Impact of Clifford operations on non-stabilizing power and quantum chaos

TL;DR

The paper develops a rigorous framework for understanding how non-stabilizerness (magic) builds up and thermalizes in circuits that interleave Clifford and non-Clifford operations. It proves a central decoupling relation for the final non-stabilizing power under random Clifford insertions, showing exponential convergence to the Haar-averaged value and enabling a simple relaxation rate that depends only on the non-Clifford powers. By analyzing two-qubit gates and operator-space dynamics, it connects non-stabilizing power with entangling power and gate-typicality, and introduces OSNP to study how Clifford operators acquire magic under evolution. The work further demonstrates how the joint behavior of $m_p$, $e_p$, and $g_t$ governs quantum-chaos transitions in brick-wall Floquet circuits, revealing that chaos arises only when these quantities are collectively large. Together, these results illuminate the resource theory of magic in realistic circuits and its relationship to quantum chaos and simulability.

Abstract

Non-stabilizerness, alongside entanglement, is a crucial ingredient for fault-tolerant quantum computation and achieving a genuine quantum advantage. Despite recent progress, a complete understanding of the generation and thermalization of non-stabilizerness in circuits that mix Clifford and non-Clifford operations remains elusive. While Clifford operations do not generate non-stabilizerness, their interplay with non-Clifford gates can strongly impact the overall non-stabilizing dynamics of generic quantum circuits. In this work, we establish a direct relationship between the final non-stabilizing power and the individual powers of the non-Clifford gates, in circuits where these gates are interspersed with random Clifford operations. By leveraging this result, we unveil the thermalization of non-stabilizing power to its Haar-averaged value in generic circuits. As a precursor, we analyze two-qubit gates and illustrate this thermalization in analytically tractable systems. Extending this, we explore the operator-space non-stabilizing power and demonstrate its behavior in physical models. Finally, we examine the role of non-stabilizing power in the emergence of quantum chaos in brick-wall quantum circuits. Our work elucidates how non-stabilizing dynamics evolve and thermalize in quantum circuits and thus contributes to a better understanding of quantum computational resources and of their role in quantum chaos.

Figures (11)

• Figure 1: Space of two-qubit unitaries, illustrated as a tetrahedron parametrized by the Euler angles $c_x$, $c_y$, and $c_z$. Each point inside the tetrahedron uniquely determines the two-qubit unitaries up to single-qubit unitaries with $c_j\in[0, \pi/2)$ for all $j\in\{x, y, z\}$. In this work, we consider the unitaries that lie along the edges Id --- CNOT, CNOT --- DCNOT, SWAP --- DCNOT, and Id --- SWAP (or $S^{\alpha}$, the fractional powers of SWAP). These edges are marked with the orange color in the figure.
• Figure 2: Non-stabilizing power of the two-qubit unitaries as the Euler angles are varied while the local unitaries are taken to be Cliffords. (a) Id --- CNOT ($c_x = c_y = 0$, $c_z$ is varied from $0$ to $\pi/2$). Note that the non-stabilizing power remains invariant if $c_z$ is replaced by either $c_x$ or $c_y$, provided the other two parameters are kept at zero. (b) CNOT --- DCNOT ($c_x = \pi/2$, $c_z = 0$, $c_y$ is varied). Along this edge, the two-qubit gates have maximal entangling power. (c) DCNOT --- SWAP ($c_x=c_y=\pi/2$, variable $c_z$). (d) Fractional powers of the SWAP operator, $-i\text{SWAP}^{2J/\pi}$ for $J\in [0, \pi/2]$. In all panels, the numerical results for the non-stabilizing power are obtained by averaging over all sixty stabilizer states in the two-qubit space. In panels (a-c), the numerical results match exactly with the analytical expectation $\sin^2(2c_j)/5$, with $c_j$ representing the parameter being varied.
• Figure 3: The setting considered in this work involves a random $n$-qubit Clifford operation $C$ sandwiched between two arbitrary non-Clifford unitaries, denoted as $U$ and $V$. We compute the average non-stabilizing power of this configuration, where the averaging is performed over the Clifford group supported over $n$-qubits.
• Figure 4: Evolution and relaxation of the non-stabilizing power in quantum circuits for $N=2$ qubits under three different setups: (i) Identical non-Clifford unitaries interleaved with random Clifford elements (blue lines, bullets), (ii) two distinct non-Clifford operations alternately interspersed with random Clifford unitaries (orange, squares), and (iii) a periodic sequence of three distinct non-Clifford operations interspersed with random Clifford operations (green, triangles). Each non-Clifford operation is of the form $U = \exp\{-i c_x \sigma_x \otimes \sigma_x / 2\}$. In the first case, we set $c_z = \pi/4$. In the second case, we use $c_z \in \{\pi/4, \pi/8\}$, and in the third case, $c_z \in \{\pi/4, \pi/8, \pi/16\}$. The markers indicate the numerical results averaged over $10^3$ independent circuit realizations, while the dashed curves represent Eq. (\ref{['corrolary']}). Inset: Relaxation dynamics of $m_p$ towards the Haar value on a semi-log scale, for $N=2$ qubits (thick curves) as well as for the case where the above two-qubit non-Clifford gates are embedded in a $4$-qubit Hilbert space as $\mathbb{I}_{2} \otimes \exp\{-i c_x \sigma_x \otimes \sigma_x\} \otimes \mathbb{I}_{2}$, for the same settings of $c_x$ as above (dotted curves). In all scenarios considered, there is an exponential approach to the Haar value.
• Figure 5: Evolution and relaxation of the non-stabilizing power in circuits when non-Clifford unitaries with different values of $m_p$ are interspersed with random Clifford operations, for $N=2$ and $N=4$. For $N=2$, we use the same unitary form as before, i.e., $U=\exp\{-ic_x\sigma_x\otimes\sigma_x\}$, but now the $c_x$ are randomly drawn at every time step. For $N=4$, we use the same two-qubit gates embedded in a four-qubit Hilbert space. As shown in the inset, the overall relaxation remains, to a good approximation, exponential despite fluctuations around the mean values. For the same set of $c_x$ values, the case of $N=4$ shows a slower relaxation rate than the case of $N=2$. The numerical results are carried out for a single realization of the set of random $c_x$ values.

Theorems & Definitions (5)

• Theorem 4.1
• Corollary 4.1.1
• proof
• Theorem
• proof