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Approximation Methods for Simulation and Equivalence Checking of Noisy Quantum Circuits

Mingyu Huang, Ji Guan, Wang Fang, Mingsheng Ying

TL;DR

This work tackles the scalability bottleneck of simulating and verifying noisy quantum circuits in the NISQ era by introducing a double-size tensor-network representation and an SVD-based low-rank approximation of noise super-operators. By retaining dominant noise components and bounding the residual error, the authors develop level-$l$ approximations with contraction complexity $O(15^l N^l)$ and an error that scales favorably with the noise rate, yielding practical performance with up to ~200 qubits and ~20 noise operators. Implemented in Google's TensorNetwork framework, the approach outperforms state-of-the-art methods (quantum trajectories, MPO-based) in both speed and memory, and provides an adaptable tool for approximate equivalence checking via Jamiołkowski fidelity. The methodology has tangible implications for quantum SDKs, hardware benchmarking, and automatic test pattern generation, enabling reliable testing and verification of large-scale noisy quantum circuits.

Abstract

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.

Approximation Methods for Simulation and Equivalence Checking of Noisy Quantum Circuits

TL;DR

This work tackles the scalability bottleneck of simulating and verifying noisy quantum circuits in the NISQ era by introducing a double-size tensor-network representation and an SVD-based low-rank approximation of noise super-operators. By retaining dominant noise components and bounding the residual error, the authors develop level- approximations with contraction complexity and an error that scales favorably with the noise rate, yielding practical performance with up to ~200 qubits and ~20 noise operators. Implemented in Google's TensorNetwork framework, the approach outperforms state-of-the-art methods (quantum trajectories, MPO-based) in both speed and memory, and provides an adaptable tool for approximate equivalence checking via Jamiołkowski fidelity. The methodology has tangible implications for quantum SDKs, hardware benchmarking, and automatic test pattern generation, enabling reliable testing and verification of large-scale noisy quantum circuits.

Abstract

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.

Paper Structure

This paper contains 28 sections, 4 theorems, 22 equations, 10 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Noisy Simulation
  3. Approximate Equivalence Checking
  4. Contributions of This Paper
  5. Organization of the Paper:
  6. Related Works
  7. Simulation of Quantum Circuits
  8. Equivalence Checking of Quantum Circuits
  9. Relation to Our Previous Work
  10. Preliminary
  11. Noisy Quantum Circuit Simulation
  12. Approximation Algorithm For Noisy Quantum Circuit Simulation
  13. Applications in Equivalence Checking
  14. Experiments
  15. Experiment Configurations:
  16. ...and 13 more sections

Key Result

Lemma 1

Suppose $A$ and $B$ are $n \times n$ matrices, and $\tilde{A}$ and $\tilde{B}$ are the tensor permutation of $A$ and $B$ as defined. If $\|A - B\| < \delta$, then we have $\|\tilde{A} - \tilde{B}\| < \sqrt{n} \delta$.

Figures (10)

  • Figure 1: Test Workflow of Classical Circuit
  • Figure 2: A 2-qubit QAOA circuit
  • Figure 3: Tensor Network Diagram for Noisy Simulation
  • Figure 4: Circuit for noisy simulation of the 2-qubit QAOA circuit in Fig. \ref{['fig:circuit_example']} with decoherence noise $\mathcal{E}_{D}$ by appending $| v \rangle = U| \psi \rangle$ to the end of the circuit, and redundant gates are canceled out.
  • Figure 5: Decomposition of $M_\varepsilon$
  • ...and 5 more figures

Theorems & Definitions (10)

  • Example 1: Decomposition of Depolarizing Noise
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1: Precision and Complexity of Algorithm \ref{['Algorithm:Approximate']}
  • proof
  • Definition 1: Jamiołkowski fidelity
  • Theorem 2: Precision and Complexity of Algorithm \ref{['Algorithm:ApproximateEC']}
  • proof