Leveraging Symmetry Merging in Pauli Propagation

TL;DR

This work tackles the challenge of efficiently simulating quantum operator dynamics by introducing symmetry merging into Pauli propagation (symmetry PP). By grouping symmetry-related Pauli strings into orbit representatives and propagating only these representatives, the method preserves exact expectation values while reducing memory usage by a factor $r = \frac{\lvert\mathcal{R}^G_n\rvert}{4^n}$ and improving numerical stability under truncation and noise. The authors provide rigorous guarantees (Theorem symmetry and Theorem space-complexity) and demonstrate substantial practical benefits on ergodic 1D Ising and all-to-all XXZ dynamics, supported by open-source code. Overall, symmetry PP offers a group-theoretic, memory-efficient approach to classical quantum-dynamics simulations that can leverage discrete symmetries like translation and permutation groups.

Abstract

We introduce a symmetry-adapted framework for simulating quantum dynamics based on Pauli propagation. When a quantum circuit possesses a symmetry, many Pauli strings evolve redundantly under actions of the symmetry group. We exploit this by merging Pauli strings related through symmetry transformations. This procedure, formalized as the symmetry-merging Pauli propagation algorithm, propagates only a minimal set of orbit representatives. Analytically, we show that symmetry merging reduces space complexity by a factor set by orbit sizes, with explicit gains for translation and permutation symmetries. Numerical benchmarks of all-to-all Heisenberg dynamics confirm improved stability, particularly under truncation and noise. Our results establish a group-theoretic framework for enhancing Pauli propagation, supported by open-source code demonstrating its practical relevance for classical quantum-dynamics simulations.

Figures (4)

• Figure 1: Schematics of symmetry-merging Pauli propagation.a) The symmetry group (e.g., $\mathbb{Z}_3$ generated by $\hat{T}$) partitions the set of all $n$-qubit Pauli strings into orbits ($\operatorname{Orb}_{\mathbb{Z}_3}$). Each orbit can be represented by a single orbit representative (pink circles). b) Instead of operating on the full space, symmetry PP evolves a minimal set of representatives $\mathcal{R}^G_n$ as in Eq. \ref{['eq:O-merging']}. After $l$-th unitary layer, the evolved operator $U_l^\dagger O U_l$ contains many Paulis. The merging of symmetry-equivalent Paulis by their orbit representatives reduces the number of distinct terms. c) Comparison of the symmetry PP (top) and standard Pauli propagation (bottom). After each symmetric layer, non-representative Paulis (gray) are merged to orbit representatives (pink), potentially recovering terms that would otherwise be truncated (crossed nodes).
• Figure 2: Reduced Paulis in $1d$ Ising dynamics. Scaling of the number of propagated Pauli strings with circuit layers $l$ for system sizes from $n = 3$ to $n=20$ (color bar). Lines plot standard PP $N$ (Standard) and circles denote the symmetry PP $\tilde{N}$ (Symmetry) under translation symmetry. Dashed lines show theoretical predictions for the number of distinct Pauli terms. Symmetry merging substantially suppresses the exponential growth of propagated Paulis, saturating to the theoretical scaling of $\frac{1}{n}(4^n + 4(n-1)))$ terms discussed in Example \ref{['ex:Zn']}.
• Figure 3: Spin expectation values of all-to-all Heisenberg dynamics. We compare predictions from standard PP (lines) and symmetry PP (markers). The total spin operator is defined in Eq. \ref{['eq:s2']} and the normalization factor $\mathbf{S}_0^2 = \langle \boldsymbol{+} \lvert \mathbf{S}^2 \rvert \boldsymbol{+} \rangle$ is computed with respect to the initial product state along $x$-axis $\hbox{$| \boldsymbol{+} \rangle$} := \hbox{$| + \rangle$}^{\otimes n}$ for $n=36$ qubits with $J_{\perp}=1$ and $\Delta=-1.8$. We also introduce depolarizing noise layers, damping the coefficient by non-identity Pauli weight $e^{-\gamma \lvert P\rvert}$ (e.g. $\lvert XXIZ\rvert=3$). The numerical experiments were performed using up to $100 \text{G}$ memory and up to three days on a single CPU-- a limit that only standard PP with the lowest cutoff $\epsilon$ reached. To achieve the most accurate result, symmetry PP uses $10$ times fewer Paulis than standard PP: $\tilde{N}\approx 2 \times 10^8$ Paulis at the end of blue circles for symmetry compared to $N\approx 2 \times 10^7$ Paulis at the end of the pink line for standard PP.
• Figure D.1: Simulation time in $1d$ Ising dynamics. Scaling of the computational time ($second$) with circuit layers $l$ for system sizes $n = 3$ to $n=20$ (color bar). Circles denote standard PP and triangles denote the symmetry PP under translation symmetry.

Theorems & Definitions (24)

• Theorem 1: Symmetry PP
• Theorem 2: Space complexity (informal)
• Example 1: Translation symmetry $\mathbb{Z}_n$ in $1d$
• Example 2: Permutation symmetry $S_n$
• Definition 1: Group (finite)
• Definition 2: Group action
• Definition 3: Group orbit
• Definition 4: Orbit representatives
• Example 3: Pauli representatives ($\mathbb{Z}_n$)
• Lemma 1: Burnside's lemma armstrong1997groups