Noise mitigation of quantum observables via learning from Hamiltonian symmetry decays

1. Problem Statement

Quantum computers in the Noisy Intermediate-Scale Quantum (NISQ) era suffer from significant noise, leading to inaccurate expectation values for observables. While Quantum Error Correction (QEC) is the long-term solution, it requires massive overheads not yet available. Quantum Error Mitigation (QEM) techniques, such as Zero Noise Extrapolation (ZNE), are used to reduce noise bias by post-processing data from noisy circuits.

However, existing ZNE methods face critical challenges:

• Unknown Decay Curves: The relationship between noise strength and observable decay is often unknown. Complex circuits may exhibit multi-exponential decays, making simple linear or exponential extrapolation unstable or inaccurate.
• Noise Amplification Uncertainty: Techniques like gate folding (used to amplify noise) are heuristic and may not scale noise linearly or accurately, leading to biased extrapolations.
• Observable Weight Dependence: The decay rate of an observable depends on its "weight" (the number of non-identity Pauli operators). Global symmetries (e.g., total magnetization) decay differently than local observables, making it difficult to use global symmetries to correct local measurements.

2. Methodology: GUESS

The authors propose GUESS (GUiding Extrapolations from Symmetry decayS), a novel QEM technique that learns the noise profile of a system using Hamiltonian symmetries to guide the extrapolation of target observables.

Core Concept

Instead of assuming a decay model (like ZNE), GUESS learns the specific decay coefficients by exploiting the fact that Hamiltonian symmetries (operators $S$ where $[H, S] = 0$) should ideally remain constant (conserved) under unitary evolution. Any observed decay in a symmetry's expectation value is purely due to noise.

Key Technical Steps

1. Hamiltonian Impurity (Perturbation):

   • Many relevant Hamiltonians (like Ising or Heisenberg models) lack local symmetries of the same weight as the target observables.
   • To solve this, the authors introduce a local impurity (a perturbation $I_{\bar{q}}$) to the target Hamiltonian $H_0$, creating a perturbed Hamiltonian $H_I = H_0 + I_{\bar{q}}$.
   • This perturbation is designed to enforce a specific symmetry $S$ (matching the weight and qubit support of the target observable $O$) while preserving the circuit structure (specifically the two-qubit gate count and depth). This ensures the noise propagation characteristics remain similar to the original circuit.

2. Learning Coefficients (Step 1):

   • The system is run on the perturbed Hamiltonian at various noise amplification factors ($G_k$).
   • Since the ideal value of the symmetry $\langle S \rangle_{ideal}$ is known (usually 1), the authors solve a constrained least-squares problem to find optimal extrapolation coefficients $\hat{x}$ that map the noisy symmetry measurements to the ideal value.
   • Constraint: $\sum |x_i| = 1$ (to avoid indeterminate systems).

3. Mitigation (Step 2):

   • The learned coefficients $\hat{x}$ are applied to the noisy measurements of the target observable (from the unperturbed Hamiltonian) to estimate its noise-free value.
   • The method assumes that the noise decay behavior of the symmetry (learned from the perturbed circuit) is representative of the target observable (in the unperturbed circuit), provided their weights and noise propagation paths are similar.

4. Statistical Post-Selection:

   • The symmetry measurements provide real-time data on the quality of specific qubits.
   • The authors use the mean and variance of the symmetry outcomes to detect outliers (defective qubits or high-noise runs) and discard unreliable data points before averaging, significantly improving result fidelity.

3. Key Contributions

• Symmetry-Guided Learning: A new paradigm for QEM that replaces heuristic decay assumptions with data-driven learning from conserved quantities.
• Hamiltonian Impurity Technique: A method to artificially create local symmetries of arbitrary weight without altering the circuit's noise propagation profile (crucial for maintaining the validity of the learned coefficients).
• Robustness to Amplification Errors: Unlike standard ZNE, GUESS does not explicitly rely on the precise knowledge of noise amplification factors ($G_k$). As long as the symmetry and target circuits use the same amplification technique, the method remains robust against scaling inaccuracies.
• Statistical Post-Selection: Utilizing symmetry outcomes to filter out noisy qubits, providing a built-in quality control mechanism for the quantum processor.

4. Experimental Results

The authors validated GUESS on the IBM Heron r2 quantum processor (ibm_basquecountry) using 100-qubit systems.

• Models Tested:

  • Transverse Field Ising Model (TFIM).
  • $XZ$-Heisenberg Model.
  • Simulated time evolution up to 44 Trotter steps (Ising) and 24 steps (Heisenberg).
  • Circuits contained up to 8,000 two-qubit (CZ) gates.

• Performance Metrics:

  • Accuracy: GUESS achieved a relative error of ~10% for circuits with up to 8,000 gates, significantly outperforming baseline ZNE and raw measurements.
  • Variance: GUESS demonstrated lower variance (sampling overhead) compared to ZNE across 20 different observables.
  • Physicality: The method drastically reduced the occurrence of non-physical results (expectation values $>1$ or $<-1$). For example, in the Ising model (weight-1), non-physical values dropped from ~81% (Exponential ZNE) to 0.07% (Linear GUESS).
  • Efficiency: It requires only twice the number of shots per observable compared to baseline ZNE (one set for the target, one for the symmetry), which is a manageable overhead compared to other learning-based methods like Clifford Data Regression (CDR).

5. Significance and Implications

• Utility-Scale Validation: This work demonstrates that high-accuracy error mitigation is possible on utility-scale circuits (100 qubits, deep circuits) on current hardware, bridging the gap between NISQ and fault-tolerant computing.
• Early Fault-Tolerance Bridge: The method is particularly suited for the early fault-tolerant era. It can be applied to logical qubits where noise amplification is difficult to characterize precisely, as it relies on symmetry conservation rather than precise noise scaling.
• Scalability: By avoiding the need for complex noise learning stages (like Probabilistic Error Cancellation) or massive training sets (like CDR), GUESS offers a scalable, low-overhead path to improving quantum simulation accuracy.
• Hardware Agnosticism: The technique leverages the fundamental physics of the system (symmetries) rather than specific hardware noise models, making it adaptable to various quantum architectures.

In conclusion, GUESS represents a significant advancement in QEM by transforming the problem of "guessing" noise decay into a "learning" problem grounded in physical symmetries, delivering high accuracy and reduced variance for large-scale quantum simulations.