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Hamiltonian Simulation in the Interaction Picture

Guang Hao Low, Nathan Wiebe

TL;DR

This work develops a low-space overhead quantum simulation method based on a truncated Dyson series and applies it in the interaction picture to exploit a Hamiltonian decomposition H = A + B. It provides rigorous error bounds and complexity analyses, demonstrating exponential improvements in gate complexity for diagonally dominant and long-range Hubbard-type Hamiltonians, and near-quadratic improvements for plane-wave quantum chemistry within a plane-wave basis. The approach also yields a quadratic improvement in the query complexity for sparse time-dependent Hamiltonians and, in the interaction picture, reduces dependence on large diagonal terms. These results offer practical pathways to more efficient quantum simulations of electronic structure, materials, and sparse Hamiltonians on future quantum hardware.

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.

Hamiltonian Simulation in the Interaction Picture

TL;DR

This work develops a low-space overhead quantum simulation method based on a truncated Dyson series and applies it in the interaction picture to exploit a Hamiltonian decomposition H = A + B. It provides rigorous error bounds and complexity analyses, demonstrating exponential improvements in gate complexity for diagonally dominant and long-range Hubbard-type Hamiltonians, and near-quadratic improvements for plane-wave quantum chemistry within a plane-wave basis. The approach also yields a quadratic improvement in the query complexity for sparse time-dependent Hamiltonians and, in the interaction picture, reduces dependence on large diagonal terms. These results offer practical pathways to more efficient quantum simulations of electronic structure, materials, and sparse Hamiltonians on future quantum hardware.

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 -site Hubbard models for time with arbitrary long-range interactions as well as reduce the cost of quantum chemistry simulations within a similar-sized plane-wave basis to from . We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.

Paper Structure

This paper contains 19 sections, 16 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. Time-dependent Hamiltonian simulation by a truncated Dyson series
  4. Input model
  5. Truncated Dyson series algorithm
  6. Discretization and truncation error
  7. Interaction picture simulation
  8. Comparison with simulation of time-independent Hamiltonians in the Schrödinger picture
  9. Application to the Hubbard model with long-ranged interactions
  10. Application to quantum chemistry in the plane-wave basis
  11. Application to sparse Hamiltonian simulation
  12. Simulation of sparse time-dependent Hamiltonians
  13. Simulation of sparse time-independent Hamiltonians in the interaction picture
  14. Conclusion
  15. Acknowledgments
  16. ...and 4 more sections

Key Result

Theorem 1

Let $H(s) : [0,t] \rightarrow \mathbb{C}^{2^{n_s}\times 2^{n_s}}$, let it be promised that $\max_{s}\|H(s)\|\le\alpha$ and $\langle\|\dot H\|\rangle=\frac{1}{t}\int^t_{0} \left\|\frac{\mathrm{d} H(s)}{ \mathrm{d} s}\right\| \mathrm{d}s$ and assume $M\in{\mathcal{O}}\left( \frac{t^2}{\epsilon}\left({

Figures (5)

  • Figure 1: Quantum circuit representation of (left) an oracle $\operatorname{HAM}$ from \ref{['eq:standard-form-TI']} encoding a time-independent Hamiltonian, (center) an oracle $\operatorname{HAM-T}$ from \ref{['def:HAM-T']} encoding a time-dependent Hamiltonian, and (right) an example implementation of $\operatorname{HAM}$ from with a linear-combination of unitaries from \ref{['eq:LCU']}. Bold horizontal lines with a backslash '$\backslash$' depict registers that in general comprise of multiple qubits. Vertical lines connecting boxes depict unitaries that act jointly on all registers covered by the boxes. A small square box marked by '$\operatorname{T}$' indicates control by a time index.
  • Figure 2: Quantum circuit representations of the gadget $V$ for probabilistically applying a sequence of operators $H_k \cdots H_2H_1$, encoded in $(\langle 0|_{a}\otimes \openone_s) U_k (| 0\rangle_{a}\otimes \openone_s) = H_k$, controlled on number state $| k\rangle_b,\; k\in\{0,1,\cdots,K\}$. Horizontal lines without a backslash depict single-qubit registers. Filled circles depict a unitary controlled by the $| 0\rangle\cdots| 0\rangle$ state.
  • Figure 3: Quantum circuit representation of (top) $\operatorname{DYS}_K$ in \ref{['eq:DYS-K-compressed']}, implemented using the compression gadget \ref{['Thm:compression_gadget']} depicted in \ref{['fig:Compression-Gadget']}; (bottom, left) a single step of time-evolution by the truncated Dyson series algorithm from \ref{['eq:LCU_TDS']} before oblivious amplitude amplification; (bottom, right) a single step of time-evolution by the truncated Dyson series algorithm from \ref{['eq:TDS_compressed']}. Note that when $\beta=2$, a single-round of oblivious amplitude amplification is used.
  • Figure 4: Quantum circuit representation of (top, left) $\operatorname{HAM}_L$ in \ref{['eq:HAM-T_k']}; (top, right) $\operatorname{DYS}_K$ in \ref{['eq:DYS-K']}; (bottom, left) a single step of time-evolution by the truncated Dyson series algorithm from \ref{['eq:LCU_TDS_duplicated']} before oblivious amplitude amplification; (bottom, right) a single step of time-evolution by the truncated Dyson series algorithm with duplicated ancilla registers . Note that when $\beta=2$, a single-round of oblivious amplitude amplification is used.
  • Figure 5: Quantum circuit representation of (top, left) an example implementation of $\operatorname{HAM}_K$ from \ref{['eq:TTS_components']} using $K$ queries to controlled-$\operatorname{HAM}$; (top, right) an example implementation of $\operatorname{HAM}_K$ with fewer ancilla qubits using the compression gadget of \ref{['Thm:compression_gadget']} ;(bottom, left) a single step of the truncated Taylor series algorithm before oblivious amplitude amplification; (bottom, right) a single step of time-evolution by the truncated Taylor series algorithm from \ref{['eq:TTS']}. Note that $\beta=2$ as a single-round of oblivious amplitude amplification is used.

Theorems & Definitions (30)

  • Definition 0: Time-independent matrix encoding
  • Definition 0: Time-dependent matrix encoding
  • Theorem 1: Hamiltonian simulation by a truncated Dyson series
  • Corollary 1: Multi-segment Hamiltonian simulation by a truncated Dyson series
  • proof
  • Lemma 1: Error from truncating and discretizing the Dyson series
  • Lemma 2: Query complexity of Hamiltonian simulation in the interaction picture
  • proof
  • Theorem 3: Gate complexity of Hamiltonian simulation in the interaction picture
  • proof
  • ...and 20 more