Scoring-based Static Variable Ordering for Decision Diagram-based Quantum Circuit Simulation

TL;DR

The paper addresses the sensitivity of DD-based quantum circuit simulators to static variable order by introducing a scoring-based heuristic that ranks qubits according to their roles in multi-qubit gates and parameterized rotations. The method computes an $O(M)$ score based on circuit features, then orders qubits to reduce DD size and runtime, yielding substantial speedups (up to $150\times$) and enabling large-scale factoring such as a $1011$-bit Shor computation on a single-core laptop in about $5$ hours. Empirical results across benchmark circuits (QPE, QFT, VQE) and Shor’s algorithm demonstrate strong performance gains and highlight the method’s practical impact for DD-based quantum circuit simulation. The work also discusses limitations, such as variability due to floating-point errors and cases where ordering may not be optimal, and outlines future directions including combining static ordering with dynamic reordering and noise-aware simulations.

Abstract

Decision diagram (DD)-based quantum circuit simulators represent quantum states and gates using DDs, enabling memory-efficient and fast simulations for some quantum circuits like Shor. Although it is known that DD size and processing time vary depending on the variable order in classical circuits, there has not been much research on the variable order under quantum circuit simulation. One existing study pointed out that dynamic reordering worsens the simulation time and numerical accuracy, and there is no comprehensive research on static orders in the context of quantum circuit simulation. Therefore, this paper proposes a scoring-based heuristic method for determining a static variable order that enables efficient DD-based quantum circuit simulation. When applied to benchmark circuits, the default original variable orders resulted in slow simulations, whereas the proposed method achieved speedups of up to 150x. Furthermore, the proposed order completed the simulation of Shor's 1011 factorization in 5 hours on a single-core laptop, although it was not completed within two days previously.

Figures (8)

• Figure 1: State vector representation in DD
• Figure 2: Example where DD reduces the memory amount significantly
• Figure 3: Different orders for "Quantum Walk (v-chain)" 11-qubit circuit
• Figure 4: Qubit position and the computation complexity
• Figure 5: QPE exact circuit