B. Gaveau, E. Mihokova, M. Roncadelli, L. S. Schulman
The derivation of the Feynman path integral based on the Trotter product formula is extended to the case where the system is in a magnetic field.
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We show that all geometric quantum gates (GQG's in short), which are quantum gates only with geometric phases, are robust against control field strength errors. As examples of this observation, we show (1) how robust composite rf-pulses in NMR are geometrically constructed and (2) a composite rf-pulse based on Trotter-Suzuki Formulas is a GQG.
In this paper, we investigate a thermal coherent state defined with the Lie-Trotter product formula under the formalism of the thermo field dynamics. In the definition of our thermal coherent state, we treat the thermalizing operator and the displacement operator symmetrically. We examine its uncertainty relation and quasiprobability distributions. Although this thermal coherent state is equivalent to the conventional ones except for different parameterizations and a phase factor, it is convenient for describing an experimental setup of the optical parametric oscillator laser.
Constant-Depth Multi-Product Formula for Trotter Error Mitigation in Near-Term Digital Quantum Simulation
研究多体动力学数字量子模拟中算法Trotter误差与物理噪声问题，提出恒定深度多乘积公式，通过确定辅助参数\(\alpha\)，抑制Trotter误差，结合物理噪声缓解，为近期硬件长时间量子动力学模拟提供关键方法。
Theory and practice of Trotter product formulas for quantum chemistry
提出对称保护随机近可积Trotter（SPRINT）框架，结合近可积性、随机化、对称保护等技术优化电子结构哈密顿量模拟，相比先前方法显著降低门数，并引入广义秩分解（GRADE）方法。
Quantum simulation promises to address many challenges in fields ranging from quantum chemistry to material science, and high-energy physics, and could be implemented in noisy intermediate-scale quantum devices. A challenge in building good digital quantum simulators is the fidelity of the engineered dynamics given a finite set of elementary operations. Here we present a framework for optimizing the order of operations based on a geometric picture, thus abstracting from the operation details and achieving computational efficiency. Based on this geometric framework, we provide two alternative s…
We study path integrals in the Trotter-type form for the Schrödinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by Albeverio and Itô in celebrated papers - of Fourier transforms of complex measures. Essentially, $V$ is bounded and has the regularity of a function whose Fourier transform is in $L^1$. Whereas the strong convergence in $L^2$ in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerne…
Unbiased Hamiltonian Simulation by Reversing Trotter Error Dynamics
提出概率性Trotter误差反转（PTER）方法，通过准概率分解反转Trotter误差的相干动力学，实现无偏哈密顿模拟，在保持简单性的同时改善门计数缩放。
Suzuki Type Estimates for Exponentiated Sums and Generalized Lie-Trotter Formulas in Banach Algebras
本文在 Banach 代数中利用 Jordan 乘积建立 Lie-Trotter 乘积公式的两类误差估计，提出两种广义公式及显式界，并由此直接导出 Suzuki 对称化指数和近似。
Practical Estimation of Trotter Error for Hamiltonian Simulation
针对量子计算中哈密顿量模拟的Trotter误差估计难题，提出理论、算法和软件工具，将误差估计的规模和精度提升数个数量级，并应用于电子和振动系统验证。
We consider the relation between so called continuous localization models - i.e. non-linear stochastic Schrödinger evolutions - and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.
