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Disentangling magic states with classically simulable quantum circuits

Gerald E. Fux, Benjamin Béri, Rosario Fazio, Emanuele Tirrito

TL;DR

This paper demonstrates that states produced by deep random Clifford circuits interleaved with non-Clifford gates can often be disentangled by transferring non-Clifford resources into the initial state, casting the evolution into a Clifford circuit acting on a magic product state (CAMPS) up to a threshold $t^* \approx N-1.607$. It provides a rigorous QEC-based condition for disentangling, proves that Pauli expectations can be computed efficiently for $t<t^*$, and supports the theory with numerical evidence showing near-linear entanglement disentangling and a controllable magic content. The work further connects these findings to approximate state designs, circuit compression, and potential extensions to Hamiltonian dynamics, while noting fundamental limits for sampling and the role of non-Clifford resources. Overall, CAMPS offers a unifying framework for understanding when Clifford-plus-non-Clifford circuits admit efficient classical descriptions and how this interacts with entanglement, nonstabilizerness, and information-design tasks.

Abstract

We show that states obtained from deep random Clifford circuits doped with non-Clifford phase gates (including T-gates and $\sqrt{\mathrm{T}}$-gates) can be disentangled completely, provided the number of non-Clifford gates is smaller or approximately equal to the number of qubits. This implies that Pauli expectation values of such states can be efficiently simulated classically, despite them exhibiting both extensive entanglement and extensive nonstabilizerness. We prove this result analytically using a quantum error correction formulation, demonstrate its applicability numerically, and discuss consequences for the disentanglability of states generated through Hamiltonian dynamics. We show that this result implies a novel representation of approximate state designs that can also facilitate their efficient generation, and we propose a novel quantum circuit compression scheme for Clifford circuits doped with non-Clifford phase gates.

Disentangling magic states with classically simulable quantum circuits

TL;DR

This paper demonstrates that states produced by deep random Clifford circuits interleaved with non-Clifford gates can often be disentangled by transferring non-Clifford resources into the initial state, casting the evolution into a Clifford circuit acting on a magic product state (CAMPS) up to a threshold . It provides a rigorous QEC-based condition for disentangling, proves that Pauli expectations can be computed efficiently for , and supports the theory with numerical evidence showing near-linear entanglement disentangling and a controllable magic content. The work further connects these findings to approximate state designs, circuit compression, and potential extensions to Hamiltonian dynamics, while noting fundamental limits for sampling and the role of non-Clifford resources. Overall, CAMPS offers a unifying framework for understanding when Clifford-plus-non-Clifford circuits admit efficient classical descriptions and how this interacts with entanglement, nonstabilizerness, and information-design tasks.

Abstract

We show that states obtained from deep random Clifford circuits doped with non-Clifford phase gates (including T-gates and -gates) can be disentangled completely, provided the number of non-Clifford gates is smaller or approximately equal to the number of qubits. This implies that Pauli expectation values of such states can be efficiently simulated classically, despite them exhibiting both extensive entanglement and extensive nonstabilizerness. We prove this result analytically using a quantum error correction formulation, demonstrate its applicability numerically, and discuss consequences for the disentanglability of states generated through Hamiltonian dynamics. We show that this result implies a novel representation of approximate state designs that can also facilitate their efficient generation, and we propose a novel quantum circuit compression scheme for Clifford circuits doped with non-Clifford phase gates.

Paper Structure

This paper contains 9 sections, 1 theorem, 9 equations, 5 figures.

Table of Contents

  1. Setup.
  2. Results.
  3. Consequences for Hamiltonian dynamics.
  4. Applications.
  5. Discussion and conclusion.
  6. Acknowledgements.
  7. End Matter
  8. Hamiltonian dynamics.
  9. Matchgate disentangling.

Key Result

Theorem 1

Let $\ket{\psi}= C \left( \ket{\varphi}\otimes\ket 0^{\otimes(N-k)}\right)$ be an $N$-qubit encoding of a $k$-qubit logical state $\ket{\varphi}$ through a Clifford unitary $C$, with $0 \leq k < N$. If Pauli operator $P\in\mathcal{P}_N$ is not a logical operator of the corresponding $[N,k]$ stabiliz where $\ket{x(\phi)}=\ket{0}$ or $\ket{x(\phi)} = e^{i \phi Y}\ket{0}$.

Figures (5)

  • Figure 1: Panel (a): For $t\lesssim N-O(1)$, the action on $\ket{0}^{\otimes N}$ of a quantum circuit, with a uniformly random global Clifford $C_i$ (i.e., deep random Clifford circuit, blue boxes) on all qubits followed by a T-gate (red boxes) at every time step equals (up to a phase) the state on the right (with $\tilde{C}$ Clifford) in almost all instances for large $N$. Panel (b): a general phase gate $e^{i\phi P}$ applied onto the logical state $\ket{\varphi}$ encoded by a Clifford unitary $C$. When the QEC conditions for $P$ hold, Theorem \ref{['th:theorem']} shows that this equals the logical state $\ket{\varphi}\otimes \ket{x(\phi)}$ encoded by a Clifford unitary $\tilde{C}$.
  • Figure 2: The evolution of the maximal entanglement entropy density of the MPS ($\mathcal{E}/N$) of the CAMPS [panel (a)] and the stabilizer Rényi entropy density ($\mathcal{M}/N$) for the Clifford + T dynamics [panel (b)] for different system sizes $N$, averaged over 256 random Clifford sequences. Panel (a) shows that the states generated by the Clifford + T circuits can be almost completely disentangled for the first $N$ time steps. The horizontal dashed line in panel (b) marks the SRE density of $\ket{T}^{\otimes N}$.
  • Figure 3: The maximal EE density of the MPS of the CAMPS ansatz versus the SRE density for the Clifford + T-gate circuit (solid lines) and the Clifford + $\sqrt{\mathrm{T}}$-gate circuit (dashed lines). These results follow from $\mathcal{M}(\sqrt{T}\ket{+})\approx 0.2075$ being the half of $\mathcal{M}(T\ket{+}) \approx 0.4150$ and that in both cases the MPS of the CAMPS ansatz can be completely disentangled for approximately $N$ time steps but not further.
  • Figure 4: The evolution of the maximal EE (a) and the SRE density (b) generated by the Ising Hamiltonian of Eq.\ref{['eq:hamiltonian-ising']} for $h_x=0.3$ and different system sizes $N$. The solid lines in panel (a) show the EE of the fully evolved state while the dotted lines show the EE of the MPS of the CAMPS ansatz. We notice that the CAMPS ansatz only partially disentangled the state at certain very early times or for very small system sizes. Also, we can see in panel (b) that the SRE density quickly grows and saturates above the SRE density of the $\ket{T}^{\otimes N}$ state, indicated with the dashed horizontal line.
  • Figure 5: The evolution of the maximal EEW and the SRE density generated by the Ising Hamiltonian of Eq.\ref{['eq:hamiltonian-ising']} for different values of $h_x$ and $N=16$. The solid lines in panels (a) and (b) show the entropy of the evolved state, while the dotted and dashed lines show the entropy of the MPS of the CAMPS ansatz and of the state back-propagated with the matchgate circuit $\bar{U}^\dagger(t)$, respectively. The horizontal dashed line in panel (b) marks the SRE density of the $\ket{T}^{\otimes N}$ state.

Theorems & Definitions (2)

  • Theorem 1
  • proof