Quantum Error Mitigation Strategies

• Quantum Error Mitigation is a set of hybrid quantum–classical strategies that suppress noise in NISQ devices without the full resource overhead of quantum error correction.
• It employs techniques like error extrapolation, which amplifies noise to reconstruct zero-noise observables, and quasi–probability decomposition, which yields unbiased estimators at the cost of increased sampling variance.
• These methods leverage rigorous mathematical frameworks and precise noise characterization protocols, offering practical improvements in circuit accuracy and efficiency on near-term quantum hardware.

Quantum error mitigation (QEM) encompasses a suite of hybrid quantum–classical strategies for reducing the systematic bias and random error in the outputs of quantum circuits executed on noisy intermediate-scale quantum (NISQ) devices. Distinct from full quantum error correction (QEC), QEM aims to “undo” or suppress the impact of noise on computational results through protocols that combine circuit manipulations, real-time data collection, and postprocessing, all while minimizing physical qubit and hardware resource overhead. The approach is justified and quantified by mathematically rigorous frameworks, with demonstrated efficacy in both numerical simulation and experimental implementation for practical quantum circuits, as established in foundational works (Endo et al., 2017, Sun et al., 2020, Xiong et al., 2020), among others.

1. Core Techniques of Quantum Error Mitigation

The foundational QEM protocols are underpinned by two complementary frameworks:

1. Error Extrapolation

This method relies on deliberately amplifying the error rate in quantum circuits—typically by inserting additional noisy gates or stretching circuit parameters—and then reconstructing the zero-noise observable via extrapolation. Traditionally, a linear fit was used, but it was shown that for depolarizing or Markovian error models, observables typically decay exponentially with the error rate, motivating the exponential extrapolation formula: $$Z(\epsilon_0) \propto e^{-N \epsilon_0} \quad \Longrightarrow \quad Z(0) = Z(\epsilon_0)^{\frac{r}{r-1}} Z(r\epsilon_0)^{\frac{1}{1-r}}$$ where $Z(\epsilon_0)$ and $Z(r\epsilon_0)$ are expectation values at native and amplified error rates, respectively, and $r$ is the amplification factor (commonly $r=2$). Exponential extrapolation captures the underlying Poisson decay more accurately than linear fits and significantly suppresses residual bias (Endo et al., 2017).

2. Quasi–Probability Decomposition (QPD)

Here, each noisy operation $O$ in the circuit is viewed as a proxy for an ideal $O^{(0)}$ through a linear expansion: $$O^{(0)} = \lambda\, O + \sum_i q_i B_i, \quad \text{or} \quad O^{(0)} = N^{-1}O, \quad N^{-1} = \sum_i q_i B_i$$ with Clifford group elements or similar as basis operators $B_i$ and quasi-probabilities $q_i$ (which may be negative). Circuits are sampled at random from this decomposition, and measurement results are rescaled by $\mathrm{sgn}(q_i)$ and an overall cost factor $C = \sum_i|q_i|$ or $\prod_\alpha C_\alpha$ for multi-step circuits (Endo et al., 2017). The procedure yields an unbiased estimator of the ideal observable but at increased variance (sampling overhead), analyzed in detail in (Xiong et al., 2020).

2. Noise Characterization and Circuit Design Protocols

The effectiveness of QEM depends critically on the accurate modeling and characterization of physical noise channels:

1. Gate Set Tomography (GST)–Based Protocols

GST is used to self-consistently estimate channels for all relevant circuit elements (gates, state preparations, and measurements). In practice, the experimenter prepares a set of $k$ linearly independent states $\{\bar{\rho}_k\}$, applies gates, and measures observables $\{\bar{Q}_j\}$ to build a complete characterization of the operational set. An important invariance property is maintained under similarity transforms—so sample-to-sample expectations remain unaltered by estimation imperfections: $${}_1\bar{Q}_j\,\bar{O}_N\dots\bar{O}_1\,{}_1\bar{\rho}_k = {}_1\hat{Q}_j\,\hat{O}_N\dots\hat{O}_1\,{}_1\hat{\rho}_k$$ This enables valid quasi–probability decompositions even under incomplete or noisy channel reconstruction (Endo et al., 2017).

2. Basis Set Decompositions for Multi-Qubit Gates

Any universal set (e.g., single-qubit Clifford operations and measurements) can be exploited so that higher-order gates are decomposed as tensor products over this set. Explicit, though often inefficient, decompositions are provided for controlled gates such as CNOTs.

