Quantum Error Mitigation at the pre-processing stage

TL;DR

This work introduces a pre-processing Quantum Error Mitigation (QEM) strategy that constructs a surrogate observable $\hat{Y}$ satisfying $\mathcal{E}^\dagger(\hat{Y})=X$, enabling noise mitigation at the circuit output without heavy post-processing. Leveraging Tensor Networks and a Pauli-Lindblad noise model, the authors formulate $\hat{Y}$ via the inverse adjoint channel and show that for Pauli observables, $\hat{Y}$ reduces to a rescaled Pauli term in a Dominant Component Approximation (DCA), yielding close-to-optimal variance (QCRB-saturating) with dramatically reduced classical cost compared to TEM. The approach achieves substantial measurement overhead reductions and orders-of-magnitude speedups (up to ~$10^6$) in TN contractions, demonstrated on Ising-Trotter dynamics, and avoids the need for IC-POVMs or shadow tomography. The results indicate that QEM via surrogate observables can effectively bridge NISQ devices toward fault-tolerant regimes, and may complement quantum error correction in future hardware. Overall, the method provides a scalable, bias-manageable, and computation-efficient path for noise mitigation in shallow-to-moderate-depth quantum circuits.

Abstract

The realization of fault-tolerant quantum computers remains a challenging endeavor, forcing state-of-the-art quantum hardware to rely heavily on noise mitigation techniques. Standard quantum error mitigation is typically based on post-processing strategies. In contrast, the present work explores a pre-processing approach, in which the effects of noise are mitigated before performing a measurement on the output state. The main idea is to find an observable $Y$ such that its expectation value on a noisy quantum state $\mathcal{E(ρ)}$ matches the expectation value of a target observable $X$ on the noiseless quantum state $ρ$. Our method requires the execution of a noisy quantum circuit, followed by the measurement of the surrogate observable $Y$. The main enablers of our method in practical scenarios are Tensor Networks. The proposed method improves over Tensor Error Mitigation (TEM) in terms of average error, circuit depth, and complexity, attaining a measurement overhead that approaches the theoretical lower bound. The improvement in terms of classical computation complexity is in the order of $\sim 10^6$ times when compared to the post-processing computational cost of TEM in practical scenarios. Such gain comes from eliminating the need to perform the set of informationally complete positive operator-valued measurements (IC-POVM) required by TEM, as well as any other tomographic strategy.

Figures (7)

• Figure 1: Representation of our noisy circuit model. (a) One-layer circuit, where $\mathcal{U}$ is a (one-layer) unitary transformation; $\rho(\theta) = \mathcal{U}(\ketbra{0}^{\otimes n})$ is the ideal output of the noiseless quantum circuit; $\mathcal{E}(\mathbin{\vcenter{\hbox{$\bullet$}}})$ is the (one-layer) channel used to model the noise that affects the ideal quantum state; $\mathcal{E}(\rho)$ is the noisy quantum state at the output of the noisy quantum circuit. (b) Multi-layer example of 4 unitary layers. $\mathcal{U}_l$ indicates the $l$-th unitary layer, such that $\mathcal{U} = \bigcirc_{l}\mathcal{U}_l$, and the noiseless quantum state is $\rho(\theta) = \mathcal{U}(\ketbra{0}^{\otimes n}$); $\mathcal{E}(\mathbin{\vcenter{\hbox{$\bullet$}}})$ is the (multi-layer) noise channel composed of noiseless unitary layers $\mathcal{U}_l$, their inverses $\mathcal{U}_l^{-1}$ and the one-layer noise map $\Lambda_l$. The dotted red lines indicate the contraction step $l$ for the middle-out contraction matrix $\mathcal{M}_l$.
• Figure 2: TN diagrams. (a) Middle-out contraction matrix $[\mathcal{M}_l^\dagger]^{-1}$ before (left) and after (right) TN contraction and compression. (b) Visual description of an MPO (left) and an MPS (right) of a 3-qubit system.
• Figure 3: A single Trotter step for the one-dimensional transverse-field Ising model. First layer consists of a set of unitary rotations $R_X(2h\delta_t)$ for every qubit and two distinct implementations of pairwise $ZZ$-rotations (for even and odd links between the qubits), each consisting of two repeated CNOT layers intervened by the unitary rotation $R_Z(-2J\delta_t)$ on controlled qubits. Model parameters are $h = 1$, $J = 0.5236$, and $\delta_t = 0.5$, reproducing the settings of TEM Filippov2023. Each unique CNOT layer is followed by a sparse Pauli-Lindblad noise ewout2023, denoted as $\Lambda_1$ and $\Lambda_2$, with sampling overhead $\gamma_1 = 1.140$ and $\gamma_2 = 1.137$, respectively.
• Figure 4: Histogram representing the probability density function of the off-diagonal matrix elements $\{[\mathcal{M}_l^\dagger]^{-1}_{k\neq i,i} \}_k$ when $P_i = Z^{\otimes 10}$ at step 18 of the Trotterization of the Ising model. The histogram is computed using fixed-width bins and normalized such that the total area under the distribution equals one. The resulting density estimate is then compared against a Cauchy (black dots) and a Normal (red dots) parametric fit. Fitting was performed by the SciPy Python library 2020SciPy-NMeth.
• Figure 5: Comparison of PEC, TEM, and the mitigation procedure proposed in this article with $\hat{Y}$. We mitigate the expectation value $\left<Z^{\otimes 10}\right>$ for a 10-qubit discrete-time evolution of the Ising model. (a) Dynamics of the expectation value $\left<Z^{\otimes 10}\right>$ with and without noise mitigation. PEC estimation is based on 300 circuits sampled from the quasi-probability representation of the inverse noise, with 10000 shots per circuit on the computational basis. TEM estimation is based on 300 circuits, with projective measurements in local Paulis $\sigma_1$, $\sigma_2$ or $\sigma_3$ for every qubit, and 10000 shots per circuit; these measurements basis are chosen with probabilities $p_1=0.001$, $p_2=0.001$ and $p_3=0.998$ to adjust for this concrete observable; mitigation using $\hat{Y}$ is performed in one circuit with a total of $3\times10^6$ shots. The bond dimension of the noise mitigation map is at most 200. (b) Sampling overhead in the numerical experiments (dots) estimated as the ratio of the noise-mitigated and unmitigated estimation errors, $\gamma = \Delta \hat{O}_\text{n.m.} / \Delta \hat{O}_\text{noisy}$, as well as their theoretical predictions (dashed lines).