Journal articles: 'Suzuki-Trotter method'

1

Wijesinhe, H. S., and K. A. I. L. Wijewardena Gamalath. "Spin Waves in Two and Three Dimensional Magnetic Materials." International Letters of Chemistry, Physics and Astronomy 49 (April 2015): 35–47. http://dx.doi.org/10.18052/www.scipress.com/ilcpa.49.35.

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The equations of motion for the dynamic properties of spin waves in three dimensions were obtained using Heisenberg model and solved for two and three dimensional lattices analytically up to an exponential operator representation. The second order Suzuki Trotter decomposition method was extended to incorporate second nearest interaction parameters into the numerical solution. Computer based simulations on systems in micro canonical ensembles in constant-energy states were used to check the applicability of this model for two dimensional lattice as well as three dimensional simple cubic and bcc lattices. In the magnon dispersion curves all or most of the spin wave components could be recognized as peaks in the dynamic structure factor presenting the variation of energy transfer with respect to momentum transfer of spin waves. Second order Suzuki Trotter algorithm used conserved the energy.

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Wijesinhe, H. S., and K. A. I. L. Wijewardena Gamalath. "Spin Waves in Two and Three Dimensional Magnetic Materials." International Letters of Chemistry, Physics and Astronomy 49 (April 7, 2015): 35–47. http://dx.doi.org/10.56431/p-7562a7.

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The equations of motion for the dynamic properties of spin waves in three dimensions were obtained using Heisenberg model and solved for two and three dimensional lattices analytically up to an exponential operator representation. The second order Suzuki Trotter decomposition method was extended to incorporate second nearest interaction parameters into the numerical solution. Computer based simulations on systems in micro canonical ensembles in constant-energy states were used to check the applicability of this model for two dimensional lattice as well as three dimensional simple cubic and bcc lattices. In the magnon dispersion curves all or most of the spin wave components could be recognized as peaks in the dynamic structure factor presenting the variation of energy transfer with respect to momentum transfer of spin waves. Second order Suzuki Trotter algorithm used conserved the energy.

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3

Tranter, Andrew, Peter J. Love, Florian Mintert, Nathan Wiebe, and Peter V. Coveney. "Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure." Entropy 21, no. 12 (December 13, 2019): 1218. http://dx.doi.org/10.3390/e21121218.

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Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.

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MIYASHITA, Seiji, and Tota NAKAMURA. "MONTE CARLO STUDIES ON FRUSTRATED QUANTUM SPIN SYSTEMS BY A NEW APPROACH TO THE NEGATIVE-SIGN PROBLEM: TRANSFER-MATRIX MONTE CARLO METHOD." International Journal of Modern Physics C 07, no. 03 (June 1996): 425–31. http://dx.doi.org/10.1142/s0129183196000375.

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A new technique for the negative sign problem in the quantum Monte Carlo method using the Suzuki-Trotter decomposition is introduced. In order to reduce the cancellation between between samples with positive and negative weights, we make use of the transfer matrix method, which has been named the Transfer-Matrix Monte Carlo method. Applications to the Heisenberg antiferromagnet on the ∆-chain and on the kagome lattice, and also to the Kondo lattice system also are given.

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LANDAU, D. P., SHAN-HO TSAI, M. KRECH, and ALEX BUNKER. "IMPROVED SPIN DYNAMICS SIMULATIONS OF MAGNETIC EXCITATIONS." International Journal of Modern Physics C 10, no. 08 (December 1999): 1541–51. http://dx.doi.org/10.1142/s0129183199001327.

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Using Suzuki–Trotter decompositions of exponential operators we describe new algorithms for the numerical integration of the equations of motion for classical spin systems. These techniques conserve spin length exactly and, in special cases, also conserve the energy and maintain time reversibility. We investigate integration schemes of up to eighth order and show that these new algorithms can be used with much larger time steps than a well established predictor–corrector method. These methods may lead to a substantial speedup of spin dynamics simulations, however, the choice of which order method to use is not always straightforward.

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Henneke, Felix, and Manfred Liebmann. "A generalized Suzuki–Trotter type method in optimal control of coupled Schrödinger equations." Computing and Visualization in Science 17, no. 6 (December 2015): 277–93. http://dx.doi.org/10.1007/s00791-016-0266-2.

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Ariasoca, Thomas Aquino, Sholihun, and Iman Santoso. "Trotter-Suzuki-time propagation method for calculating the density of states of disordered graphene." Computational Materials Science 156 (January 2019): 434–40. http://dx.doi.org/10.1016/j.commatsci.2018.10.016.

