Quantum State Designs via Magic Teleportation

TL;DR

The paper tackles the problem of generating quantum state designs using Clifford circuits doped with finite magic, by analyzing the projected ensemble obtained after partial Pauli measurements. It introduces the Magic-Induced Design Ansatz (MIDA), linking the convergence to k-designs to the Stabilizer Renyi Entropies M_k of the pre-measurement state, and validates this via analytical arguments and large-scale numerics using the frame potential. For deep circuits, the projected ensemble approaches a k-design with an exponential dependence on M_k, while in shallow circuits the emergence of randomness exhibits magic-teleportation-like transport, with depth scaling dictated by locality and long-range connectivity. The findings provide a principled route to realize highly random quantum state designs in near-term fault-tolerant devices, leveraging a controlled amount of magic to achieve Haar-like randomness efficiently.

Abstract

We investigate how non-stabilizer resources enable the emergence of quantum state designs within the projected ensemble. Starting from initial states with finite magic and applying resource-free Clifford circuits to scramble them, we analyze the ensemble generated by performing projective Pauli measurements on a subsystem of the final state. Using both analytical arguments and large-scale numerics, we show that the projected ensemble converges towards a state $k$-design with an error that decays exponentially with the $k$-th Stabilizer Renyi Entropy of the pre-measurement state, via a Magic-Induced Design Ansatz (MIDA) that we introduce. We identify a universal scaling form, valid across different classes of magic initial states, and corroborate it through numerical simulations and analytical calculations of the frame potential. For finite-depth Clifford unitaries, we show that the timescales at which state designs emerge are controlled by the transport of magic. We identify a ``magic teleportation'' mechanism whereby non-Clifford resources injected locally spread through Clifford scrambling and measurements across distances beyond the lightcone. Our results demonstrate how a small and controlled amount of magic suffices to generate highly random states, providing a systematic route toward generating quantum state designs in early fault-tolerant devices.

Figures (7)

• Figure 1: Scheme of the main protocol used in this work. (a) A Clifford operator $\mathcal{C}$ is applied onto an initial state $\ket{\psi_0}$ with known finite magic injected somewhere in partition $B$. The projected ensemble in $A$ is obtained by measuring out the degrees of freedom of subsystem $B$ in the $Z$ basis. The vertical lines represent qubit degrees of freedom. (b) The Clifford operator $\mathcal{C}$ will be mainly decomposed in a brickwork circuit of random 2-qubit Clifford gates with open boundary conditions and depth (time) $t$. On the right, a table that synthesises the main findings of the work. For deep circuits, the Haar distance is zero for $k=1$ in all cases. The shallow circuits section assumes the magic is injected at a distance $O(L)$ from partition $A$.
• Figure 2: (a) Projected ensemble normalized distance to Haar $\Delta_A^{(k)}$, as a function of the initial number of magic qubits $N_P$ with phase $\theta \in \{\frac{\pi}{4}, \frac{\pi}{9}\}$. Different values of $k$ are explored (see colorbar on the right). (b) Decay rate of the Haar distance with the number of initial magic qubits, i.e. $\Delta_A^{(k)} \sim \exp(-\gamma N_P)$. We plot $\gamma$ as a function of the phase $\theta$, for different $k$. Dashed lines represent the $k$-th Stabilizer Rényi Entropy density of the pre-measurement state $M_k(\ket{\psi})/N_P$, up to a normalization constant. Dash-dotted lines represent the $2$-Stabilizer Rényi Entropy density, $M_2(\ket{\psi})/N_P$, which clearly fails to capture the behavior of the data. (c,d,e,f) Projected ensemble normalized distance to Haar $\Delta^{(k)}_A$ as a function of $(k-1)M_k(\ket{\psi})$. For visualization purposes, the $y$-axis is rescaled by $k/\mu$ (see Eq. \ref{['eq_mu']}). The blue curves correspond to initial product phase states with $N_P \in \{1,\dots, 6\}$ and $\theta \in [0, \pi/4]$. The orange curves correspond to random initial states with fixed magic $M \in [0, 2]$ (protocol in SMAT). The system size is $L = 22$ in all plots, while $L_A = 1$ in (c), $L_A = 2$ in (d), $L_A = 4$ in (e), and $L_A = 6$ in (a,b,f).
• Figure 3: (a,b) Projected ensemble normalized distance to Haar $\Delta_A^{(k)}$, as a function of circuit depth $t$. In the initial state $\ket{\psi_0^{\rm (left)}}$ of panel (a), magic is localized near subsystem $A$, whereas in the initial state $\ket{\psi_0^{\rm (right)}}$ of panel (b), it is concentrated on the opposite side. In both cases, $N_P = 1$ and $\theta = \pi/4$. In (b,inset) the saturation timescales are extracted by fitting $\Delta_A^{(k)} \sim \exp(-\tau t)$. (c,d) Average Von Neumann magic in $A$ (blue) and second frame potential (orange), as a function of time $t$. For visualization purposes, both quantities are normalized to take values in $[0,1]$. A Clifford circuit with long-range connectivity $\alpha = 1$ (c) and $\alpha = 3$ (d) is used to produce the pre-measurement state (see inset in (c)). Different curves represent different values of $k$ and $L$, with fixed $L_A = 1$. Different choices of initial states and $L_A$ (with the same scaling of the distance between the magic injection and subsytem $A$) yield qualitatively identical behaviors.
• Figure 4: (a) Reduced chi-square as a function of $L$, for different $k$ and $L_A$. (b) Reduced chi-square as a function of $L_A$, for different $k$ and fixed $L = 22$. In dashed, $\chi^2 = 1$ signalling the expected value for perfectly compatible datasets. $N_0 = 80$ different initial states with $M \in [0.2, 2.1]$ were used for the computation of $\chi^2$ (states with $M \ll 1$ removed due to disproportionately affecting $\chi^2$ with low statistical uncertainties, revealing mostly finite-size errors in $\mu$).
• Figure 5: (a) Scheme for a modified version of the protocol in Fig. \ref{['fig_protocol_scheme']}. In this case, the initial state is just a stabilizer product state such as $\ket{0}^{\otimes L}$. Instead, the magic is injected upon rotation of the measurement basis with $N_P$ phase shift gates $P(\theta)$ and Hadamard gates $H$. The measurements are performed in the $Z$ basis and the projected ensemble is obtained in subsystem $A$. (b) Time-dependent comparison for $\Delta_A^{(k)}$ between the protocols of (a) (in blue) and Fig. \ref{['fig_protocol_scheme']} with initial state $\ket{\psi_0^{\rm right}}$ (in orange), for different $k$. We fix $L_A = N_P = 1$, $\theta = \pi/4$, $L = 16$, although other parameter choices result in the same matching behaviour between the protocols.