Hamiltonian Simulation in the Interaction Picture

Abstract

We present a low-space overhead simulation algorithm based on the truncated Dyson series for time-dependent quantum dynamics. This algorithm is applied to simulating time-independent Hamiltonians by transitioning to the interaction picture, where some portions are made time-dependent. This can provide a favorable complexity trade-off as the algorithm scales exponentially better with derivatives of the time-dependent component than the original Hamiltonian. We show that this leads to an exponential improvement in gate complexity for simulating some classes of diagonally dominant Hamiltonian. Additionally we show that this can reduce the gate-complexity scaling for simulating $N$-site Hubbard models for time $t$ with arbitrary long-range interactions as well as reduce the cost of quantum chemistry simulations within a similar-sized plane-wave basis to $\tilde{\mathcal{O}}(N^{2}t)$ from $\tilde{\mathcal{O}}(N^{11/3}t)$. We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.

arXiv comment30 pages, 5 figures. v2: Fix some errors in equations and figures

Cited by in corpus (34)

• Quantum computational chemistry
• Quantum computing enhanced computational catalysis
• Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization
• A Trailhead for Quantum Simulation of SU(3) Yang-Mills Lattice Gauge Theory in the Local Multiplet Basis
• Nearly optimal lattice simulation by product formulas
• Increasing the representation accuracy of quantum simulations of chemistry without extra quantum resources
• Simulating Quantum Materials with Digital Quantum Computers
• Provably accurate simulation of gauge theories and bosonic systems
• Time-dependent unbounded Hamiltonian simulation with vector norm scaling
• Time-marching based quantum solvers for time-dependent linear differential equations
• Quantum Algorithms for Estimating Physical Quantities using Block-Encodings
• Simulating key properties of lithium-ion batteries with a fault-tolerant quantum computer
• Hybridized Methods for Quantum Simulation in the Interaction Picture
• Reducing molecular electronic Hamiltonian simulation cost for Linear Combination of Unitaries approaches
• Quantum algorithm for time-dependent Hamiltonian simulation by permutation expansion
• Quantum algorithm for time-dependent differential equations using Dyson series
• Quantum computation of stopping power for inertial fusion target design
• Quantum Regularized Least Squares
• Quantum simulation of battery materials using ionic pseudopotentials
• Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation
• Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian
• Tailoring Term Truncations for Electronic Structure Calculations Using a Linear Combination of Unitaries
• Fault-tolerant quantum algorithms for quantum molecular systems: A survey
• Analysis of quantum Krylov algorithms with errors
• On quantum algorithms for the Schrödinger equation in the semi-classical regime
• Success of digital adiabatic simulation with large Trotter step
• Minimum Trotterization Formulas for a Time-Dependent Hamiltonian
• Exponentially faster implementations of Select(H) for fermionic Hamiltonians
• On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks
• Efficient and practical Hamiltonian simulation from time-dependent product formulas
• TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry
• Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators
• Some Error Analysis for the Quantum Phase Estimation Algorithms
• Chemically Motivated Simulation Problems are Efficiently Solvable by a Quantum Computer