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Limits of Clifford Disentangling in Tensor Network States

Sergi Masot-Llima, Piotr Sierant, Paolo Stornati, Artur Garcia-Saez

Abstract

Tensor network methods leverage the limited entanglement of quantum states to efficiently simulate many-body systems. Alternatively, Clifford circuits provide a framework for handling highly entangled stabilizer states, which have low magic and are thus also classically tractable. Clifford tensor networks combine the benefits of both approaches, exploiting Clifford circuits to reduce the classical complexity of the tensor network description of states, with promising effects on simulation approaches. We study the disentangling power of Clifford transformations acting on tensor networks, with a particular emphasis on entanglement cooling strategies. We identify regimes where exact or heuristic Clifford disentanglers are effective, explain the link between the two approaches, and characterize their breakdown as non-Clifford resources accumulate. Additionally, we prove that, beyond stabilizer settings, no Clifford operation can universally disentangle even a single qubit from an arbitrary non-Clifford rotation. Our results clarify both the capabilities and fundamental limitations of Clifford-based simulation methods.

Limits of Clifford Disentangling in Tensor Network States

Abstract

Tensor network methods leverage the limited entanglement of quantum states to efficiently simulate many-body systems. Alternatively, Clifford circuits provide a framework for handling highly entangled stabilizer states, which have low magic and are thus also classically tractable. Clifford tensor networks combine the benefits of both approaches, exploiting Clifford circuits to reduce the classical complexity of the tensor network description of states, with promising effects on simulation approaches. We study the disentangling power of Clifford transformations acting on tensor networks, with a particular emphasis on entanglement cooling strategies. We identify regimes where exact or heuristic Clifford disentanglers are effective, explain the link between the two approaches, and characterize their breakdown as non-Clifford resources accumulate. Additionally, we prove that, beyond stabilizer settings, no Clifford operation can universally disentangle even a single qubit from an arbitrary non-Clifford rotation. Our results clarify both the capabilities and fundamental limitations of Clifford-based simulation methods.
Paper Structure (14 sections, 1 theorem, 40 equations, 11 figures)

This paper contains 14 sections, 1 theorem, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Clifford-based simulation methods
  4. Entanglement cooling in Clifford enhanced Tensor Networks
  5. Limitations of CTNs
  6. Heuristic disentangling with higher locality Clifford gates
  7. Optimality of 2-local heuristic
  8. Limits on exact entanglement cooling
  9. Entanglement build-up with non-Clifford rotations
  10. Conclusions
  11. Exact entanglement cooling
  12. Complete proof for the No cooler disentanglement
  13. Simplifed proof
  14. General proof

Key Result

Theorem 3.1

Let be an $N$-qubit rotation with $\alpha=\cos(\theta),\beta=-i\sin(\theta)$, and an initial state where $\ket{\Psi}$ is an $N-1$ qubit state and $\ket{\phi_n}$ is a $1$-qubit state. If a unitary $\mathcal{U}$ is applied to $\left( \alpha I + \beta P_1 \dots P_n \right) \ket{\Psi}\otimes\ket{\phi_n}$, for arbitrary $\theta$,$\ket{\Psi}$, that results in a state of the form $\ket{\tilde{\Psi}}\o

Figures (11)

  • Figure 1: Simulations with Clifford enhanced TN (MPS in the example). In a), we can find a Clifford unitary $\mathcal{C}$ that reduces bond dimension $\chi$ of an MPS $\ket{\nu}$, yielding $\ket{\nu'}$. Any Clifford gate in the circuit, such as those marked i), can be absorbed into $\mathcal{C}$ to obtain $\mathcal{C}'$, whereas non-Clifford rotations must be conjugated with $\mathcal{C}$ before being contracted to $\ket{\nu'}$. The search for a better $\mathcal{C}$ can be repeated at any step of the circuit simulation. In b), the Hamiltonian evolution of a quantum state with MPS and time steps $dU$ can similarly be improved with a disentangling Clifford gate that decreases $\chi$ of $\ket{\nu}$, whereas the conjugate is contracted with the Hamiltonian, giving $dU'$.
  • Figure 2: Heuristic $K$-local entanglement cooling illustrated for $k=2$. Proceeding in a sweep over the MPS, each pair (group of $k$) of sites are contracted with the elements of $\mathcal{C}_2$ ($\mathcal{C}_k$), to find the one with minimum entanglement entropy $S$, which is chosen for that step. The procedure stops after $d$ sweeps, what we call depth, and returns the resulting Clifford gate $C^d_2$ ($C^d_k$).
  • Figure 3: Progression of the maximal entanglement $max(S)$ across all sites of an sTN during the simulation of a Clifford+$T$-gates circuit, for different sizes $N$, and up to $3N$$T$-gates. We compare circuits where the entanglement cooling in Fig. \ref{['fig:ent_cool_heur']} was used (in blue) to those where it was not (red), and we normalize both axes with $N/2$ to compare different sizes better. Dashed blue lines correspond to the entropy bound $S_b(N)$Page93entropy for finite $N$, in ascending vertical order.
  • Figure 4: Comparison between the average (solid lines) entanglement of $m=100$ circuits using heuristic optimization with $k=2$ (red) vs $k=3$ (blue) on a sTN simulating a circuit of $12$ qubits with up to $24$$T$-gates. In dashed, partially transparent lines we show each individual run, whereas the error bars display the standard error of the mean $\sigma/\sqrt{m}$.
  • Figure 5: Entropy (y axis) of a state constructed with Clifford and $x$ T-gates (x axis), after being disentangled by the $2$-local heuristic with different sweep depths $d$, in different colors. The error bars represent the standard error (SE) of the mean ($\sigma/\sqrt{N}$) for the two extreme cases (depth 1 and depth 5).
  • ...and 6 more figures

Theorems & Definitions (5)

  • Definition 1: Clifford Tensor Networks
  • Definition 2: $K$-local entanglement cooling
  • Definition 3: Exact entanglement cooling
  • Theorem 3.1
  • Definition : Exact entanglement cooling