Trotter Approximation Research Articles

Probing Quantum Efficiency: Exploring System Hardness in Electronic Ground State Energy Estimation.

We consider the question of how correlated the system hardness is between classical algorithms of electronic structure theory in ground state estimation and quantum algorithms. To define the system hardness for classical algorithms, we employ empirical criterion based on the deviation of electronic energies produced by coupled cluster and configuration interaction methods from the exact ones along multiple bonds dissociation in a set of molecular systems. For quantum algorithms, we have selected the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) methods. As characteristics of the system hardness for quantum methods, we analyzed circuit depths for the state preparation, the number of quantum measurements needed for the energy expectation value, and various cost characteristics for the Hamiltonian encodings via Trotter approximation and linear combination of unitaries (LCU). Our results show that the quantum resource requirements are mostly unaffected by classical hardness, with the only exception being the state preparation part, which contributes to both VQE and QPE algorithm costs. However, there are clear indications that constructing the initial state with a significant overlap with the true ground state is easier than obtaining the state with an energy expectation value within chemical precision. These results support optimism regarding the identification of a molecular system where a quantum algorithm excels over its classical counterpart, as quantum methods can maintain efficiency in classically challenging systems.

Comparison of current quantum devices for quantum computing of Heisenberg spin chain dynamics

We simulate the time-dependent dynamics of a chain of three interacting spins described by the Heisenberg XXX Hamiltonian using two types of different quantum computers: Quantinuum H1-1, a trapped-ion quantum computer, and ibm_prague, a superconducting qubit type quantum computer. The time evolution of the wave function is accomplished by the Suzuki–Trotter approximation. We find that very accurate results are obtained using H1-1 after applying error mitigation by renormalization of the time-dependent spin-state populations as well as using ibm_prague after renormalization, dynamical decoupling, Pauli twirling, and measurement error mitigation of the populations.

Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation

Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.

Rate of Convergence in Trotter's Approximation Theorem and Its Applications

The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate of convergence in Trotter’s theorem under some constraints. In the present paper, a new rate of convergence in Trotter’s theorem in full generality is given. Moreover, we see that this rate of convergence works well to obtain quantitative estimates for some limit theorems in probability theory.

On the Convergence of a Novel Time-Slicing Approximation Scheme for Feynman Path Integrals

Abstract In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman’s path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. The corresponding Schrödinger propagator belongs to the class of generalized metaplectic operators, a fact that naturally motivates the introduction of a manageable time-slicing approximation scheme consisting of operators of the same type. By leveraging on this design and techniques of wave packet analysis we are able to prove several convergence results with precise rates in terms of the mesh size of the time slicing subdivision, even stronger then those that can be achieved under the same assumptions using the standard Trotter approximation scheme. In particular, we prove convergence in the norm operator topology in $L^2$, as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.

Quantum algorithms for quantum dynamics: A performance study on the spin-boson model

Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.

Edgeworth Expansions for Centered Random Walks on Covering Graphs of Polynomial Volume Growth

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends on not only geometric features of the underlying graphs but also the modified harmonic embedding of the graph into a certain nilpotent Lie group. Moreover, we apply the rate of convergence in Trotter's approximation theorem to establish the Berry-Esseen type bound for the random walks.

Jastrow-type Decomposition in Quantum Chemistry for Low-Depth Quantum Circuits.

We propose an efficient O(N2)-parameter ansatz that consists of a sequence of exponential operators, each of which is a unitary variant of Neuscamman's cluster Jastrow operator. The ansatz can also be derived as a decomposition of T2 amplitudes of the unitary coupled cluster with generalized singles and doubles, which gives a near full-CI energy. The proposed ansatz therefore can reproduce the uCCGSD energy, or rather will reach the exact full-CI energy because of the exponential operator product form. Because the cluster Jastrow operators are expressed by a product of number operators and the derived Pauli operator products, namely, the Jordan-Wigner strings, are all commutative, it does not require the Trotter approximation to implement to a quantum circuit and should be a good candidate for the variational quantum eigensolver algorithm of a near-term quantum computer. The accuracy of the ansatz was examined for dissociation of a nitrogen dimer, and compared with other existing O(N2)-parameter ansatzs. Not only the original ansatzs defined in the second-quantization form but also their Trotter-splitting variants, in which the cluster amplitudes are optimized to minimize the energy obtained with a few, typically single, Trotter steps, were examined by quantum circuit simulators.

