Trotter Formula Research Articles

The constant of point–line incidence constructions

We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(P,L)≥(c+o(1))|P|2/3|L|2/3, with c≈1.27. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erdős's construction.

State-dependent Trotter limits and their approximations

This paper gives sufficient conditions to conclude the validity of digital simulations involving the Trotter product formula for finite discretizations of continuous operators. Essentially, it depends on the state-dependent Trotter error, for which it establishes explicit bounds that are also of independent interest.

Simulating time evolution on distributed quantum computers

We study a variation of the Trotter-Suzuki decomposition, in which a Hamiltonian exponential is approximated by an ordered product of two-qubit operator exponentials such that the Trotter step size is enhanced for a small number of terms. Such decomposition directly reflects hardware constraints of distributed quantum computers, where operations on monolithic quantum devices are fast compared to entanglement distribution across separate nodes using interconnects. We simulate non-equilibrium dynamics of transverse-field Ising and XY spin chain models and investigate the impact of locally increased Trotter step sizes that are associated with an increasingly sparse use of the quantum interconnect. We find that the overall quality of the approximation depends smoothly on the local sparsity and that the proliferation of local errors is slow. As a consequence, we show that fast local operations on monolithic devices can be leveraged to obtain an overall improved result fidelity even on distributed quantum computers where the use of interconnects is costly.

Stochastic quantum Krylov protocol with double-factorized Hamiltonians

We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator ${e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\tau}}$ as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm on near-term quantum devices, we provide numerical benchmarks for a variety of molecular systems with circuit-based state-vector simulators including the effects of sampling noise, achieving ground-state energy errors of less than $1 \mathrm{kcal} {\mathrm{mol}}^{\ensuremath{-}1}$ with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.

Fast classical simulation of Hamiltonian dynamics by simultaneous diagonalization using Clifford transformation with parallel computation

Simulating quantum many-body dynamics is important both for fundamental understanding of physics and practical applications for quantum information processing. Therefore, classical simulation methods have been developed so far. Specifically, the Trotter-Suzuki decomposition can analyze a highly complex quantum dynamics, if the number of qubits is sufficiently small so that main memory can store the state vector. However, simulation of quantum dynamics via Trotter-Suzuki decomposition requires a huge number of steps, each of which accesses the state vector, and hence the simulation time becomes impractically long. To settle this issue, we propose a technique to accelerate simulation of quantum dynamics via simultaneous diagonalization of mutually commuting Pauli groups, which is also attracting a lot of attention to reduce the measurement overheads in quantum algorithms. We group the Hamiltonian into mutually commuting Pauli strings, and each of them is diagonalized in the computational basis via a Clifford transformation. Since diagonal operators are applied on the state vector simultaneously with minimum memory access, this method successfully uses performance of highly parallel processors such as Graphics Processing Units (GPU). Compared to both CPU and GPU implementations using fast quantum circuit simulator “qulacs,” the numerical experiments have shown that our method provides a few tens of times acceleration.

AN INCIDENCE RESULT FOR WELL-SPACED ATOMS IN ALL DIMENSIONS

AbstractWe prove an incidence result counting the k-rich $\delta $ -tubes induced by a well-spaced set of $\delta $ -atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi–Trotter theorem and holds in all dimensions $d \geq 2$ . We also prove an analogue of Beck’s theorem for $\delta $ -atoms and $\delta $ -tubes as an application of our result.

Operator-norm Trotter product formula on Banach spaces

Proof of the operator-norm convergent Trotter product formula on a Banach space is unexpectedly elaborate and a few of known results are based on assumption that at least one of the semigroups involved into this formula is holomorphic. Here we present an example of the operator-norm convergent Trotter product formula on a Banach space, where this condition is relaxed to demand that involved semigroups are contractive. Bibliography: 21 titles.

Operator-norm Trotter product formula on Banach spaces

Proof of the operator-norm convergent Trotter product formula on a Banach space is unexpectedly elaborate and a few of known results are based on assumption that at least one of the semigroups involved into this formula is holomorphic. Here we present an example of the operator-norm convergent Trotter product formula on a Banach space, where this condition is relaxed to demand that involved semigroups are contractive.

Towards simulating time evolution of specific quantum many-body system by lower counts of quantum gates

In the noisy intermediate-scale quantum (NISQ) era, quantum computers have become important tools to simulate the real-time evolution of strong correlated many-body systems. The general scheme of designing quantum circuits to simulate the quantum dynamics is based on the Trotter-Suzuki decomposition technique, which has its limitation because the accuracy of evolution process depends on the size of trotter steps when the evolution operator is decomposed as quantum gates. Going beyond the limitation of the Trotter-Suzuki scheme, we design a parameterized quantum circuit named α-circuit with simple determined size and the only one control parameter θ to simulate the real-time evolution of the specific XXX Heisenberg model with the specific initial state . The α-circuit can accurately generate the time-evolution results by tuning parameter θ, which means the circuit can also be regarded as a good state preparation machine (SPM).

