MPStab: an hybrid stabilizers tensor-network quantum circuit simulator

Technical Summary of "MPStab: a hybrid stabilizers tensor-network quantum circuit simulator"

Problem Statement
Simulating quantum systems on classical computers faces a fundamental bottleneck: the exponential growth of the state space dimension with the number of qubits. While statevector simulations are exact, they become intractable beyond a few dozen qubits. Existing structured algorithms attempt to bypass this by exploiting specific circuit properties, but they face distinct limitations:

• Tensor Networks (e.g., MPS): Efficient for systems with bounded entanglement (local interactions, short-time dynamics) but fail when entanglement grows rapidly, requiring an exponential increase in bond dimension.
• Stabilizer Methods: Efficient for Clifford circuits (polynomial time) by tracking Pauli stabilizers but lack universality; they cannot efficiently handle non-Clifford gates (e.g., $T$ gates, arbitrary rotations) which introduce "non-stabilizerness" (magic).

The challenge is to simulate circuits that contain both significant entanglement (requiring tensor networks) and non-Clifford operations (requiring stabilizer tracking), regimes where neither method alone is efficient.

Methodology: The HSMPO Formalism
The paper introduces mpstab, a Python library implementing the Hybrid Stabilizer-Matrix Product Operator (HSMPO) formalism. The core methodology relies on decomposing a universal quantum circuit $U$ into alternating blocks of Clifford unitaries ($C_j$) and single-qubit "magic" rotations ($R_j$):
$$U = C_0 R_1 C_1 R_2 \dots R_N C_N$$

The algorithm exploits the property that Clifford conjugation maps Pauli strings to Pauli strings. By "pushing" Clifford gates past magic rotations via conjugation ($C_j R_j C_j^\dagger = \tilde{R}_j$), the circuit is rewritten as a sequence of "dressed" rotations $\tilde{R}_j$ followed by a residual Clifford evolution.

• Dressed Rotations: Each $\tilde{R}_j$ is a rotation around a (potentially non-local) Pauli string. Crucially, any such rotation admits an exact Matrix Product Operator (MPO) representation with a fixed bond dimension of 2, regardless of the non-locality of the Pauli string.
• Hybrid Evolution:
  1. Backpropagation (Heisenberg Picture): Clifford operations are applied to the observables using efficient tableau arithmetic (stabilizer formalism). This is exact and scales polynomially.
  2. Forward Evolution (Schrödinger Picture): The dressed rotations $\tilde{R}_j$ are applied to the quantum state (represented as an MPS) as MPOs. The state is compressed after each step to maintain a maximum bond dimension $\chi$.
• Expectation Values: The final expectation value is computed by contracting the evolved MPS with the backpropagated observable.

Key Contributions

1. mpstab Library: The authors present an open-source, modular Python package that implements the HSMPO framework. It integrates with the Qibo ecosystem, allowing users to define circuits using standard syntax while leveraging hybrid simulation.
2. Modular Architecture: The package separates the simulation into independent engines:
   • Stabilizer Engines: Handles Clifford backpropagation (e.g., via Stim).
   • Tensor Network Engines: Handles MPS/MPO evolution (e.g., via Quimb with Jax backend).
   • This modularity allows users to swap backends, enable automatic differentiation, or integrate custom engines.
3. Error Mitigation Features: The library includes functionality to generate "surrogate" circuits by probabilistically replacing magic gates with Clifford gates, a feature useful for data-driven error mitigation techniques like Clifford Data Regression.
4. Complexity Analysis: The authors derive a computational cost model comparing HSMPO to standard MPS. They show that HSMPO offers a polynomial-time advantage for circuits dominated by Clifford gates with moderate non-Clifford content, provided the required bond dimension for a target fidelity is significantly lower in the hybrid approach than in pure MPS.

Results and Benchmarks
The paper benchmarks mpstab against a pure tensor network simulator (Quimb) using a custom ansatz that independently controls the amount of entanglement (via CNOT ladder depth) and "magic" content (via the fraction of non-Clifford rotations, $\beta$).

• Cost Validation: Wall-clock measurements confirm the analytical cost model. A crossover point exists where mpstab becomes more efficient than pure MPS; this occurs when the circuit has a high ratio of Clifford to non-Clifford gates and the required bond dimension is large.
• Fidelity Advantage: At a fixed bond dimension $\chi$, mpstab consistently achieves higher fidelity than pure MPS, particularly as circuit depth increases. The stabilizer component absorbs the entanglement generated by Clifford gates exactly, leaving the MPS to represent only the entanglement from the dressed non-Clifford rotations.
• Scalability: The advantage of mpstab is most pronounced in regimes with low non-Clifford gate counts ($N_{nC}$) and high Clifford counts ($N_C$). As the system size increases, the advantage persists only if the magic retention $\beta$ is tuned such that the total number of magic gates does not scale prohibitively.

Significance
The paper positions mpstab as a practical tool for extending the boundaries of classical quantum simulation. It does not claim to simulate arbitrary large-scale quantum computers but rather targets specific regimes where classical computation remains viable: circuits with substantial Clifford structure and moderate non-Clifford injection.

The significance lies in:

1. Bridging the Gap: It effectively combines the efficiency of stabilizer methods for entanglement generation with the universality of tensor networks for non-Clifford operations.
2. Practical Utility: By integrating with Qibo and offering modular engines, it facilitates the use of hybrid simulation in variational algorithms and error mitigation workflows.
3. Regime Identification: The authors provide a concrete "phase map" (based on Clifford vs. non-Clifford gate counts and bond dimensions) that guides users on when to employ mpstab over standard simulators.

The work concludes that mpstab is the preferred simulator for Clifford-dominated circuits with moderate non-Clifford content, where it achieves higher accuracy at lower computational cost compared to pure tensor network approaches.