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Hutchinson, J., J. P. Keating, and F. Mezzadri. "On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems." Advances in Mathematical Physics 2015 (2015): 1–18. http://dx.doi.org/10.1155/2015/652026.

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We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.

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9

Paraniak, Mikołaj M., and Berthold-Georg Englert. "Quantum Dynamical Simulation of a Transversal Stern–Gerlach Interferometer." Symmetry 13, no. 9 (September 8, 2021): 1660. http://dx.doi.org/10.3390/sym13091660.

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Originally conceived as a thought experiment, an apparatus consisting of two Stern–Gerlach apparatuses joined in an inverted manner touched on the fundamental question of the reversibility of evolution in quantum mechanics. Theoretical analysis showed that uniting the two partial beams requires an extreme level of experimental control, making the proposal in its original form unrealizable in practice. In this work, we revisit the above question in a numerical study concerning the possibility of partial-beam recombination in a spin-coherent manner. Using the Suzuki–Trotter numerical method of wave propagation and a configurable, approximation-free magnetic field, a simulation of a transversal Stern–Gerlach interferometer under ideal conditions is performed. The result confirms what has long been hinted at by theoretical analyses: the transversal Stern–Gerlach interferometer quantum dynamics is fundamentally irreversible even when perfect control of the associated magnetic fields and beams is assumed.

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10

Wijesinhe, H. S., and K. A. I. L. Wijewardena Gamalath. "Spin Waves in One Dimensional Magnetic Material." International Letters of Chemistry, Physics and Astronomy 47 (February 2015): 24–39. http://dx.doi.org/10.18052/www.scipress.com/ilcpa.47.24.

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Using Heisenberg model, the equations of motion for the dynamic properties of spin waves in three dimensions were obtained and solved analytically up to an exponential operator representation. Second order Suzuki Trotter decomposition method with evolution operator solution was applied to obtain the numerical solutions by making it closer to real spin systems. Computer based simulations on systems in micro canonical ensembles in constant-energy states were used to check the applicability of this model for one dimensional lattice by investigating the occurrence, temperature dependence and spin-spin interaction dependence of the spin waves. A visualization technique was used to show the existence of many spin wave components below the Curie temperature of the system. In the magnon dispersion curves all or most of the spin wave components could be recognized as peaks in the dynamic structure factor. Energy conservation of the algorithm is also shown.

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11

Wijesinhe, H. S., and K. A. I. L. Wijewardena Gamalath. "Spin Waves in One Dimensional Magnetic Material." International Letters of Chemistry, Physics and Astronomy 47 (February 24, 2015): 24–39. http://dx.doi.org/10.56431/p-0ssdan.

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Using Heisenberg model, the equations of motion for the dynamic properties of spin waves in three dimensions were obtained and solved analytically up to an exponential operator representation. Second order Suzuki Trotter decomposition method with evolution operator solution was applied to obtain the numerical solutions by making it closer to real spin systems. Computer based simulations on systems in micro canonical ensembles in constant-energy states were used to check the applicability of this model for one dimensional lattice by investigating the occurrence, temperature dependence and spin-spin interaction dependence of the spin waves. A visualization technique was used to show the existence of many spin wave components below the Curie temperature of the system. In the magnon dispersion curves all or most of the spin wave components could be recognized as peaks in the dynamic structure factor. Energy conservation of the algorithm is also shown.

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12

PREDESCU, CRISTIAN. "SPATIALLY-DISCRETIZED HIGH-TEMPERATURE APPROXIMATIONS AND THEIR O(N) IMPLEMENTATION ON A GRID." Journal of Theoretical and Computational Chemistry 05, no. 02 (June 2006): 255–80. http://dx.doi.org/10.1142/s0219633606002246.

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We consider the problem of performing imaginary-time propagation of wavefunctions on a grid. We demonstrate that spatially-continuous high-temperature approximations can be discretized in such a way that their convergence order is preserved. Requirements of minimal computational work and reutilization of data then uniquely determine the optimal grid, quadrature technique, and propagation method. It is shown that the optimal propagation technique is O(N), with respect to the grid size. The grid technique is utilized to compare the Monte Carlo efficiency of the Trotter–Suzuki approximation against a recently introduced fourth-order high-temperature approximation, while circumventing the issue of statistical noise, which usually prevents such comparisons from being carried out. We document the appearance of a systematic bias in the Monte Carlo estimators that involve temperature differentiation of the density matrix, bias that is due to the dependence of the eigenvalues on the inverse temperature. This bias is negotiated more successfully by the short-time approximations having higher convergence order, which leads to non-trivial computational savings.

