Probing quantum complexity via universal saturation of stabilizer entropies

TL;DR

This work analyzes how nonstabilizerness, quantified by the α-Stabilizer Rényi entropies $M_\alpha$, saturates under two paradigms: Clifford circuits doped with $qN$ T-gates and random Hamiltonian evolution. It derives exact results for $\alpha=2$ in the Clifford+T model, and develops analytic approximations for general $\alpha$ in random-evolution scenarios, revealing a universal, phase-transition-like saturation at critical points $q_{c,\alpha}$ or $t_{c,\alpha}$. The study shows that $\alpha<1$ captures Clifford-simulation complexity, while $\alpha>1$ measures proximity to stabilizer states and Pauli-based certification costs, with $M_2$ providing direct fidelity-estimation bounds. The findings highlight fundamentally different scaling behaviors and universal features across $\alpha$, offering a new lens to assess quantum complexity and the feasibility of classical simulations and certification tasks.

Abstract

Nonstabilizerness or `magic' is a key resource for quantum computing and a necessary condition for quantum advantage. Non-Clifford operations turn stabilizer states into resourceful states, where the amount of nonstabilizerness is quantified by resource measures such as stabilizer Rényi entropies (SREs). Here, we show that SREs saturate their maximum value at a critical number of non-Clifford operations. Close to the critical point SREs show universal behavior. Remarkably, the derivative of the SRE crosses at the same point independent of the number of qubits and can be rescaled onto a single curve. We find that the critical point depends non-trivially on Rényi index $α$. For random Clifford circuits doped with T-gates, the critical T-gate density scales independently of $α$. In contrast, for random Hamiltonian evolution, the critical time scales linearly with qubit number for $α>1$, while is a constant for $α<1$. This highlights that $α$-SREs reveal fundamentally different aspects of nonstabilizerness depending on $α$: $α$-SREs with $α<1$ relate to Clifford simulation complexity, while $α>1$ probe the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. As technical contributions, we observe that the Pauli spectrum of random evolution can be approximated by two highly concentrated peaks which allows us to compute its SRE. Further, we introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, where we provide its exact SRE. Our results opens up new approaches to characterize the complexity of quantum systems.

Figures (11)

• Figure 1: Overview of main results. We study two models: a) Circuits composed of randomly chosen Clifford circuits $U_\text{C}$ doped with $N_\text{T}=qN$ T-gates, where $N$ is the number of qubits and $q$ the T-gate density. b) Evolution in time $t$ for random Hamiltonians starting from an initial stabilizer state. c)$\alpha$-stabilizer Rényi entropy (SRE) $M_\alpha$ increases monotonously with $q$ and $t$ until converging to a constant. For large $N$ we observe a sharp transition to maximal SRE $M_\alpha^\text{max}$ (horizontal dashed line) for a critical T-gate density $q_{\text{c},\alpha}$ or critical time $t_{\text{c},\alpha}$ (vertical dashed line). The derivative of $M_\alpha$ crosses at the critical point for all $N$ and the dynamics close to the critical point can be mapped onto a single curve (see Fig.\ref{['fig:CliffordT_renyi']} and Fig. \ref{['fig:randomEvol_renyi2']}). d) Critical T-gate density $q_{\text{c},\alpha}$ and time $t_{\text{c},\alpha}$ as function of Rényi index $\alpha$. Both $q_{\text{c},\alpha}$ and $t_{\text{c},\alpha}$ vary non-monotonously with $\alpha$ as they probe different aspects of nonstabilizerness complexity. In particular, critical time $t_{\text{c},\alpha}$ changes its scaling from constant ($\alpha<1$) to $t_{\text{c},\alpha}\sim \sqrt{N}$ ($\alpha>1$).
• Figure 2: Universal behavior of $2$-SRE for random Clifford + T circuits. a)$m_2$ against T-gate density $q=N_\text{T}/N$ for different $N$. Black vertical dashed line is transition T-gate density $q_{\text{c},2}=\ln(2)/\ln(4/3)$b) Derivative of SRE in respect to T-gate density $\partial_q m_2$ against $q$ with universal crossing for all $N$ at critical $q_{\text{c},2}=\ln(2)/\ln(\frac{4}{3})$. c) Collapse of $\partial_q m_2$ against $q$ when shifted by $q_{\text{c},2}$ and scaled by $N$. Close to the critical density $q_{\text{c},2}$, curves for different $N$ intersect at a single point and can be mapped onto each other by a simple rescaling, which is hallmark of universality.
• Figure 3: Saturation of $\alpha$-SREs for random Clifford circuits doped with T-gates. a)$M_\alpha$ divided by $\alpha$-SRE per T-gate $M_\alpha^\text{T}$ against T-gate density $q$. The crosses denote the critical T-gate density $q_{\text{c},\alpha}$ as derived in Eq. (\ref{['eq:transitionTgate']}). b)$M_\alpha$ divided by maximal SRE $M_\alpha^\text{max}$ from Eq. (\ref{['eq:renyimax']}) derived for $N\gg1$. We show $N=14$ qubits and average over 20 random instances, where we note that due to self-averaging the variance in SRE is small.
• Figure 4: Universal behavior of $2$-SRE for random basis evolution model Eq. (\ref{['eq:random_CliffGUE']}). a)$M_2$ against time $t$ for different $N$. b) Derivative in respect to $t^2$ of SRE $\partial_{t^2} M_2$ against $t$. Black vertical dashed line is critical time $t_\text{c}^2=\frac{1}{4}N\ln(2)$. c)$\partial_{t^2} M_2$ against $t^2$ shifted by $t_\text{c}^2$. Hallmark of universality is the crossing of all curves at the critical point $t_\text{c}^2$ and collapse to a single curve.
• Figure 5: $\alpha$-SREs of random Hamiltonian evolution. a) Pauli spectrum plotted as histogram, where we show the probability $C$ of observing the Pauli expectation values $\beta_\sigma^2=\bra{\psi}\sigma\ket{\psi}^2$. We show different $t$ of GUE evolution for $N=10$ averaged over 20 random instances. For clarity, we do not show the trivial identity operator $\bra{\psi}I\ket{\psi}^2=1$ of the spectrum. For $t\gg1$, Pauli spectrum is mostly concentrated around $\beta_\sigma^2=2^{-N}\approx 10^{-3}$. b) SRE $M_\alpha$ against $t$ for different $\alpha$ and $N=12$, where dashed lines are approximations Eq. (\ref{['eq:RenyiAlphaGtr1']}), Eq. (\ref{['eq:RenyiAlphaLess1tsmall']}), Eq. (\ref{['eq:RenyiAlphaLess1tlarge']}) and Eq. (\ref{['eq:RenyiAlpha1']}). The sudden change of the dashed line for $\alpha<1$ is due to the change from Eq. (\ref{['eq:RenyiAlphaLess1tsmall']}) to Eq. (\ref{['eq:RenyiAlphaLess1tlarge']}) at $2^{N(1-\alpha)}(1-F^2)^\alpha=1$. c) SRE $M_\alpha$ against $N$ for $t=10^{-2}$. Dashed line is fit with approximations. Our model accurately describes $\alpha$-SREs of random evolution as function of $N$ and $t$, allowing us to predict overall scaling and critical time $t_{\text{c},\alpha}$.