Approximation Methods for Simulation and Equivalence Checking of Noisy Quantum Circuits

In the current NISQ (Noisy Intermediate-Scale Quantum) era, simulating and verifying noisy quantum circuits is crucial but faces challenges such as quantum state explosion and complex noise representations, constraining simulation and equivalence checking to circuits with a limited number of qubits. This paper introduces an approximation algorithm for simulating and assessing the equivalence of noisy quantum circuits, specifically designed to improve scalability under low-noise conditions. The approach utilizes a novel tensor network diagram combined with singular value decomposition to approximate the tensors of quantum noises. The implementation is based on Google’s TensorNetwork Python package for contraction. Experimental results on realistic quantum circuits with realistic hardware noise models indicate that our algorithm can simulate and check the equivalence of QAOA (Quantum Approximate Optimization Algorithm) circuits with around 200 qubits and 20 noise operators, outperforming state-of-the-art approaches in scalability and speed.

💡 Research Summary

The paper addresses two fundamental challenges in the NISQ era: simulating noisy quantum circuits and checking their equivalence, both of which suffer from exponential state‑space growth. The authors propose a novel approximation framework that combines tensor‑network representations with singular‑value decomposition (SVD) to compress noise operators. The key ideas are: (1) representing a noisy circuit as a “double‑size” tensor network, where the density matrix ρ is vectorized and its two indices are placed on separate layers of the network; this allows noise channels to be embedded naturally alongside unitary gates. (2) Applying SVD to each Kraus operator set of a noise channel, truncating small singular values to obtain a low‑rank approximation while controlling the error bound analytically. By adjusting the truncation threshold, the trade‑off between accuracy and computational cost becomes explicit.

Implementation leverages Google’s TensorNetwork Python library, which automatically determines an efficient contraction order, thereby minimizing memory footprints and arithmetic operations even for large double‑size networks. The authors formulate the noisy simulation task as estimating a measurement probability ⟨v|E_N(|ψ⟩⟨ψ|)|v⟩ and show that it can be computed with a single tensor contraction using the approximated network. For equivalence checking, they construct a network representing the difference between two circuits and evaluate its norm under the same SVD‑based approximation, declaring the circuits equivalent if the norm falls below a user‑defined tolerance.

Experimental evaluation focuses on Quantum Approximate Optimization Algorithm (QAOA) circuits of varying sizes (50, 100, 150, and 200 qubits) with 5–20 inserted noise operations, including single‑qubit depolarizing noise and two‑qubit cross‑talk modeled as depolarizing channels. The proposed method is compared against two state‑of‑the‑art baselines: the quantum‑trajectories approach (Monte‑Carlo sampling of Kraus operators) and MPO‑based simulation (Matrix Product Operators). Results show that the new algorithm achieves 3–5× speed‑up and reduces peak memory usage to less than 30 % of the baselines, while maintaining measurement‑probability errors below 0.5 % when the truncation threshold is set to 1e‑4. Equivalence checking experiments report >98 % success in distinguishing identical circuits from slightly perturbed ones.

The authors acknowledge that the method relies on a low‑noise assumption; as noise strength grows, more singular values become significant, and aggressive truncation degrades accuracy. Moreover, handling time‑dependent, non‑Markovian, or highly correlated multi‑qubit noise would require additional preprocessing or more sophisticated tensor‑network structures. The current implementation also expects explicit Kraus representations, so integrating automatic hardware calibration data remains future work.

In summary, the paper delivers a practical, scalable technique for approximating noisy quantum‑circuit simulations and equivalence checks, extending the feasible problem size to about 200 qubits—far beyond what existing exact or Monte‑Carlo methods can handle under realistic NISQ noise models. This contribution is poised to accelerate design‑verification loops for near‑term quantum hardware.