Unbiased observable estimation with approximate channels in fault-tolerant quantum computation

TL;DR

This work tackles the bias in observable estimates introduced by coherent unitary errors and approximate gate synthesis in quantum circuits. It introduces an unbiased estimator built from a linear mixture of noisy channels, augmented by Pauli twirling and circuit-level mixing, to recover the ideal channel's expectation values. The approach yields accurate observable estimates for moderate circuit sizes and gate-decomposition errors, with a controlled but exponential shot-cost overhead that limits scalability; it is especially valuable for reducing T-gate counts in early fault-tolerant scenarios and mitigating near-term coherent errors. Overall, the method provides a resource-aware path to bias mitigation in noisy quantum devices, with demonstrated applicability to Ising-model dynamics and potential extensions to broader noise models and device-specific calibrations.

Abstract

Unitary errors, such as those arising from fault-tolerant compilation of quantum algorithms, systematically bias observable estimates. Correcting this bias typically requires additional resources, such as an increased number of non-Clifford gates. In this work, we present an alternative method for correcting bias in the expectation values of observables. The method leverages a decomposition of the ideal quantum channel into a probabilistic mixture of noisy quantum channels. Using this decomposition, we construct unbiased estimators as weighted sums of expectation values obtained from the noisy channels. We provide a detailed analysis of the method, identify the conditions under which it is effective, and validate its performance through numerical simulations. In particular, we demonstrate unbiased observable estimation in the presence of unitary errors by simulating the time dynamics of the Ising Hamiltonian. Our strategy offers a resource-efficient way to reduce the impact of unitary errors, improving methods for estimating observables in noisy near-term quantum devices and fault-tolerant implementation of quantum algorithms.

Figures (11)

• Figure 1: Estimating the expectation value $\braket{\mathcal{O}}$ in the presence of unitary errors along the $X$-, $Y$-, and $Z$-axes. Unitary error, modeled by Eq. \ref{['eq:unstructured_error']} and applied at every parametrized gate in the circuit (Eq. \ref{['eq:noisy_app_circuit']}), causes significant deviation from the exact value (dashed blue line). The noisy expectation value (dashed red line) demonstrates this deviation. By employing our method, the estimated expectation value moves closer to the exact one. Further improvement is observed when combining the proposed strategy with Pauli twirling, as indicated by the brown dashed line closely aligning with the blue dashed line. However, the variance of the expectation value distribution increases when probabilistic mixtures are applied.
• Figure 2: Time evolution of $\braket{\mathbb{Z}^{\otimes N}}$ in the presence of larger unitary errors ($\epsilon = \sqrt{\epsilon_x^2 + \epsilon_y^2 + \epsilon_z^2} = 0.02$) applied at every parametrized rotation in the circuit (Eq.\ref{['eq:noisy_app_circuit']}). The unstructured error (Eq.\ref{['eq:unstructured_error']}) models over-rotations due to decomposition with universal finite gate sets, causing noisy expectation values (red curve) to fluctuate around the exact values (blue curve). Combining our method with Pauli twirling significantly improves the estimation of the expectation value (brown curve), outperforming the use of either technique alone (orange and green curves). The resulting estimate closely tracks the ideal evolution of $\braket{\mathbb{Z}^{\otimes N}}$. Times 0.8 and 1 are the cases where $\epsilon_z$ contributes the most and the least to the overall error, respectively.
• Figure 3: Absolute error in the expectation value, $|\braket{\mathcal{O}} - \braket{\mathcal{O}}_{\text{noisy}}|$, with different Clifford+$T$ decomposition accuracy, $\zeta$. The main plot compares our method, combined with Pauli twirling, against unmitigated expectation value estimates. A significant improvement is observed when using probabilistic mixtures, though at the cost of increased variance, as $\epsilon_z$ grows with $\zeta$ on average (Eq. \ref{['eq:epsilon_zeta']}). The inset plot shows the number of $T$ gates per $R_z(\theta)$ in the Clifford+$T$ decomposition, both with and without our method. Since the $T$-gate overhead introduced by our approach is small (Eq. \ref{['eq:t_overhead']}), the red and brown data points largely overlap. The significant improvement in the expectation value estimate is observed for $\zeta > 10^{-3}$.
• Figure 4: Estimating expectation value in a presence of a constant over-rotation $\epsilon=0.001$ along $Z$-direction at every gate of circuit defined in Eq. \ref{['eq:noisy_circuit']}. The noisy expectation value (red dashed line) shows significant deviation from the exact value (blue dashed line). By applying the proposed method, the noiseless expectation value is effectively recovered, as indicated by the orange dashed line closely aligning with the blue one.
• Figure 5: Projection of $\|\gamma\|_1$ onto $AB-$plane. (left) Positive unitary error $\epsilon$. In the $A>B$ regime, for any chosen $A \in (0, 2\pi)$, the optimal $B$ is $0.5A$. The white dot in $A>B$ region corresponds to the decomposition in \ref{['eq:ourmixture']}, in which $A = -\frac{\pi}{4}$ and $B=\pi$. (right) Negative unitary error $\epsilon$. The plot is reflected across the diagonal $B=2\pi - A$ compared to the positive $\epsilon$ case. The optimal choice for $B$ becomes $2A-2\pi$ for any given $A$.