On the complexity of implementing Trotter steps

Abstract

Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner. However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques. We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms. We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law. Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques. We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank. Combining with a tighter error analysis, we show that it suffices to use $\left({\eta }^{1/3}{n}^{1/3}+\frac{n^{2/3}}{{\eta }^{2/3}}\right)n^{1+o(1)}$ gates to simulate uniform electron gas with $n$ spin orbitals and $\eta$ electrons in second quantization in real space, asymptotically improving over the best previous work. We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation. We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic $n$-qubit $2$-local Hamiltonians with commuting terms require at least $\Omega (n^2)$ gates to evolve with accuracy $\epsilon =\Omega (1/poly(n))$ for time $t=\Omega (\epsilon )$. Our proof is based on a gate-efficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-$2$ subspace, which may be of independent interest. Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.

69 pages, 7 figures. Tightened gate complexity analysis with no amplitude amplification prefactors. Discussed ancilla space complexity. Added master theorem analysis of the recursive low-rank algorithm. Enhanced version of the article published in PRX Quantum at https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.4.020323

References in corpus (12)

• Quantum algorithm for solving linear systems of equations
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Synthesis of Quantum Logic Circuits
• Strongly correlated 2D quantum phases with cold polar molecules: controlling the shape of the interaction potential
• Polynomial-time quantum algorithm for the simulation of chemical dynamics
• Entanglement entropy for the long range Ising chain
• Cold polar molecules in 2D traps: Tailoring interactions with external fields for novel quantum phases
• Simulating chemistry efficiently on fault-tolerant quantum computers
• Efficient synthesis of universal Repeat-Until-Success circuits
• Repulsive shield between polar molecules
• 1D Quantum Liquids with Power-Law Interactions: a Luttinger Staircase with Polar Molecules
• State-dependent error bound for digital quantum simulation of driven systems

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