Error Estimates and Higher Order Trotter Product Formulas in Jordan-Banach Algebras
本研究将Trotter-Suzuki近似分析拓展到Jordan-Banach代数，解决了此前二阶Trotter公式误差估计存在性的公开问题，验证了其在自旋模拟中的效果，为Jordan代数在算子动力学中的应用提供新思路。
Beyond Trotterization: Variational Product Formulas for Quantum Simulation
提出一种变分替代Trotter-Suzuki分解的方法，通过全局作用量原理优化变分参数，在保持时间演化幺正性的同时降低误差，适用于任意初态，相比高阶Suzuki方案精度更高且门数更少。
Trotterization in Quantum Theory
本文从数学视角阐述了量子力学中处理非交换算子指数的Trotterization概念，给出了Trotter乘积公式的数学表述与正式证明草图，并展示了其在降低量子电路复杂性方面的效用。
Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation
本文扩展Trotter误差理论至多乘积公式，实现任意时间区间上1-范数误差的二次方缩减，并引入动态多乘积公式及极小极大估计方法以增强鲁棒性，适用于近期量子模拟。
We consider the dynamics $t\mapstoτ_t$ of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that $τ_t$ can be efficiently approximated by a product of $n$ automorphisms, each of them being an alternating product generated by the individual terms. For any integer $m$, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as $n^{-m}$. Our bounds hold in norm, pointwise for algebra elements that are sufficiently well ap…
We consider Hamiltonian simulation using the first order Lie-Trotter product formula under the assumption that the initial state has a high overlap with an energy eigenstate, or a collection of eigenstates in a narrow energy band. This assumption is motivated by quantum phase estimation (QPE) and digital adiabatic simulation (DAS). Treating the effective Hamiltonian that generates the Trotterized time evolution using rigorous perturbative methods, we show that the Trotter step size needed to estimate an energy eigenvalue within precision $ε$ using QPE can be improved in scaling from $ε$ to $ε^…
Ordering-Aware Theory of Trotter Error
本文系统研究乘积公式中哈密顿量项排序对模拟精度的影响，提出排序敏感的Trotter误差界，并利用局域结构将排序优化简化为图问题，证明顺序排序优于砖墙电路，确立排序为提升模拟精度的重要自由度。
Generation of a coherent squeezed like state defined with the Lie-Trotter product formula using a nonlinear photonic crystal
本文提出利用含二阶非线性极化率的非线性光子晶体与谐振腔生成由 Lie-Trotter 乘积公式定义的相干压缩类态，实现 15.9 dB 压缩水平，并定量评估了分束后光束的纠缠。
We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some …
A Monte Carlo approach to bound Trotter error
本文提出一种蒙特卡洛方法，通过无符号问题算子的谱范数上界来界定Trotter误差，在多种哈密顿量上界限紧致，并显著降低均匀电子气和线性并烯模型的模拟成本。
Exponentially Reduced Circuit Depths Using Trotter Error Mitigation
本研究针对数字量子模拟中乘积公式的Trotter误差问题，严谨分析Richardson外推与多项式插值缓解技术，证明其可实现精度的指数级提升，还优化了时间复杂度且无需分数Trotter步，更准确刻画了该类误差缓解技术。
On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic Applications
本文研究了在Banach空间强算子拓扑下证明Trotter乘积公式与量子Zeno效应定量界的假设条件，通过连续嵌入的Banach空间层级推导了Suzuki高阶界，并展示了其在玻色子应用中的自然实例。
Joint Mitigation of Algorithmic and Physical Errors in Noisy Hamiltonian Simulation
提出联合外推策略，将噪声强度与Trotter步长关联，通过单次Richardson外推同时消除物理噪声和Trotter误差，并在超导量子计算机及100比特SPD模拟中验证其高效性。
State-dependent Trotter Limits and their approximations
本文研究量子模拟中连续自由度的离散化问题，通过建立状态依赖Trotter误差的显式上界，给出了保证近似离散化物理有效性的充分条件。
On the commutator scaling in Hamiltonian simulation with multi-product formulas
本文提出了一种改进的多积公式，通过引入截断阶数的嵌套交换子，实现了在系统大小 $N$ 上的高效 Hamiltonian 模拟，同时保持 $1/\varepsilon$ 的 polylog 缩放。
Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas
研究针对哈密顿量模拟中乘积公式电路规模与精度成多项式关系的问题，提出高阶嵌套对易子补偿（HNCC）算法，通过特定展开和随机采样补偿误差，实现多对数精度依赖，降低门数，优于其他基于乘积公式的方法。
Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry
针对Trotter误差对输入态依赖性及无界算子应用不明确的问题，提出含高阶乘积公式的误差估计通用理论，通过氢原子案例解析化学模拟误差，揭示胖尾态误差标度特性及高阶Trotter层级失效情况，解析与数值结果一致。