Bravyi-Kitaev Superfast simulation of electronic structure on a quantum computer.

Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of the BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of a hydrogen molecule we have compared BKSF, Jordan-Wigner, and Bravyi-Kitaev transforms under the Trotter approximation. The gate count to implement BKSF is lower than Jordan-Wigner but higher than Bravyi-Kitaev. We considered different orderings of the exponentiated terms and found lower Trotter errors than the previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to the further study of the BKSF algorithm for quantum simulation.

Combination of the pair density approximation and the Takahashi–Imada approximation for path integral Monte Carlo simulations

We construct an accurate imaginary time propagator for path integral Monte Carlo simulations for heterogeneous systems consisting of a mixture of atoms and molecules. We combine the pair density approximation, which is highly accurate but feasible only for the isotropic interactions between atoms, with the Takahashi–Imada approximation for general interactions. We present finite temperature simulations results for energy and structure of molecules–helium clusters X4He20 (XHCCH and LiH) which show a marked improvement over the Trotter approximation which has a 2nd-order time step bias. We show that the 4th-order corrections of the Takahashi–Imada approximation can also be applied perturbatively to a 2nd-order simulation.

Linearized auxiliary fields Monte Carlo technique: Efficient sampling of the fermion sign

We introduce a method that combines the power of both the lattice Green' function Monte Carlo (LGFMC) with the auxiliary field quantum Monte Carlo (AFQMC) techniques, and allows us to compute exact ground-state properties of the Hubbard model for $U\ensuremath{\lesssim}4t$ on finite clusters. Thanks to LGFMC, one obtains unbiased zero temperature results, not affected by the so-called Trotter approximation of the imaginary time propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$. At the same time, the AFQMC formalism yields a remarkably fast convergence in $\ensuremath{\tau}$ before the fermion sign problem becomes prohibitive. As an application we report ground-state energies of the Hubbard model at $U/t=4$ with up to 100 sites.

Gamma-type operators and the Black–Scholes semigroup

We study Gamma-type operators from the analytic and probabilistic viewpoint in the setting of weighted continuous function spaces and estimate the rate of convergence of their iterates towards their limiting semigroup, providing, in this way, a quantitative version of the classical Trotter approximation theorem. The semigroup itself has some interest, since it is generated by the Black–Scholes operator, frequently occurring in the theory of option pricing in mathematical finance.

Finite Trotter Approximation to the Averaged Mean Square Distance in the Anderson Model

We prove that a finite Trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice ℤd exhibits at most a diffusive behavior in dimensions d≥3 as long as the Fourier transform of the single-site probability, \(\hat{\mu }\), is in L2(ℝ).

A quantitative version of Trotter's approximation theorem

A quantitative version, based on modified K-functionals, of the classical Trotter's theorem concerning the approximation of C 0 -semigroups is presented. The result is applied to the study of the degree of convergence of the iterated Bernstein operators on the N-dimensional simplex to their limiting semigroup.

A ‘path-by-path’ monotone extrapolation sequence for Feynman path integral calculations of quantum mechanical free energies

Feynman path integral methods based on the Trotter approximation represent paths by a set of P discrete points. We prove that the M-point partition function is an upper bound of the P-point one if M is a divisor of P. Also for this case, we show that, during calculations using P-point paths, it is possible – at negligible additional cost – to obtain M-point estimators of the partition function that, for each individual path, converge monotonically. This permits accurate extrapolation to infinite P, which greatly improves the accuracy of calculations of thermodynamic quantities.

