | 0 | 3-21G | 26 |

H_{2}O_{2} | 1 | 0 | STO-3G | 24 |

H_{4}C_{2} | 1 | 0 | STO-3G | 28 |

He_{1} | 1 | 0 | STO-3G | 2 |

K_{1} | 2 | 0 | STO-3G | 26 |

N_{1} | 4 | 0 | STO-3G | 10 |

Ne_{1} | 1 | 0 | STO-3G | 10 |

O_{1} | 3 | 0 | STO-3G | 10 |

O_{2} | 1 | 0 | STO-3G | 20 |

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**Figure 1.**Cumulative density plot of hydrogen ordering errors for one Trotter step, using a bond length of $0.7414$ Å and an STO-3G atomic basis. The vertical line denotes an error of 1 kcal/mol. Approximately $20\%$ of the orderings achieve this error or lower for the first-order Trotter–Suzuki approximation. Around $80\%$ of orderings achieve an error of 0.005 Hartree, approximately half that of the worst possible ordering.

**Figure 2.**The incompatibility graph of the Jordan–Wigner and Bravyi–Kitaev Hamiltonians, with the totally commuting set removed. Nodes correspond to Hamiltonian terms, edges correspond to non-commutativity between terms. Two independent sets are clearly revealed, with the XY-set colored blue and the Z-set colored red.

**Figure 3.**Distribution of Trotter errors by ordering for varying bond lengths for H

_{2}in a minimal basis. (

**a**): versus absolute Trotter error. As the bond length decreases, both the Trotter error and the dependence of the Trotter error on the ordering chosen increase. (

**b**): versus the Trotter error as a percentage of the ground state energy. The same trend as with the absolute Trotter error is observed, although the ordering dependence at extremely low bond length is accentuated.

**Figure 4.**Trotter errors for the dataset of molecular Hamiltonians using a magnitude ordering. The vertical bar indicates chemical accuracy. Most of the systems achieve chemical accuracy with one Trotter step. (

**a**): versus the number of spin-orbitals. Most of the high-error results are for low numbers of spin-orbitals. (

**b**): versus the number of terms in the Hamiltonian. Again, most of the high-error results are for low numbers of terms. (

**c**): versus the maximum nuclear charge. All of the high-error systems are for systems with exclusively light atoms, and the overall trend roughly follows the predictions of prior literature.

**Figure 5.**Statistics of the fully commuting sets of terms found in the coloring of the Hamiltonians in the dataset. (

**a**): number of fully commuting sets versus the number of terms in the Hamiltonian. (

**b**): number of independent sets divided by the number of terms, versus the number of spin-orbitals. A roughly linear trend is observed, indicating a $\Theta \left(\right)open="("\; close=")">{N}^{3}$ scaling. (

**c**): average number of terms in each fully commuting subset for a given Hamiltonian. (

**d**): standard deviation of number of terms in each fully commuting subset for a given Hamiltonian. The increasing variance in group sizes could be problematic for ordering purposes.

**Figure 6.**Trotter error of the depleteGroups and equaliseGroups orderings relative to a magnitude ordering. Upper plots are linear within $\pm {10}^{-6}$. (

**a**): by number of qubits. (

**b**): by maximum nuclear charge. Again, the magnitude ordering is preferable in most cases; however, for systems with period three atoms, the depleteGroups and equaliseGroups are best. (

**c**): frequency of “ordering orders”, being the sequence of ordering performance. The distribution here is relatively flat.

**Figure 7.**The Trotter error operator norm versus true Trotter error, for various orderings. (

**a**): hydrogen molecule in a minimal basis. Two hundred and fifty bins are used. Loose correlation is observed, although there is ambiguity for a true Trotter error of less than 0.001 a.u. (

**b**): helium hydride in a minimal basis, using 100,000 random orderings. One thousand bins are used. Little obvious correlation is observed.

**Figure 9.**Trotter error of the errorOperator ordering relative to a magnitude ordering. Upper plots are linear between $\pm {10}^{-6}$. (

**a**): by number of qubits. (

**b**): by maximum nuclear charge. As with the previous orderings, the variance between Trotter ordering schemes is low for systems involving heavy atoms. In these cases, the errorOperator ordering performs roughly commensurately with the magnitude ordering. (

**c**): frequency of “ordering orders”, being the sequence of ordering performance. The distribution here is relatively flat.

**Table 1.**Optimal Trotter orderings for the Hydrogen molecule in a STO-3G basis, for varying bond length, using a second-order Trotter–Suzuki approximation and a Jordan–Wigner mapping. Each ordering proceeds from top to bottom. The ordering changes as bond length increases, although, prior to the asymptotic limit, all orderings are of the form of alternating between commuting sets. At asymptotic separation, it is preferable to simulate sets in sequence, due to the symmetry of the coefficients.