The book deals with a stochastic formulation of path integration in real time, by rotating the_space_ variables over exp(i pi/4). Preliminary chapters deal with quantum and classical mechanics, probability theory and stochastic calculus, and the conventional approach to path integration through the Trotter product formula. Subsequently, the stochastic formulation in complex space is established, and applied to semi-classical expansions, nonlinear oscillations, quantum dynamics on analytic manifolds, dissipative systems (i.e., those which are open in that they interact with an environment), tun…
A Quantum Algorithm with Polylogarithmic Depth per Trotter Step for the Extended Hubbard Model
提出Q2FMM算法，利用快速多极方法将长程相互作用分层粗粒化，并通过可逆量子电路实现垃圾信息清除，使单个Trotter步的电路深度随系统规模呈多对数增长。
Dual channel multi-product formulas
本文提出双通道多产品公式，通过减少Trotter误差缩放，实现更高效的量子模拟精度。
The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for sys…
Trotter error and gate complexity of the SYK and sparse SYK models
研究SYK及稀疏SYK模型的Trotter误差和门复杂度，推导出一阶和更高阶公式的门复杂度估计，并探讨了模拟任意输入态的时间演化对复杂度的影响。
Better product formulas for quantum phase estimation
本文针对量子能量估计任务，利用Trotter误差理论和基于Magnus展开的方法，将乘积公式的误差分析推广至任意阶，发现了一种误差二次优于Trotter-Suzuki的9项二阶公式，并在低能子空间中证明了二次渐近加速。
Efficient and practical Hamiltonian simulation from time-dependent product formulas
本文提出基于乘积公式的量子时间演化算法，针对多尺度哈密顿量在门复杂度上优于Trotter公式，数值模拟显示在强场Ising模型中性能提升一个数量级。
Beyond Commutativity: Redesigning Trotter Decomposition via Local Symmetry
本文提出基于局部 SU(2) 对称性的新分解原则，通过将哈密顿量项分组为局部三粒子簇，有效抑制交换性分解中的误差和电路开销，提升量子模拟的精度和效率。
Numerical path integral approach to quantum dynamics and stationary quantum states
本文研究Feynman路径积分数值方法在实时量子动力学中的应用，比较Trotter核与精确核，并提出基于非相干实时传播求解基态和定态的新方法。
Randomized product formulas beyond optimal deterministic scaling
本文针对分离能标的哈密顿量，提出两类随机化乘积公式，在两种访问模型下分别实现$\mathcal O(\alpha^2)$和$\mathcal O(\alpha^3)$误差标度，优于确定性公式下界，数值模拟验证了门数量的减少。
Entanglement-Dependent Error Bounds for Hamiltonian Simulation
本研究通过纠缠熵与Trotter-Suzuki公式误差的关系，为哈密顿量模拟提供了更精确的资源估计方法，尤其在不同维度系统中表现出显著的改进。
Suzuki-Trotter Decompositions and other Methods for Quantum Time Evolution
本文综述量子时间演化的Suzuki-Trotter分解等方法，聚焦近年高效算法与误差界改进，并讨论噪声硬件下的量子计算应用及多种替代方案。
We present a decomposition scheme based on Lie-Trotter-Suzuki product formulae to represent an ordered operator exponential as a product of ordinary operator exponentials. We provide a rigorous proof that does not use a time-displacement superoperator, and can be applied to non-analytic functions. Our proof provides explicit bounds on the error and includes cases where the functions are not infinitely differentiable. We show that Lie-Trotter-Suzuki product formulae can still be used for functions that are not infinitely differentiable, but that arbitrary order scaling may not be achieved.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
Product formula approximations of the time-evolution operator on quantum computers are of great interest due to their simplicity, and good scaling with system size by exploiting commutativity between Hamiltonian terms. However, product formulas exhibit poor scaling with the time $t$ and error $ε$ of simulation as the gate cost of a single step scales exponentially with the order $m$ of accuracy. We introduce well-conditioned multiproduct formulas, which are a linear combination of product formulas, where a single step has polynomial cost $\mathcal{O}(m^2\log{(m)})$ and succeeds with probabilit…
Quantum Algorithm for Vibronic Dynamics: Case Study on Singlet Fission Solar Cell Design
