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arXiv:2308.15100

The first-order Trotter decomposition in the dynamical-invariant basis

Takuya Hatomura

Published Aug 29, 2023• 0 citations
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Abstract

The Trotter decomposition is a basic approach to Hamiltonian simulation (digital quantum simulation). The first-order Trotter decomposition is the simplest one, whose deviations from target dynamics are of the first order of a small coefficient in terms of the infidelity. In this paper, we consider the first-order Trotter decomposition in the dynamical-invariant basis. By using a state-dependent inequality, we point out that deviations of this decomposition are of the second order of a small coefficient. Moreover, we also show that this decomposition includes a useful example, i.e., digital implementation of shortcuts to adiabaticity by counterdiabatic driving.

DOI
10.48550/arxiv.2308.15100

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Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.

•Dec 9, 2019•39 citations
The Trotter decomposition is a basic approach to Hamiltonian simulation (digital quantum simulation). The first-order Trotter decomposition is the simplest one, whose deviations from target dynamics are of the first order of a small coefficient in terms of the infidelity. In this paper, we consider the first-order Trotter decomposition in the dynamical-invariant basis. By using a state-dependent inequality, we point out that deviations of this decomposition are of the second order of a small coefficient. Moreover, we also show that this decomposition includes a useful example, i.e., digital implementation of shortcuts to adiabaticity by counterdiabatic driving.
Trotter decomposition is the basis of the digital quantum simulation. Asymmetric and symmetric decompositions are used in our GPU demonstration of the digital adiabatic quantum simulations of (2[Formula: see text]+[Formula: see text]1)-dimensional quantum [Formula: see text] lattice gauge theory. The actual errors in Trotter decompositions are investigated as functions of the coupling parameter and the number of Trotter substeps in each step of the variation of coupling parameter. The relative error of energy is shown to be equal to the Trotter error usually defined in terms of the evolution operators. They are much smaller than the order-of-magnitude estimation. The error in the symmetric decomposition is much smaller than that in the asymmetric decomposition. The features of the Trotter errors obtained here are useful in the experimental implementation of digital quantum simulation and its numerical demonstration.
A higher-order Suzuki-Trotter decomposition or Trotterization can be exploited to mitigate the Trotter error in digital quantum simulation. This work revisits the second-order symmetric Trotterization in terms of the Trotter error, where quantum many-body spin dynamics of the transverse-field Ising model is simulated. While the work presents a pedagogical way to exploit a real quantum computer, the effectiveness of the symmetric Trotterization is evaluated in a prototype superconducting quantum device on IBM Quantum Experience. It turns out that the symmetric Trotterization does not provide higher accuracy than the first-order Trotterization in the testbed using the transverse-field Ising model. The result indicates that apart from the quantum errors, such as logical gate error and readout error, the use of a higher-order Trotterization should be circumspect, and the Trotter error would play an insignificant role in particular applications in an early stage of realized noisy intermediate-scale quantum (NISQ) devices.
Simulating quantum dynamics beyond the reach of classical computers is one of the main envisioned applications of quantum computers. The most promising quantum algorithms to this end in the near term are the simplest, which use the Trotter formula and its higher-order variants to approximate the dynamics of interest. The approximation error of these algorithms is often poorly understood, even in the most basic and topical cases where the target Hamiltonian decomposes into two realizable terms: H=H_{1}+H_{2}. Recent studies have reported anomalously low approximation error with unexpected scaling in such cases, which they attribute to quantum interference between the errors from different steps of the algorithm. Here, we provide a simpler picture of these effects by relating the Trotter formula to its second-order variant for such H=H_{1}+H_{2} cases. Our method generalizes state-of-the-art error bounds without the technical caveats of prior studies, and elucidates how each part of the total error arises from the underlying quantum circuit. We compare our bound to the true error numerically, and find a close match over many orders of magnitude in the simulation parameters. Our findings further reduce the required circuit depth for the least experimentally demanding quantum simulation algorithms, and illustrate a useful method for bounding simulation error more broadly.
Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2}δt^{2}) instead of O(T^{2}δt^{2}) expected from general Trotter error bounds, where δt is the time step and T is the total time. This result suggests a self-healing mechanism and explains why, despite increasing T, infidelities for fixed-δt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.