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Towards Hamiltonian Simulation with Decision Diagrams

Aaron Sander, Lukas Burgholzer, Robert Wille

TL;DR

Investigation of Decision Diagrams for Hamiltonian simulation shows that DDs indeed may offer a promising new data structure which, for certain examples, can provide orders of magnitudes of improvement compared to the state-of-the-art, yet also comes with its own, fundamentally different, limitations.

Abstract

This paper proposes a novel approach to Hamiltonian simulation using Decision Diagrams (DDs), which are an exact representation based on exploiting redundancies in representations of quantum states and operations. While the simulation of Hamiltonians has been studied extensively, scaling these simulations to larger or more complex systems is often challenging and may require approximations or new simulation methods altogether. DDs offer such an alternative that has not yet been applied to Hamiltonian simulation. In this work, we investigate the behavior of DDs for this task. To this end, we review the basics of DDs such as their construction and present how the relevant operations for Hamiltonian simulation are implemented in this data structure -- leading to the first DD-based Hamiltonian simulation approach. Based on several series of evaluations and comparisons, we then discuss insights about the performance of this complementary approach. Overall, these studies show that DDs indeed may offer a promising new data structure which, for certain examples, can provide orders of magnitudes of improvement compared to the state-of-the-art, yet also comes with its own, fundamentally different, limitations.

Towards Hamiltonian Simulation with Decision Diagrams

TL;DR

Investigation of Decision Diagrams for Hamiltonian simulation shows that DDs indeed may offer a promising new data structure which, for certain examples, can provide orders of magnitudes of improvement compared to the state-of-the-art, yet also comes with its own, fundamentally different, limitations.

Abstract

This paper proposes a novel approach to Hamiltonian simulation using Decision Diagrams (DDs), which are an exact representation based on exploiting redundancies in representations of quantum states and operations. While the simulation of Hamiltonians has been studied extensively, scaling these simulations to larger or more complex systems is often challenging and may require approximations or new simulation methods altogether. DDs offer such an alternative that has not yet been applied to Hamiltonian simulation. In this work, we investigate the behavior of DDs for this task. To this end, we review the basics of DDs such as their construction and present how the relevant operations for Hamiltonian simulation are implemented in this data structure -- leading to the first DD-based Hamiltonian simulation approach. Based on several series of evaluations and comparisons, we then discuss insights about the performance of this complementary approach. Overall, these studies show that DDs indeed may offer a promising new data structure which, for certain examples, can provide orders of magnitudes of improvement compared to the state-of-the-art, yet also comes with its own, fundamentally different, limitations.
Paper Structure (19 sections, 33 equations, 4 figures)

This paper contains 19 sections, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian Simulation
  3. Quantum Dynamics
  4. Product Formula
  5. Ising Model
  6. Heisenberg Model
  7. Motivation
  8. General Problem
  9. Related Work
  10. Decision Diagrams
  11. Representing Quantum States
  12. Representing Quantum Operators
  13. Fundamental Operations on Decision Diagrams
  14. Experimental Evaluations
  15. Application of DDs to Hamiltonian Simulation Circuits
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Redundancy landscapes for the Ising and Heisenberg model with $L=12$ for $n=1$ and $n=2$ Trotter step applications of the selected angles.
  • Figure 2: Growth of node count in the decision diagrams for the Ising model ($J=1$, $g=0.001$) and Heisenberg model ($J_x=J_y=J_z=1, h=1)$ with timestep size $\delta t = 0.1$.
  • Figure 3: Runtime scaling of various methods for simulating the Ising model $J=1$, $g=0.001$, and timestep size $\delta t = 0.1$. Each corresponds to this being performed with $n$ Trotter steps.
  • Figure 4: Selected examples and their runtime requirement for different methods

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9