Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry

Abstract

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization, and thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely.

50 pages, 9 figures, 3 tables

References in corpus (39)

• Efficient simulation of one-dimensional quantum many-body systems
• Quantum computational chemistry
• Quantum Chemistry in the Age of Quantum Computing
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Towards Quantum Chemistry on a Quantum Computer
• Elucidating Reaction Mechanisms on Quantum Computers
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• Mathematical theory and numerical methods for Bose-Einstein condensation
• A Theory of Trotter Error
• Effective Hamiltonians, prethermalization and slow energy absorption in periodically driven many-body systems
• Gate count estimates for performing quantum chemistry on small quantum computers
• Even more efficient quantum computations of chemistry through tensor hypercontraction
• Low Depth Quantum Simulation of Electronic Structure
• Is there evidence for exponential quantum advantage in quantum chemistry?
• Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization
• XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations
• Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation
• Multi-scale Entanglement Renormalization Ansatz in Two Dimensions: Quantum Ising Model
• Hamiltonian simulation in the low-energy subspace
• Provably accurate simulation of gauge theories and bosonic systems
• Time-dependent unbounded Hamiltonian simulation with vector norm scaling
• Grid-based methods for chemistry simulations on a quantum computer
• Optimised Trotter Decompositions for Classical and Quantum Computing
• One bound to rule them all: from Adiabatic to Zeno
• Dynamical Decoupling of Unbounded Hamiltonians
• Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation
• Efficient geometric integrators for nonadiabatic quantum dynamics. I. The adiabatic representation
• Universal Atom Interferometer Simulator -- Elastic Scattering Processes
• Gradient Symplectic Algorithms for Solving the Radial Schrodinger Equation
• State-dependent error bound for digital quantum simulation of driven systems
• On the pointwise convergence of the integral kernels in the Feynman-Trotter formula
• State-dependent Trotter Limits and their approximations
• Optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and nonlinearity
• Unification of Random Dynamical Decoupling and the Quantum Zeno Effect
• Path integral in a magnetic field using the Trotter product formula
• Trotter error with commutator scaling for the Fermi-Hubbard model
• Non-Markovian noise that cannot be dynamically decoupled by periodic spin echo pulses
• Complexity of Digital Quantum Simulation in the Low-Energy Subspace: Applications and a Lower Bound
• Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

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