The Trotter product formula for nonlinear Fokker–Planck flows

One proves herein that the flow S(t), generated by the nonlinear Fokker–Planck equation ρt−Δβ(ρ)+div(a(ρ)ρ)=0 in (0,∞)×Rd, is expressed by the Trotter product formulaS(t)ρ0=limn→∞⁡(SA1(tn)SA2(tn))nρ0 inL1(Rd), where SA1(t) is the flow (continuous semigroup) generated in L1(Rd) by the nonlinear diffusion operator A1(ρ)=−Δβ(ρ), while SA2(t) is that generated in L1(Rd) by the conservation law operator A2(ρ)=div(a(ρ)ρ) defined in the entropy sense. As an application, one obtains a split-product formula for the McKean–Vlasov stochastic differential equation associated with the Fokker–Planck equation.

Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules.

In this work, we investigate the capability of known quantum computing algorithms for fault-tolerant quantum computing to simulate the laser-driven electron dynamics of excitation and ionization processes in small molecules such as lithium hydride, which can be benchmarked against the most accurate time-dependent full configuration interaction (TD-FCI) calculations. The conventional TD-FCI wave packet propagation is reproduced using the Jordan-Wigner transformation for wave function and operators and the Trotter product formula for expressing the propagator. In addition, the time-dependent dipole moment, as an example of a time-dependent expectation value, is calculated using the Hadamard test. To include non-Hermitian operators in the ionization dynamics, a similar approach to the quantum imaginary time evolution (QITE) algorithm is employed to translate the propagator, including a complex absorption potential, into quantum gates. The computations are executed on a quantum computer simulator. By construction, all quantum computer algorithms, except for the QITE algorithm used only for ionization but not for excitation dynamics, would scale polynomially on a quantum computer with fully entangled qubits. In contrast, TD-FCI scales exponentially. Hence, quantum computation holds promises for substantial progress in the understanding of electron dynamics of excitation processes in increasingly large molecular systems, as has already been witnessed in electronic structure theory.

Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method

First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited by the order of the differences in L1 space. Therefore, how to design a computational scheme for the first-order linear IDEs with computational efficiency becomes an urgent problem to be solved. To this end, a polynomial approximation scheme based on the shifted Legendre spectral collocation method is proposed in this paper. First, we transform the first-order linear IDEs into an Cauchy problem for consideration. Second, by decomposing the system operator, we rewrite the Cauchy problem into a more general form for approximating. Then, by using the shifted Legendre spectral collocation method, we construct a computational scheme and write it into an abstract version. The convergence of the scheme is proven in the sense of L1-norm by employing Trotter-Kato theorem. At the end of this paper, we summarize the usage of the scheme into an algorithm and present some numerical examples to show the applications of the algorithm.

Viscoelastic and viscoacoustic modelling using the Lie product formula

ABSTRACTWe present efficient and accurate modelling of seismic wavefields in anelastic media. We use a first‐order viscoelastic wave equation based on the generalized Robertsson's formulation. In our work, viscoacoustic and viscoelastic wave equations use the standard linear solid mechanism. To numerically solve the first‐order wave equation, we employed a scheme derived from the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices, and each exponential matrix term is approximated by the Taylor series expansion. The accuracy of the proposed scheme is evaluated by comparison with the analytical solution for a homogeneous medium. We also present simulations of some geological models with different structural complexities, whose results confirm the accuracy of the proposed scheme and illustrate the attenuation effect on the seismic energy during its propagation in the medium. Our results demonstrate that the numerical scheme employed can be used to extrapolate wavefields stably for even larger time steps than those usually used by traditional finite‐difference schemes.

Time evolution of an infinite projected entangled pair state: A gradient tensor update in the tangent space

Time evolution of an infinite 2D many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, $D$, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures -- like the one used in the full update (FU), the simple update (SU), and their intermediate neighborhood tensor update (NTU) -- and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update (GTU) is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.

An Incidence Estimate and a Furstenberg Type Estimate for Tubes in mathbb {R}^2

We study the delta -discretized Szemerédi–Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both problems, we construct sharp examples that share common features.

One bound to rule them all: from Adiabatic to Zeno

We derive a universal nonperturbative bound on the distance between unitary evolutions generated by time-dependent Hamiltonians in terms of the difference of their integral actions. We apply our result to provide explicit error bounds for the rotating-wave approximation and generalize it beyond the qubit case. We discuss the error of the rotating-wave approximation over long time and in the presence of time-dependent amplitude modulation. We also show how our universal bound can be used to derive and to generalize other known theorems such as the strong-coupling limit, the adiabatic theorem, and product formulas, which are relevant to quantum-control strategies including the Zeno control and the dynamical decoupling. Finally, we prove generalized versions of the Trotter product formula, extending its validity beyond the standard scaling assumption.

First-Order Trotter Error from a Second-Order Perspective.

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.

Efficient world-line-based quantum Monte Carlo method without Hubbard–Stratonovich transformation

By precisely writing down the matrix element of the local Boltzmann operator ({mathrm{e}}^{-tau h}, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard–Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.

The Multivariate Schwartz--Zippel Lemma

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \cdots \times S_m \subset \mathbb{C}^n$ be the multigrid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of the combinatorial nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo--Lipton--Schwartz--Zippel lemma holds, except for a special family of polynomials that we call $\lambda$-reducible. This yields a simultaneous generalization of the Szemerédi--Trotter theorem and the Schwartz--Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $\lambda$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a Cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $\lambda$-reducible polynomials, which we leave as an open problem.

Projection Method for Droplet Dynamics on Groove-Textured Surface with Merging and Splitting

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.