Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness

TL;DR

An efficient classical algorithm is introduced that exactly computes stabilizer R\'enyi entropies and stabilizer nullity for generic many-body wavefunctions of $N$ qubits and develops a Monte-Carlo estimator for stabilizer R\'enyi entropies together with a Clifford-based variance-reduction scheme that suppresses sampling fluctuations.

Abstract

Quantum magic, quantified by nonstabilizerness, measures departures from stabilizer structure and underlies potential quantum speedups. We introduce an efficient classical algorithm that exactly computes stabilizer Rényi entropies and stabilizer nullity for generic many-body wavefunctions of $N$ qubits. The method combines the fast Walsh-Hadamard transform with an exact partition of Pauli operators. It achieves an exponential speedup over direct approaches, reducing the average cost per sampled Pauli string from $O(2^N)$ to $O(N)$. Building on this framework, we further develop a Monte-Carlo estimator for stabilizer Rényi entropies together with a Clifford-based variance-reduction scheme that suppresses sampling fluctuations. We benchmark the accuracy and efficiency on ensembles of random magic states, and apply the method to random Clifford circuits with doped $T$ gates, comparing different doping architectures. Our approach applies to arbitrary quantum states and provides quantitative access to magic resources both encoded in highly entangled states and generated by long-time nonequilibrium dynamics.

Figures (3)

• Figure 1: (a) Runtime for computing $M_2$ of random magic states by brute-force versus FWHT-based Pauli enumeration; dashed lines indicate $\sim\mathcal{O}(2^{3N})$ and $\sim\mathcal{O}(2^{2N})$ scaling. (b) Probability density of the normalized partial moment $m_{2;x}/\mathbb{E}_x[m_{2;x}]$ for Haar-random states (black: Gaussian fits); inset shows $\mathrm{Std}_x(m_{2;x})/\mathbb{E}_x(m_{2;x})$ as a function of $N$. (c) Normalized standard deviation $\mathrm{Std}_x(m_{2;x})/\mathbb{E}_x(m_{2;x})$ versus Clifford depth $N_C$ for $|\psi_m\rangle=|T\rangle^{\otimes N_T}\otimes|0\rangle^{\otimes(N-N_T)}$ at several $N$. (d) Fluctuations of $M_2$ estimates for highly entangled $\mathcal{C}|\psi_m\rangle$ at $N=16$: MC+FWHT (sampling $\{P_{x,z}\}_z$ with $\mathcal{N}=10^4$) versus Metropolis--Hastings sampling of Pauli strings ($1.6\times 10^6$ samples). The Metropolis--Hastings data use $10\times$ more CPU time.
• Figure 2: (a) Quantum circuit begin with a $N$-qubit random Clifford product state $$, followed by $N_C$ layers of random two-qubit Clifford gates and $T$ gates on all qubits. (b) Stabilizer Rényi entropy density $M_2/N$ of the left-figure state as a function of $N_C$ for different system sizes $N$. The data are averaged over $80$ random instances. For $N=14$, $16$, we use the exact FWHT algorithm; for $N=20$, the number of Monte Carlo samples is $\mathcal{N}=2 \times 10^4$.
• Figure 3: (a) Ensemble-averaged $M_2$ versus the number of scramble-$T$-injection cycles for different numbers $N_C$ of random two-qubit Clifford layers per cycle. Inset: the gap $M_2^{\rm Haar}-\mathbb{E}[M_2]$ on a logarithmic scale. (b, c) Two $T$-doping architectures at fixed injection rate (one $T$ gate per Clifford layer on average), illustrated for $N=6$: (b) bursty injection with $N_B = N/2 = 3$ parallel $T$ gates applied after $N_B =3$ Clifford layers; (c) uniform injection with $T$ gates spread among the Clifford layers ($N_B = 1$). (d) $\mathbb{E}[M_2]$ versus the total number of injected $T$ gates for several block sizes $N_B$. Labels such as "10T/10L" denote blocks of 10 $T$ gates applied after 10 Clifford layers (and similarly for "4T/4L" and "1T/1L"). Inset: $M_2^{\rm Haar}-\mathbb{E}[M_2]$ on a logarithmic scale. Parameters: $N=20$, MC sample size $\mathcal{N}=2\times10^5$, and $80$ circuit realizations per data point.