Quantum Statistics of Multidimensional Nonlinear Oscillators

This paper presents two perturbation approaches to calculating the quantum statistics of multidimensional nonlinear oscillators in a simple, economic way. These are the power series expansion formalism and the system-specific split operator method. The former involves a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Both approaches are outlined with some improvements, and their possible limitations are discussed. The relative efficacy of the approaches is tested on two typical models that are often used by researchers as benchmarks for different numerical methods. Although no system-specific reference system is involved in the power series expansion of the Boltzmann operator, it quite accurately describes the quantum statistics in the entire temperature range, even though the coupling strength is rather large. Yet another important advantage of this approach is that it is essentially analytical, and therefore its numerical implementation does not require any computational effort. This makes the power series expansion technique particularly attractive for treating many-body problems. In contrast, the split operator method is found to be efficient only in the limit of separable motion. Otherwise, this approach results in a less than optimally efficient approximation for the density matrix, or even one which is worse than the standard Trotter approximation.

Time dependent transport equation with continuous solutions (I) convergence of discrete ordinates approximation

Abstract The time dependent transport equation with periodic boundary conditions is discussed in the space of continuous functions. A suitable subspace is constructed and it is shown that the corresponding operator A generates a strongly continuous semigroup T F (t) in F = D(A). By virtue of Trotter's approximation theorem of strongly continuous semigroups, the convergence of the discrete ordinates approximation for the time dependent transport equation is rigorously obtained.

Improved short-time propagator for repulsive inverse-power-law potentials

Recently Makri and Miller have shown that the standard Trotter approximation for the short-time propagator K( x 0, x, t) can be greatly improved by writing K( x 0, x, t) in exponential form and then systematically expanding the exponent in powers of the time t. The Makri-Miller method is based on the assumption that in the semiclassical limit, there is a unique classical path which connects the initial point x 0 to the final point x in the time t. We show that for potentials such as repulsive inverse power-laws, which admit two classical paths connecting the given end points in the given time, further improvement can be made by expanding the exponents of the two terms corresponding to the two paths. Numerical tests for the inverse square potential, for which analytic results are available for comparison, show that such further improvements can be substantial. The tests are carried out for real, imaginary, and complex times.

Calculation of specific heat and susceptibilities with the use of the Trotter approximation

The Trotter approximation has been used extensively in quantum Monte Carlo simulations. We consider here different ways of calculating the general zero-field susceptibility \ensuremath{\chi} and the specific heat C when a Trotter approximation is used, deriving analytically the error dependence on the imaginary-time increment \ensuremath{\Delta}\ensuremath{\tau} as well as the low-temperature behavior. We find that certain definitions of \ensuremath{\chi} and C exhibit spurious divergences at low temperatures, and suggest the most appropriate ways to extract these quantities. We test our general predictions on two models, and discuss the implications of our results for numerical simulations.

Quantum Monte Carlo and transfer-matrix calculations for one-dimensional easy-plane ferromagnets.

We have applied previously used quantum Monte Carlo (QMC) techniques to obtain numerically the thermodynamics of two well-studied quasi-one-dimensional (1D) easy-plane ferromagnetic models, in the presence of an applied magnetic field in the easy plane. The checkerboard decomposition form of the Trotter approximation to the partition function has been used. Internal energy, specific heat, magnetization, and susceptibility have been obtained for model Hamiltonians believed appropriate for spin S=(1/2) [(${\mathrm{C}}_{6}$${\mathrm{H}}_{11}$${\mathrm{NH}}_{3}$)${\mathrm{CuBr}}_{3}$ (CHAB)] and S=1 (${\mathrm{CsNiF}}_{3}$), in temperature and field ranges where classical theories have predicted solitonlike kink excitations. The S=(1/2) QMC calculations are verified and superseded by a numerically exact quantum transfer-matrix (QTM) technique. Results for the temperature dependence of the peak in the specific heat versus field are compared with available experimental results. For the model applicable to CHAB, it is found that there is no value of the easy-plane anisotropy parameter from 4% to 10% for which the QTM calculation can adequately reproduce the experimentally obtained peak height and position. On the other hand, the QMC results for the model assumed for ${\mathrm{CsNiF}}_{3}$ do roughly reproduce the temperature dependence of the experimental peak positions, but not the peak heights. However, statistical errors present in our QMC data are large, and a better method is still needed for computing the quantum statistical mechanics of S=1 systems.