Faster Quantum Simulations Unlock New Materials and Drug Discoveries

Scientists tackling the simulation of quantum many-body systems face a persistent challenge due to the exponential growth of computational complexity. Belal Abouraya from the German University in Cairo, Jirawat Saiphet from the University of Tübingen, and Fedor Jelezko et al. present a new method to improve the efficiency of matrix product states (MPS), a key technique for modelling one-dimensional quantum systems. Their research introduces a streamlined approach to simulating time-dependent Hamiltonians, achieving second-order convergence, a significant improvement over standard first-order methods. Demonstrating this advancement with simulations of nitrogen-vacancy colour centres in diamond, the team shows a reduction in average error by a factor of approximately 1000, potentially enabling more accurate and scalable modelling for future quantum technologies.

High-order quadrature improves time-dependent quantum many-body simulations

Researchers have developed a new method for simulating the complex behaviour of quantum many-body systems, addressing a long-standing challenge in physics and quantum information science. These systems are notoriously difficult to model due to the exponential growth in computational requirements as the system size increases.
This work introduces an efficient augmentation to existing matrix product state (MPS) algorithms, enabling more accurate and faster simulations of time-dependent quantum dynamics. The proposed technique achieves second-order convergence, a significant improvement over the first-order convergence of standard methods currently employed.

The breakthrough centres on replacing the instantaneous Hamiltonian used in conventional MPS time-evolution solvers with a carefully calculated average Hamiltonian derived from a high-order quadrature rule. This approach, inspired by classical numerical integration techniques like Simpson’s rule, allows for a more precise approximation of the time evolution operator over short time intervals.

Crucially, the method requires minimal alterations to established tensor network numerical libraries, facilitating its immediate implementation and widespread adoption. Numerical simulations demonstrate substantial performance gains, with average error reductions of up to 1000 observed in systems of few nitrogen-vacancy (NV) centres in diamond.

This advancement has direct implications for the development of practical and scalable quantum technologies. The research team applied their method to simulate the dynamics of a chain of single spins associated with NV colour centres, materials possessing remarkable physical properties suitable for quantum registers and sensors operating at ambient temperature.

For systems comprising a few NV centres, the new method improved average error by a factor of approximately 1000 for moderate time step sizes. Furthermore, simulations with larger systems of NV centres revealed a consistent improvement in average error, reaching a factor of around 50. This work establishes a pathway towards efficient simulation of quantum many-body systems influenced by time-dependent Hamiltonians, opening new avenues for exploring complex quantum phenomena and designing advanced quantum devices.

Second-order time evolution via averaged Hamiltonians within matrix product state algorithms

Matrix product states (MPS) serve as the foundational technique for simulating one-dimensional quantum many-body systems. This work introduces a refined method to enhance MPS algorithms for simulating the dynamics of time-dependent Hamiltonians, achieving second-order convergence compared to the first-order convergence of standard approaches.

The core innovation lies in substituting the instantaneous Hamiltonian within fixed-order MPS time-steppers with a carefully calculated average Hamiltonian, derived using a high-order quadrature rule based on Simpson’s rules. This substitution provides a more accurate approximation of the time evolution operator over short time intervals.

Implementation of this method requires minimal alteration to existing tensor network numerical libraries, facilitating seamless integration with established MPS time evolution solvers such as time-evolving block decimation (TEBD). Numerical simulations were performed on a chain of single spins representing nitrogen-vacancy (NV) color centers in diamonds, a system with potential for scalable quantum technologies.

The dynamics were calculated by computing the time-evolution of the quantum state using the modified MPS algorithm and the evolution operator, exp −i ∫Δt 0 H(τ)dτ, where Δt represents a sufficiently small time-step. Analysis revealed a substantial improvement in accuracy, with the average error reduced by a factor of approximately 1000 for systems comprising a few NV centers and moderate step sizes.

For larger systems containing approximately 50 NV centers, the method consistently improved the average error by a factor of around 50. These results demonstrate the efficacy of the proposed approach in efficiently simulating quantum many-body systems influenced by time-dependent Hamiltonians, paving the way for advancements in quantum control and spectroscopy.

Improved time-dependent Hamiltonian simulations using modified matrix product state methods

Simulating the dynamics of a chain of single spins associated with nitrogen-vacancy colour centres in diamonds, the research demonstrates an average error improvement by a factor of approximately 1000 for systems with few NV centres, achieved using moderate step sizes. Numerical simulations were performed with parameters including a coupling parameter g(3) of 53 KHz and g(N) of 100 KHz, alongside magnetic field parameters B(3) z,1 of 42.82 G, B(3) z,2 of 88.31 G, and B(3) z,3 of 82.88 G.

The resonant frequencies were set to ω(3) of 2.797GHz and ω(N) of 2.75GHz, with a gamma e value of -28.025GHz/T and a zeta value of 0, and a Di value of 2.87GHz. The study employed the sesolve numerical solver from the QuTiP Python library, maintaining maximum absolute and relative errors of 10−12 and 10−10 respectively to ensure accurate error estimations.

The work introduces a modification to commonly used matrix product state (MPS) time-steppers for time-dependent Hamiltonians, replacing the instantaneous Hamiltonian with a Simpson-rule average across each step. This averaging restores the native third-order local accuracy of third-order integrators, such as second-order time-evolving block decimation (TEBD) and matrix product operator with WII (WII).

For a system of three NV centres, the Simpson stepper exhibited improved error rates compared to the Riemann stepper, with only a slight increase in runtime attributable to the increased number of function evaluations. Analysis of system scalability revealed that for varying system sizes, the Simpson stepper consistently yielded lower propagation error than the standard Riemann substitution, while incurring a modest increase in runtime.

Specifically, the error ratio, defined as the average error from the Riemann stepper divided by that of the Simpson stepper, was evaluated across different system sizes. Results indicate that the Simpson stepper’s performance holds for larger systems, demonstrating better error rates across all sizes tested. The maximum bond dimension was maintained at 3 for three NV centres and 16 for N NV centres, ensuring truncation errors remained negligible during simulations.

Second-order convergence enhances time evolution of quantum many-body systems

Researchers have developed a refined method for simulating the dynamics of quantum many-body systems, addressing a longstanding challenge in physics and quantum information science. This advancement builds upon existing tensor network algorithms, specifically matrix product states, to improve both the accuracy and convergence rate of simulations involving time-dependent Hamiltonians.

The proposed technique achieves second-order convergence, a notable improvement over the first-order convergence of standard methods currently in use. Application of this method to a chain of nitrogen-vacancy (NV) centers in diamonds demonstrated a substantial reduction in error, approximately a factor of 1000, for systems with a limited number of centers and moderate step sizes.

This model is relevant to the development of practical and scalable quantum technologies. The improvement stems from replacing the instantaneous Hamiltonian within a single-step propagator with a Simpson-rule average of the Hamiltonian across the step, restoring third-order local accuracy to integrators.

Numerical results confirm that the Simpson stepper consistently yields lower propagation errors than the standard Riemann substitution, with only a modest increase in computational time due to the increased number of function evaluations. The authors acknowledge that the method relies on the existence and boundedness of the first four derivatives of the Hamiltonian.

Simulations were conducted with a maximum bond dimension of 3 and 16 for systems of three and numerous NV centers, respectively, to minimise truncation error. Future research may focus on extending the method to larger systems and exploring its performance with more complex Hamiltonians, potentially unlocking more efficient simulations of quantum many-body systems under time-dependent influences and furthering the development of quantum technologies.