Simulating Quantum Walk Hamiltonians without Pauli Decomposition

TL;DR

The paper addresses efficient simulation of continuous-time quantum walks (CTQWs) on sparse graphs without relying on Pauli decompositions. It introduces matching decomposition, which partitions graph edges into commute-friendly matchings, and a novel iterative graph compression that reduces qubits and control overhead, followed by a circuit construction and Trotterization over the matchings. Empirical results show substantial resource savings over Pauli-based pipelines—up to $43\%$ fewer CX gates and $54\%$ shallower circuits on connected graphs—with comparable operator-norm accuracy, and the authors derive conditions under which the matching decomposition can exactly simulate CTQWs when the matchings commute. The work provides a practical circuit-assembly primitive for CTQW simulation on sparse graphs and motivates graph-structure-aware synthesis and further analysis of commuting decompositions' role in exact quantum simulations.

Abstract

In this work, we present a new algorithm for generating quantum circuits that efficiently implement continuous time quantum walks on arbitrary simple sparse graphs. The algorithm, called matching decomposition, works by decomposing a continuous-time quantum walk Hamiltonian into a collection of exactly implementable Hamiltonians corresponding to matchings in the underlying graph followed by a novel graph compression algorithm that merges edges in the graph. Lastly, we convert the walks to a circuit and Trotterize over these components. The dynamics of the walker on each edge in the matching can be implemented in the circuit model as sequences of CX and CRx gates. We do not use Pauli decomposition when implementing walks along each matching. Furthermore, we compare matching decomposition to a standard Pauli-based simulation pipeline and find that matching decomposition consistently yields substantial resource reductions, requiring up to 43% fewer controlled gates and up to 54% shallower circuits than Pauli decomposition across multiple graph families. Finally, we also present examples and theoretical results for when matching decomposition can exactly simulate a continuous-time quantum walk on a graph.

Figures (7)

• Figure 1: Matching decomposition and space reduction example. The original 4-vertex graph $G$ with edges $\{(00, 01), (10, 11), (00, 11), (01, 10)\}$ decomposes into two matchings. $M_0$ (red) contains Hamming distance 1 edges $\{(00, 01), (10, 11)\}$ which differ only in qubit 0, compressing to a single edge $(0,1)$ with active qubits $\mathcal{A} = \{0\}$ and weight-reducing qubits $\mathcal{W} = \{\}$. $M_1$ (blue) contains Hamming distance 2 edges $\{(00, 11), (01, 10)\}$, compressing to edge $(0,1)$ with $\mathcal{A} = \{1\}$ and $\mathcal{W} = \{0\}$. The non-empty $\mathcal{W}$ for $M_1$ indicates that CX gates are required for the basis transformation.
• Figure 2: Trotterized circuit for the example graph with $N$ Trotter steps. The $R_x(2t/N)$ gate on $q_0$ implements $M_0$. The CX-$R_x(2t/N)$-CX sequence implements $M_1$, where the CX gates perform the basis transformation required by $\mathcal{W} = \{0\}$. The entire sequence is repeated $N$ times.
• Figure 3: The 2-norm of the operator difference $\|e^{-iAt} - U_{\text{Trotter}}\|_2$ between the exact CTQW evolution and Trotterized approximations for the 8-vertex connected graph dataset. Lines show mean values over all graphs for matching decomposition (blue, circles) and Pauli decomposition (orange, squares), with shaded regions indicating $\pm 1$ standard deviation. Line styles distinguish evolution times: solid ($t=0.1$), dashed ($t=0.5$), and dash-dot ($t=1.0$). Trotter steps range from $N=10$ to $N=100$. Both methods exhibit comparable convergence rates, with errors scaling as $O(t^2/N)$.
• Figure 4: CX gate counts and circuit depths for the matching and Pauli decompositions on the connected graph datasets. Shaded regions indicate standard deviation across graphs.
• Figure 5: CX gate counts and circuit depth for matching decomposition (blue) and Pauli decomposition (orange) on Erdős-Rényi random graphs with $p = 0.01$. Matching decomposition requires 25%, 33%, and 31% fewer CX gates at $N=32$, 64, and 128 respectively, and produces 37%, 41%, and 49% shallower circuits.

Theorems & Definitions (18)

• Definition 1: XOR Mask
• Definition 2: Active Qubits
• Definition 3: Weight-Reducing Qubits
• Definition 4: Mergeable edges
• Definition 5: Compressed Edge
• Definition 6: Bit Deletion Operator
• Lemma 1
• proof
• Lemma 2
• proof