3. Mathematical Structure and Scaling

The theoretical foundation of QEM leverages linear algebraic, probabilistic, and matrix representation frameworks:

Pauli Transfer Matrix Notation

A quantum state $\rho$ may be represented as a real column vector ${}_1\rho$, observables $Q$ as row vectors, and operations $O$ as matrices, so circuit output becomes: $$Q = {}_1 Q\, {}_1 \rho,\quad \mathrm{Tr}[Q\,O_N\circ\cdots\circ O_1(\rho)] = {}_1 Q\,O_N\cdots O_1\,{}_1\rho$$

Monte Carlo Estimation and Sampling Overhead

For QPD-based mitigation, the estimator

$$Q^{(0)} = \sum_l q_l\,{}_1 Q^{(l)} O_\mathrm{tot}^{(l)}\,{}_1\rho^{(l)}$$

is realized by sampling circuits labeled by $l$, multiplying by $\mathrm{sgn}(q_l)$ and a normalization $C$. The variance and thus the number of experimental runs increases by $C^2$ compared to the non-mitigated case. For extrapolation methods, repeated circuit executions at various noise rates are combined as per the extrapolation formula.

Scaling Considerations

Sampling overhead compounds multiplicatively with each circuit layer. For a single channel of error rate $\epsilon$, lower and upper bounds for the sampling overhead factor (SOF) are

$$\gamma_C \ge \frac{4\epsilon}{(1-\epsilon)^2},\qquad \gamma_C \le 4\epsilon \cdot \frac{1-\epsilon}{(1-2\epsilon)^2}$$

for Pauli and depolarizing channels, respectively (Xiong et al., 2020). Circuit concatenations or error correction code amalgamations further modulate the effective error and SOF.

4. Comparative Performance and Cost-Benefit Analysis

Efficacy in Simulated Benchmarks

Large-scale numerical simulations using the SWAP test (involving up to 19 qubits) demonstrate that both QEM approaches dramatically reduce the systematic error relative to raw, noisy circuits. Exponential extrapolation approaches recover values within $0.5 \pm 0.0111$ (compared to theoretical $0.5$), with variance significantly suppressed over linear fitting. QPD methods eliminate bias entirely (yielding centering around $0.5$), though at the cost of a broadening (increased sampling noise) (Endo et al., 2017).

Resource Considerations

While QEM does not increase the required qubit count or circuit depth, it requires additional sampling or shot runs. For Pauli and depolarizing errors, sample overheads are minimized but can still accumulate rapidly with depth or when many noisy gates are present. Coding–QEM hybrid schemes are beneficial when circuit size exceeds a threshold determined by the error rate and code performance (Xiong et al., 2020).

Relationship to Alternative Strategies

QEM is situated as an expedient near-term alternative to QEC, which requires large hardware and circuit overhead. QEM enables significant circuit “volume boost” (e.g., by a factor $\frac{1-10}{\epsilon}$ in accessible two-qubit gate count for accuracy $\epsilon$) and is not restricted by error thresholds. The cost is a sampling overhead with exponential dependence on the product of circuit volume and average gate infidelity, but with typically “mild” exponential rates, allowing finite quantum advantage for practically relevant problems (Aharonov et al., 21 Mar 2025).

5. Protocol Adaptability, Generalizations, and Limitations

Applicability to Noise and Circuit Models

The techniques are most rigorously justified for localized, Markovian error channels. GST-based modeling enables protocols to adapt even when imperfect prior knowledge exists. QEM generalizes to circuits constructed from universal basis gate sets. More elaborate stochastic QEM and trajectory–based frameworks extend mitigation to analog and strongly coherent/continuous-time noise regimes (Sun et al., 2020), and hybrid extrapolations can suppress errors from residual model estimation uncertainties.

Known limitations

The methods rely on accurate error channel knowledge and a manageable degree of noise correlations and circuit non-Markovianity. Sampling overhead can become prohibitive at high error rates or large circuit depths. QPD-based methods, in particular, pay for bias elimination with increased estimator variance.

6. Practical Implementation and Experimental Performance

Implementation Workflow

1. Characterize noise channels via GST or tomographically complete measurements.
2. Decompose non-ideal operations into linear combinations over basis set(s).
3. Construct QEM circuits by randomly inserting correction gates for QPD, or by amplifying noise and recording observables for extrapolation.
4. Collect measurement data and rescale observables or apply sign-weighted averages according to the prescribed protocols.
5. Analyze sampling overheads and tune the protocol (e.g., choose error amplification factors or circuit segmentations) for optimal resource use.

Experimental Realizations

The core principles have been validated in controlled simulations and offer a clear path to near-term experimental adoption on existing NISQ devices, as they do not require more qubits or gate depth than the original algorithmic circuits. The protocols are resource-efficient for moderate error rates and scalable to the range of a few tens (and in some cases, up to 50+) qubits under realistic noise assumptions.

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In summary, the primary QEM protocols offer a robust mathematical and operational framework for reducing both bias and variance in observables extracted from noisy circuits, leveraging error model estimation (via GST), stochastic sampling, and algebraic extrapolation/formal decomposition. They present a practical, generalized, and largely hardware-agnostic methodology for observable-level error reduction, supporting more accurate quantum computation in the NISQ regime without full-fledged error correction infrastructure (Endo et al., 2017, Xiong et al., 2020).