[Paper Review] Hamiltonian Simulation in the Interaction Picture

Guang Hao Low, Nathan Wiebe|arXiv (Cornell University)|May 2, 2018

Scientific Computing and Data Management53 citations

TL;DR

This paper presents a low-space overhead algorithm for simulating time-dependent quantum dynamics via a truncated Dyson series, applied in the interaction picture to achieve better gate complexity for certain Hamiltonians, including diagonally dominant and Hubbard models with long-range interactions.

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 $\widetilde{\mathcal{O}}(N^2t)$ from $\widetilde{\mathcal{O}}(N^{11/3}t)$. We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.

Motivation & Objective

• Motivate and develop a low-space overhead method for time-dependent quantum dynamics using the truncated Dyson series.
• Show how switching to the interaction picture can yield favorable complexity trade-offs for certain Hamiltonians.
• Provide rigorous analysis and circuit constructions that improve gate and space costs over Schrödinger-picture methods.
• Demonstrate practical advantages with applications to Hubbard models with long-range interactions and plane-wave electronic structure.
• Extend the framework to sparse, time-dependent Hamiltonians and discuss complexity-theoretic implications.

Proposed method

• Develop a general time-dependent simulation algorithm based on the truncated Dyson series for H(s) with spectral norm alpha and average rate-of-change ⟨||Ḣ||⟩.
• Encode Hamiltonians with block-encoding oracles HAM and HAM-T following the standard-form input model.
• Analyze truncation and discretization errors to show a cost O(alpha t log(alpha t/ε)/log log(alpha t/ε)) queries in the time-dependent setting.
• Apply the truncated Dyson series to the interaction picture where H = A + B and H_I(t) = e^{iAt}Be^{-iAt}, enabling simulation with cost scaling favoring ||B|| rather than ||A||.
• Derive multi-segment strategies for long-time evolution and provide explicit gate, qubit, and query complexities.
• Present complexity improvements for sparse time-dependent Hamiltonians and diagonally dominant cases, including Hubbard models and plane-wave electronic structure.

Experimental results

Research questions

• RQ1 How can a truncated Dyson series be rigorously analyzed for time-dependent Hamiltonians to achieve controllable error with quantum circuits?
• RQ2 What are the gate and space complexity costs when simulating in the interaction picture versus the Schrödinger picture for specific Hamiltonian structures?
• RQ3 Can the interaction picture reduce dependence on large A-norm terms and amplify efficiency proportional to the smaller B-norm in A+B?
• RQ4 What are the implications of this approach for long-range interacting Hubbard models and plane-wave electronic structure in terms of gate counts?
• RQ5 How does the approach extend to sparse time-dependent Hamiltonians and what are the resulting complexity improvements?

Key findings

• For time-dependent Hamiltonians, the Dyson-series-based simulation achieves a cost of O(alpha t log(alpha t/ε)/log log(alpha t/ε)) queries and reduced qubit overhead relative to prior methods.
• Simulation in the interaction picture can yield exponential improvements in gate complexity for some diagonally dominant Hamiltonians by scaling with ||B|| rather than ||A||.
• Applied to the N-site Hubbard model with long-range interactions, the method attains gate complexity ~ Õ(N^2 t), representing an almost quadratic improvement over prior Õ(N^{11/3} t) scaling.
• In plane-wave basis electronic-structure contexts, the method reduces simulations to ~ Õ(N^2 t) gates, closely approaching quadratic improvement.
• A quadratic improvement in query complexity is shown for sparse time-dependent Hamiltonians within the standard query model.
• The framework provides rigorous bounds and explicit circuit constructions, including multi-segment strategies for long-time evolution.