提出基于乘积公式的量子算法模拟振动哈密顿量时间演化，支持任意电子态和振动模式，并集成到单线态裂变太阳能电池材料发现流程，显著降低实现成本。
Optimization of path-integral tensor-multiplication schemes in open quantum systems
针对开放量子系统中路径积分张量乘法方案因张量大小随量子记忆长度指数增长而受限的问题，提出基于矩阵表示和奇异值分解的优化方法，显著降低计算时间和内存使用，并实现精确值的快速外推。
Some improvements to product formula circuits for Hamiltonian simulation
本文通过Trotter-Suzuki分解提出三项改进以优化基态能量估计算法的乘积公式实现，包括缩小线路模板、并行化对易受控旋转及更高效的并行调度，这些改进可与其他优化相结合。
Two-parameter families of matrix product operator integrals of motion in Heisenberg spin chains
本文研究了Heisenberg自旋链中不同各向异性情况下双参数矩阵积算符守恒量族的构造方法，并探讨了其在Floquet协议中的应用。
We propose a modification of the Faddeev-Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato-Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few ex…
A correspondence between the Rabi model and an Ising model with long-range interactions
Rabi模型与具有长程相互作用的Ising模型之间存在对应关系，通过Trotter公式将两者联系起来，揭示了微扰展开与自旋域数量展开的等价性。
Path integral Monte Carlo in a discrete variable representation with Gibbs sampling: dipolar planar rotor chain
本文提出了一种基于离散变量和吉布斯采样的路径积分蒙特卡洛方法，用于研究偶极子平面转子链的基态性质，并通过对比不同方法验证了该方法的优越性。
In this short communication I generalize the method of obtaining quasi-Feynman formulas described in my previous paper on that topic. The theorem presented allows to obtain the solution to the Cauchy problem for the Schrödinger equation with the Hamiltonian decomposed to a finite sum of operators. The concept of Chernoff tangency is used, and the solution is written in the form of a quasi-Feynman formula as before. Theorem proven is compared to known approximation theorems: Trotter's, Chernoff's, Butko-Schilling-Smolyanov's.
The cavity method is a well established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [arXiv:0706.4391] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics. As a side result we also…
Impact of the phonon environment on the nonlinear quantum-dot-cavity QED. I. Path-integral approach
提出基于路径积分、Trotter 分解与连接集团展开的半解析方法，揭示声子环境在强耦合和高温下对非线性 QD–腔 QED 的非马尔可夫效应及 JC 阶梯重整化。
Multi-Product Hamiltonian Simulation with Explicit Commutator Scaling
本文对良态多乘积公式进行严格复杂度分析，显式建立交换子标度及近最优时间精度依赖，证明其在特定应用中优于二阶乘积公式，并在某些参数区间超越后Trotter方法。
Average-case Speedup for Product Formulas
本研究针对量子模拟中的乘积公式（Trotter化）方法，证明其对绝大多数输入态的特罗特误差标度优于最坏情况估计，通过非交换鞅的一致光滑性不等式得到典型情况门复杂度提升，为平均情况量子算法研究开辟道路。
Fast mixing of operator-loop path-integral quantum Monte Carlo for stoquastic XY Hamiltonians
本文证明算子环路径积分量子蒙特卡洛方法对stoquastic XY模型具有多项式混合时间，并据此在多项式时间内估计配分函数，显著改进先前算法。
Heat kernel for the quantum Rabi model
本文通过直接求值 Trotter-Kato 乘积公式并结合无穷对称群的轨道积分与调和分析，首次给出了量子 Rabi 模型热核的显式解析封闭公式。
In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also hav…
Approximating exponentials of commutators by optimized product formulas
针对嵌套对易子哈密顿量演化高效实现研究不足的问题，构造了3到6阶的优化乘积公式来近似任意两算子对易子的指数，减少了所需指数数量，可用于递归提阶，有望提升量子控制及相关量子算法的效率。
Fully analytic formulas, which do not involve any numerical integration, are derived for the discretized influence functionals of a very extensive assortment of spectral distributions. For Feynman integrals derived using the Trotter splitting and Strang splitting, we present general formulas for the discretized influence functionals in terms of proper integrals of the bath response function. When an analytic expression exists for the bath response function, these integrals can almost always be evaluated analytically. In cases where these proper integrals cannot be integrated analytically, nume…
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