The effort to generate matrix exponentials and associated differentials, required to determine the time evolution of quantum systems, frequently constrains the evaluation of problems in quantum control theory, variational circuit compilation, or Monte Carlo sampling. We introduce two ideas for the time-efficient approximation of matrix exponentials of linear multiparametric Hamiltonians. We modify the Suzuki-Trotter product formula from an approximation to an interpolation scheme to improve both accuracy and walltime. This allows us to achieve high fidelities within a single interpolation step, which can be computed directly from cached matrices. Furthermore, we define the interpolation on a grid of system parameters, and show that the interpolation infidelity converges with fourth-order accuracy in the number of interpolation bins. Published by the American Physical Society 2024
The effort to generate matrix exponentials and associated differentials, required to determine the time evolution of quantum systems, frequently constrains the evaluation of problems in quantum control theory, variational circuit compilation, or Monte Carlo sampling. We introduce two ideas for the time-efficient approximation of matrix exponentials of linear multiparametric Hamiltonians. We modify the Suzuki-Trotter product formula from an approximation to an interpolation scheme to improve both accuracy and walltime. This allows us to achieve high fidelities within a single interpolation step, which can be computed directly from cached matrices. Furthermore, we define the interpolation on a grid of system parameters, and show that the interpolation infidelity converges with fourth-order accuracy in the number of interpolation bins. Published by the American Physical Society 2024
Abstract Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area.In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previouslypublished alternatives. They can also be used as basic methods in recursive procedures to increasethe order of approximation.We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.
Compared with time-independent Hamiltonians, the dynamics of generic quantum Hamiltonians H ( t ) are complicated by the presence of time ordering in the evolution operator. In the context of digital quantum simulation, this difficulty prevents a direct adaptation of many time-independent simulation algorithms for time-dependent simulation. However, there exists a framework within the theory of dynamical systems that eliminates time ordering by adding a “clock” degree of freedom. In this work, we provide a computational framework, based on this reduction, for encoding time-dependent dynamics as time-independent systems. As a result, we make two advances in digital Hamiltonian simulation. First, we create a time-dependent simulation algorithm based on performing qubitization on the augmented clock system and, in doing so, provide the first qubitization-based approach to time-dependent Hamiltonians that goes beyond Trotterization of the ordered exponential. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials and then propose and analyze an algorithm based on these formulas. Unlike other algorithms of similar accuracy, the multiproduct approach achieves commutator scaling, meaning that this method outperforms existing methods for physically local time-dependent Hamiltonians with sufficient smoothness. Our work reduces the disparity between time-dependent and time-independent simulation and indicates a step toward optimal quantum simulation of time-dependent Hamiltonians. Published by the American Physical Society 2024
Matrix exponentiation (ME) is widely used in various fields of science and engineering. For example, the unitary dynamics of quantum systems is described by exponentiation of Hamiltonian operators. However, despite a significant attention, the numerical evaluation of ME remains computationally expensive, particularly for large dimensions. Often this process becomes a bottleneck in algorithms requiring iterative evaluation of ME. Here we propose a method for approximating ME of a single operator with a bounded coefficient into a product of certain discrete operators. This approach, which we refer to as Rapid Exponentiation using Discrete Operators (REDO), is particularly efficient for iterating ME over large numbers. We describe REDO in the context of a quantum system with a constant as well as a time-dependent Hamiltonian, although in principle, it can be adapted in a more general setting. As concrete examples, we choose two applications. First, we incorporate REDO in optimal quantum control algorithms and report a speed-up of several folds over a wide range of system size. Secondly, we propose REDO for numerical simulations of quantum dynamics. In particular, we study exotic quantum freezing with noisy drive fields.
We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.
We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation, and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates, which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.
The advent of near-term digital quantum computers could offer us an exciting opportunity to investigate quantum many-body phenomena beyond that of classical computing. To make the best use of the hardware available, it is paramount that we have methods that accurately simulate Hamiltonian dynamics for limited circuit depths. In this paper, we propose a method to classically optimize unitary brickwall circuits to approximate quantum time evolution operators. Our method is scalable in system size through the use of tensor networks. We demonstrate that, for various three-body Hamiltonians, our approach produces quantum circuits that can outperform trotterization in both their accuracy and the quantum circuit depth needed to implement the dynamics, with the exact details being dependent on the Hamiltonian. We also explain how to choose an optimal time step that minimizes the combined errors of the quantum device and the brickwall circuit approximation. Published by the American Physical Society 2024