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13

Nii, N. "A New Method to Solve the Diffusion Equations under Approximation B in Electron-Photon Cascade Theory: By Using Suzuki-Trotter Formula." Progress of Theoretical Physics 87, no. 4 (April 1, 1992): 891–910. http://dx.doi.org/10.1143/ptp/87.4.891.

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14

Figueroa-Manrique, S., and K. Rodríguez-Ramírez. "Towards a quantum Monte Carlo for lattice systems." Suplemento de la Revista Mexicana de Física 1, no. 3 (August 22, 2020): 23–30. http://dx.doi.org/10.31349/suplrevmexfis.1.3.23.

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In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.

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15

Kivlichan, Ian D., Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, et al. "Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization." Quantum 4 (July 16, 2020): 296. http://dx.doi.org/10.22331/q-2020-07-16-296.

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Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with N<105 fermionic modes can be performed with roughly O(1) and O(N2) T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.

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16

Lötstedt, Erik, Lidong Wang, Ryuhei Yoshida, Youyuan Zhang, and Kaoru Yamanouchi. "Error-mitigated quantum computing of Heisenberg spin chain dynamics." Physica Scripta 98, no. 3 (February 24, 2023): 035111. http://dx.doi.org/10.1088/1402-4896/acbcac.

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Abstract We simulate the time-dependent dynamics of a three-site spin chain described by the Heisenberg XXX Hamiltonian. The quantum circuit representing the time-dependent wave function is constructed using the Suzuki-Trotter approximation, and is executed on the quantum computer ibm_kawasaki. At each time step, the density matrix of the three-qubit state is reconstructed by state tomography. By applying four different mitigation methods, Clifford data regression, Pauli twirling, density matrix purification, and density matrix orthogonalization, we demonstrate that accurate time-dependent populations and density matrices can be calculated on noisy superconducting-qubit type quantum computers.

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17

Herasymenko, Y., and T. E. O'Brien. "A diagrammatic approach to variational quantum ansatz construction." Quantum 5 (December 2, 2021): 596. http://dx.doi.org/10.22331/q-2021-12-02-596.

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Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.

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Mansuroglu, Refik, Timo Eckstein, Ludwig Nützel, Samuel A. Wilkinson, and Michael J. Hartmann. "Variational Hamiltonian simulation for translational invariant systems via classical pre-processing." Quantum Science and Technology, January 10, 2023. http://dx.doi.org/10.1088/2058-9565/acb1d0.

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Abstract The simulation of time evolution of large quantum systems is a classically challenging and in general intractable task, making it a promising application for quantum computation. A Trotter-Suzuki approximation yields an implementation thereof, where a higher approximation accuracy can be traded for an increased gate count. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict eﬃcient quantum circuits for time evolution of translationally invariant quantum systems. Our strategy can improve upon the Trotter-Suzuki accuracy by several orders of magnitude. It translates into a reduction in gate count and hence gain in overall ﬁdelity at the same algorithmic accuracy. This is important in NISQ-applications where the ﬁdelity of the output state decays exponentially with the number of gates. The performance advantage of our classical assisted strategy can be extended to open boundaries with translational symmetry in the bulk. We can extrapolate our method to beyond classically simulatable system sizes, maintaining its total ﬁdelity advantage over a Trotter-Suzuki approximation making it an interesting candidate for beyond classical time evolution.

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Wang, J., W. Pan, and D. Y. Sun. "Efficient world-line-based quantum Monte Carlo method without Hubbard–Stratonovich transformation." Scientific Reports 12, no. 1 (May 17, 2022). http://dx.doi.org/10.1038/s41598-022-12259-5.

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AbstractBy precisely writing down the matrix element of the local Boltzmann operator ($${\mathrm{e}}^{-\tau h}$$ e - τ h , where $$h$$ 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.

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Bakthavatchalam, Tamil Arasan, Suriyadeepan Ramamoorthy, Malaikannan Sankarasubbu, Radha Ramaswamy, and Vijayalakshmi Sethuraman. "Bayesian Optimization of Bose-Einstein Condensates." Scientific Reports 11, no. 1 (March 3, 2021). http://dx.doi.org/10.1038/s41598-021-84336-0.

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AbstractMachine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction ($$\frac{1}{50}$$ 1 50 th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions $$36 \times $$ 36 × faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems.

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Journal articles on the topic 'Suzuki-Trotter method'

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