With the target of efficiently solving the quantum-mechanical Schrödinger equation on future quantum computers, first- and second-order product formulas for the time-evolution operator are proposed, designed to exploit hierarchies in the operator norms of the individual Hamiltonian components. Inspired by the THRIFT product formula, these decomposition schemes achieve a suppression of the leading error term by a factor of alpha<1 with respect to the standard Lie--Trotter--Suzuki splitting of the same order, where alpha quantifies the norm ratio between the dominant and subdominant parts of the Hamiltonian. Unlike THRIFT, the proposed formulas retain the general product-formula structure, making them applicable to Hamiltonians whose terms act on an arbitrary number of qubits. The advantages of this approach are demonstrated through numerical experiments on engineered two-dimensional spin-1/2 XY model systems and isolated molecules within the double-factorization framework.
AI Summary
The paper studies product-formula simulation of Hamiltonians $H=H_0+\alpha\sum_{k=1}^L H_k$, with a dominant, efficiently exponentiable $H_0$ and small $|\alpha|$. It seeks THRIFT-like suppression of leading errors while using only exponentials of individual Hamiltonian components, rather than grouped terms $H_0+\alpha H_k$.
The first-order “Partial–Strang” formula sandwiches a first-order splitting of the perturbation between two half-steps of $H_0$. Triangle inequalities and commutator bounds give local error $O(\alpha^2t^2)+O(|\alpha|t^3)$, compared with $O(|\alpha|t^2)$ for ordinary first-order splitting. A second-order construction combines an existing five-factor, two-operator splitting with symmetric splittings of the perturbation. Choosing $\lambda=1/2-\sqrt{3}/6\approx0.211325$ cancels the cubic error linear in $\alpha$, leaving a leading cubic contribution proportional to $\alpha^2$. It uses $4L+1$ exponentials, versus $2L+1$ for second-order Suzuki splitting, and is compared with an existing related optimized formula.
Exact classical simulations benchmark a nine-spin XY lattice and five STO-3G molecular Hamiltonians in double-factorized form. Molecular tests report first-order state-error reductions of $4.5$–$8$ relative to Lie–Trotter, using 100 steps to total time one atomic unit.
The stated scaling advantages concern short steps and fixed decompositions. Partial–Strang loses quadratic $\alpha$ scaling when its $|\alpha|t^3$ residual dominates. Molecular gains depend on commutators and fragment counts, not norm ratios alone. Benchmarks use small systems, Frobenius or state-distance errors, and no quantum hardware; the circuit discussion does not quantify end-to-end resource savings.
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Error found: The first-order formula has an additional O(|alpha| t^3) error, so its quadratic-alpha behavior requires sufficiently small steps relative to alpha, as the appendix acknowledges.
The second-order formula generally retains an O(|alpha| t^5) error, so its quadratic-alpha behavior is conditional even though its leading t^3 error coefficient has the claimed suppression.
Equation 'second_order_alpha' incorrectly equates a full perturbation step with a half-step product, although the final second-order construction uses the half steps correctly.
The molecular benchmark's Frobenius-norm ratios do not directly establish the operator-norm hierarchy assumed in the theoretical analysis.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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17 pages, 6 figures, submitted to Academia Quantum
With the target of efficiently solving the quantum-mechanical Schrödinger equation on future quantum computers, first- and second-order product formulas for the time-evolution operator are proposed, designed to exploit hierarchies in the operator norms of the individual Hamiltonian components. Inspired by the THRIFT product formula, these decomposition schemes achieve a suppression of the leading error term by a factor of alpha<1 with respect to the standard Lie--Trotter--Suzuki splitting of the same order, where alpha quantifies the norm ratio between the dominant and subdominant parts of the Hamiltonian. Unlike THRIFT, the proposed formulas retain the general product-formula structure, making them applicable to Hamiltonians whose terms act on an arbitrary number of qubits. The advantages of this approach are demonstrated through numerical experiments on engineered two-dimensional spin-1/2 XY model systems and isolated molecules within the double-factorization framework.
The paper studies product-formula simulation of Hamiltonians H=H0+α∑k=1LHk, with a dominant, efficiently exponentiable H0 and small ∣α∣. It seeks THRIFT-like suppression of leading errors while using only exponentials of individual Hamiltonian components, rather than grouped terms H0+αHk.
The first-order “Partial–Strang” formula sandwiches a first-order splitting of the perturbation between two half-steps of H0. Triangle inequalities and commutator bounds give local error O(α2t2)+O(∣α∣t3), compared with O(∣α∣t2) for ordinary first-order splitting. A second-order construction combines an existing five-factor, two-operator splitting with symmetric splittings of the perturbation. Choosing λ=1/2−3/6≈0.211325 cancels the cubic error linear in α, leaving a leading cubic contribution proportional to α2. It uses 4L+1 exponentials, versus 2L+1 for second-order Suzuki splitting, and is compared with an existing related optimized formula.
Exact classical simulations benchmark a nine-spin XY lattice and five STO-3G molecular Hamiltonians in double-factorized form. Molecular tests report first-order state-error reductions of 4.5–8 relative to Lie–Trotter, using 100 steps to total time one atomic unit.
The stated scaling advantages concern short steps and fixed decompositions. Partial–Strang loses quadratic α scaling when its ∣α∣t3 residual dominates. Molecular gains depend on commutators and fragment counts, not norm ratios alone. Benchmarks use small systems, Frobenius or state-distance errors, and no quantum hardware; the circuit discussion does not quantify end-to-end resource savings.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.
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.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
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.
(Suzuki-)Trotter decompositions, splitting methods, (Lie) product formulae... The most common numerical methods for the time evolution of quantum systems come with many names. And they are used practically everywhere with applications ranging from the solution of classical equations of motion and various Monte Carlo simulations to the real and imaginary time evolution on classical as well as quantum computers. Here we review the state of the art of said methods, focussing especially on the progress made over the last few years. We highlight recently discovered efficient time evolution algorithms and explain how best to use them in practice. A central part of this work is the estimation of error bounds that has improved greatly within the past decade. The relevance of time evolution methods for quantum computing is discussed with a focus on noisy hardware. Finally, a comprehensive overview of generalisations, related methods and alternatives to Trotterization is provided. This includes time-dependent Hamiltonian dynamics, processed methods, multi-product formulae, symplectic integrators, TDVP for tensor networks, quantum signal processing, Crouch-Grossman methods and more. The overall perspective in this work is that of a theoretical physicist. All mathematical proofs as well as some technical details are omitted for easier readability. Instead, this review serves as a hands-on guide and, of course, as a starting point for references that provide further details.