Bond length (Å) | ||||
---|---|---|---|---|

0.3707 | 0.7414 | 1.1121 | 1.4828 | 10.000 |

0.24197 ${\sigma}_{1}^{z}$ | 0.17120 ${\sigma}_{1}^{z}$ | −0.10205 ${\sigma}_{2}^{z}$ | −0.03780 ${\sigma}_{2}^{z}$ | 0.09021 ${\sigma}_{3}^{y}{\sigma}_{2}^{x}{\sigma}_{1}^{x}{\sigma}_{0}^{y}$ |

0.04084 ${\sigma}_{3}^{y}{\sigma}_{2}^{x}{\sigma}_{1}^{x}{\sigma}_{0}^{y}$ | 0.04532 ${\sigma}_{3}^{y}{\sigma}_{2}^{x}{\sigma}_{1}^{x}{\sigma}_{0}^{y}$ | 0.05100 ${\sigma}_{3}^{y}{\sigma}_{2}^{x}{\sigma}_{1}^{x}{\sigma}_{0}^{y}$ | 0.05711 ${\sigma}_{3}^{y}{\sigma}_{2}^{x}{\sigma}_{1}^{x}{\sigma}_{0}^{y}$ | 0.09021 ${\sigma}_{3}^{x}{\sigma}_{2}^{y}{\sigma}_{1}^{y}{\sigma}_{0}^{x}$ |

0.24197 ${\sigma}_{0}^{z}$ | 0.17120 ${\sigma}_{0}^{z}$ | −0.10205 ${\sigma}_{3}^{z}$ | −0.03780 ${\sigma}_{3}^{z}$ | −0.09021 ${\sigma}_{3}^{x}{\sigma}_{2}^{x}{\sigma}_{1}^{y}{\sigma}_{0}^{y}$ |

−0.04084 ${\sigma}_{3}^{y}{\sigma}_{2}^{y}{\sigma}_{1}^{x}{\sigma}_{0}^{x}$ | −0.04532 ${\sigma}_{3}^{y}{\sigma}_{2}^{y}{\sigma}_{1}^{x}{\sigma}_{0}^{x}$ | 0.05100 ${\sigma}_{3}^{x}{\sigma}_{2}^{y}{\sigma}_{1}^{y}{\sigma}_{0}^{x}$ | 0.05711 ${\sigma}_{3}^{x}{\sigma}_{2}^{y}{\sigma}_{1}^{y}{\sigma}_{0}^{x}$ | −0.09021 ${\sigma}_{3}^{y}{\sigma}_{2}^{y}{\sigma}_{1}^{x}{\sigma}_{0}^{x}$ |

−0.48079 ${\sigma}_{3}^{z}$ | −0.22279 ${\sigma}_{3}^{z}$ | 0.12533 ${\sigma}_{0}^{z}$ | 0.09462 ${\sigma}_{0}^{z}$ | 0.03964 ${\sigma}_{0}^{z}$ |

−0.04084 ${\sigma}_{3}^{x}{\sigma}_{2}^{x}{\sigma}_{1}^{y}{\sigma}_{0}^{y}$ | −0.04532 ${\sigma}_{3}^{x}{\sigma}_{2}^{x}{\sigma}_{1}^{y}{\sigma}_{0}^{y}$ | −0.05100 ${\sigma}_{3}^{y}{\sigma}_{2}^{y}{\sigma}_{1}^{x}{\sigma}_{0}^{x}$ | −0.05711 ${\sigma}_{3}^{y}{\sigma}_{2}^{y}{\sigma}_{1}^{x}{\sigma}_{0}^{x}$ | 0.03964 ${\sigma}_{1}^{z}$ |

−0.48079 ${\sigma}_{2}^{z}$ | −0.22279 ${\sigma}_{2}^{z}$ | 0.12533 ${\sigma}_{1}^{z}$ | 0.09462 ${\sigma}_{1}^{z}$ | 0.03964 ${\sigma}_{3}^{z}$ |

0.04084 ${\sigma}_{3}^{x}{\sigma}_{2}^{y}{\sigma}_{1}^{y}{\sigma}_{0}^{x}$ | 0.04532 ${\sigma}_{3}^{x}{\sigma}_{2}^{y}{\sigma}_{1}^{y}{\sigma}_{0}^{x}$ | −0.05100 ${\sigma}_{3}^{x}{\sigma}_{2}^{x}{\sigma}_{1}^{y}{\sigma}_{0}^{y}$ | −0.05711 ${\sigma}_{3}^{x}{\sigma}_{2}^{x}{\sigma}_{1}^{y}{\sigma}_{0}^{y}$ | 0.03964 ${\sigma}_{2}^{z}$ |

**Table 2.**The molecular dataset used. Note that most of the systems involving a non-minimal basis set were H

_{2}and HeH

^{+}systems, as specified in Appendix B. The polyatomic category includes molecules, ions and radicals.

Qubits | 1–10 | 11–20 | 21–30 | Total |
---|---|---|---|---|

Polyatomic | 1 | 14 | 5 | 20 |

Atoms | 4 | 6 | 0 | 10 |

Ions | 2 | 2 | 0 | 4 |

Other bases | 4 | 4 | 2 | 10 |

Total | 11 | 26 | 7 | 44 |

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Tranter, A.; Love, P.J.; Mintert, F.; Wiebe, N.; Coveney, P.V.
Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure. *Entropy* **2019**, *21*, 1218.
https://doi.org/10.3390/e21121218

**AMA Style**

Tranter A, Love PJ, Mintert F, Wiebe N, Coveney PV.
Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure. *Entropy*. 2019; 21(12):1218.
https://doi.org/10.3390/e21121218

**Chicago/Turabian Style**

Tranter, Andrew, Peter J. Love, Florian Mintert, Nathan Wiebe, and Peter V. Coveney.
2019. "Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure" *Entropy* 21, no. 12: 1218.
https://doi.org/10.3390/e21121218