Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits

TL;DR

This work examines how adding non-Clifford $T$-gates to deep random Clifford circuits affects both spectral complexity and non-stabilizerness. By constructing a brick-wall Clifford+$T$ ensemble and analyzing phase correlations, level spacing, and stabilizer Rényi entropy, the authors show that $T$-gates rapidly induce chaotic, RMT-like spectral statistics with $O(1)$ gates, while magic generation exhibits a more nuanced transition that ultimately approaches Haar-like behavior as $N_T$ grows proportionally with the system size $N$. The study reveals a linear growth of average magic with $N_T$ for small $N_T$, a saturation near Haar values for large $N_T$, and a predicted phase transition at a critical density $n_T^*\approx 2.41$. These results highlight that magic serves as a more sensitive indicator of complexity than spectral degeneracies in doped Clifford circuits. The findings have implications for understanding the resources required for universal quantum computation and the classical simulability of quantum dynamics.

Abstract

We study the emergence of complexity in deep random $N$-qubit $T$-gate doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the non-stabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. $T$-gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford+$T$ ensemble, and determine the distribution of magic, as well as the average non-stabilizing power of the quantum circuit ensemble. In the dilute limit, $N_T \ll N$, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of $T$-gates, $N_T$, and displays a discrete distribution for small $N_T$. At $N_T\approx N$, the distribution becomes quasi-continuous, and for $N_T\gg N$ it converges to that of Haar-random unitaries, and averages to a finite magic density, $m_2$, $\lim_{N\to\infty} \langle m_2 \rangle_\text{Haar} = 1$. This is in contrast to the spectral transition, where ${\cal O} (1)$ $T$-gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.

Figures (9)

• Figure 1: (a) Unitary operator $U \in \mathcal{U}(2^N)$ operating on a $N$-qubit register $\{q_1, q_2, \ldots, q_N\}$. (b) Schematic construction of a brick-wall random Clifford$+T$ gates circuit. Each layer includes $2$-qubit Clifford gates $C \in \mathcal{C}_2$ acting on neighboring qubits and (for doped circuits) a layer of randomly injected$T$-gates. Typical circuits considered in the following contain $D=5N$ such layers (see text for details).
• Figure 2: (a) Parity-integrated probability $P(L)$ for $N = 16$. (b) Integrated probabilities $I(L/L_{\mathrm{max}})$ for $N = \{2, 3, \dots, 16\}$ (from blue to red). The vertical black lines denote representative rational values $L/L_{\mathrm{max}}=p/q$.
• Figure 3: (a,b) Representative eigenvalue distribution of the Clifford operator $\mathbb C$ on the unit circle, corresponding to orbits of different length $L$ and parity $\tau$. (c,d) Phase correlation function for $N = 3$ and $N = 7$ random Clifford circuits. While collecting the data, we used box sizes $\Delta \Theta = 2\pi / 16000$ and $\Delta \Theta = 2\pi / 256000$. Note the common structure with the highest peaks located at rational multiples of $\pi$ (see text for a discussion). Histograms were obtained by sampling up to $N_s=2^{16}$ circuits.
• Figure 4: (a) Comparison of a representative correlation function $\chi(\Theta)$ for a random Clifford circuit of $N=5$ qubits with $N_T=20$ injected $T$-gates and of depth $D=25$ with (blue) against the corresponding analytical result from RMT for the CUE from Eq. (\ref{['chiRMT']}) (grey). (b) Evolution of $\chi(\Theta)$ with an increasing number $N_T$ of $T$-gates for a random Clifford circuit of $N=5$ qubits, and (c) suppression of the weight $\omega(\pi)$ of the delta peak at $\Theta=\pi$ with the number of $T$-gates for different values of the circuit depth, $D=5$, $D=15$, and $D=25$. (d) Suppression of the weight of the $\Theta=\pi$ peak as a function of $N_T$ for different values of $N$ and $D=5N$.
• Figure 5: Evolution of the correlation function for $N = 5$ qubits with the number of $T$-gates $N_T$. We used box sizes $\Delta \Theta = 2\pi / 128000$.