Pauli Check Extrapolation (PCE)

• Pauli Check Extrapolation (PCE) is a quantum framework that leverages Pauli operations to both mitigate errors and efficiently encode optimization variables.
• The error mitigation variant extrapolates expectation values via linear or exponential models, reducing exponential post-selection overhead while preserving signal fidelity.
• The encoding variant, including Warm-Start PCE, optimizes combinatorial problems by mapping binary variables onto fewer qubits, enhancing convergence and performance.

Pauli Check Extrapolation (PCE) encompasses two distinct frameworks in current quantum information literature. The first, introduced in "Pauli Check Extrapolation for Quantum Error Mitigation" (Langfitt et al., 20 Jun 2024), addresses quantum error mitigation by extrapolating along the noise-suppression axis generated by Pauli Check Sandwiching (PCS). The second, as detailed in "Warm-Starting PCE for Traveling Salesman Problem" (Carmo et al., 17 Sep 2025), refers to Pauli Correlation Encoding, a variational encoding strategy for qubit-efficient mapping of binary optimization variables, as well as its warm-start extension for combinatorial optimization. The following exposition separately details the mechanisms, theoretical models, and practical considerations of both error mitigation extrapolation and Pauli-based correlation encodings.

1. Pauli Check Sandwiching and Its Limitations

Pauli Check Sandwiching (PCS) is foundational to error mitigation schemes employing local parity checks. For a circuit $U$ on $n$ qubits, each layer is interleaved with randomly selected single-qubit Pauli gates $P_j \in \{I, X, Y, Z\}$. Specifically, each qubit experiences a "check-in" Pauli before its $U$-layer operation and a "check-out" Pauli after. In an error-free scenario, these checks cancel, leaving the logical output unchanged. PCS enables error filtering by post-selecting on the absence of Pauli-trigger events (i.e., no check flips). This post-selection process filters a subset of error trajectories, bringing measured observables $E_{\text{PCS}} = \Tr[O\,\rho_{\text{PCS}}]$ closer to the ideal $E_{\text{ideal}} = \Tr[O\,|\psi\rangle\langle\psi|]$. However, the post-selection probability decays exponentially in the number of inserted check pairs, rendering sample overhead and additional gate-induced noise the principal bottlenecks at large $n$ (Langfitt et al., 20 Jun 2024).

2. Extrapolation Schemes in Pauli Check Extrapolation

Pauli Check Extrapolation (PCE) circumvents the exponential post-selection overhead by extrapolating expectation values as a function of the number of check pairs. Rather than pushing the acceptance rate to zero, the protocol runs with $n \in \{0,1,2, \dotsc, n_{\max}\}$ checks, accumulating observable measurements $E(n)$. These data are fit to an ansatz and extrapolated to the "maximum check" or infinite-check ($n \rightarrow n_{\max}$) regime, unlike Zero-Noise Extrapolation (ZNE), which extrapolates to a hypothetical zero-noise point (Langfitt et al., 20 Jun 2024).

2.1 Linear Extrapolation Model

For circuits not exhibiting strong non-Markovianity, a linear ansatz $E(n) = a n + b$ suffices. Parameters $(a, b)$ are extracted from two points, classically $E(0), E(1)$, and the extrapolated value is $E_{\text{lin}} = a n_{\max} + b$.

2.2 Exponential Extrapolation (Markovian Model)

A Markovian noise model yields exponential bias suppression:

$E(n) = E_\infty + (E(0) - E_\infty) e^{-\alpha n}$

where $E_\infty$ is the infinite-check limit and $\alpha > 0$ the effective per-check noise suppression rate. Three evaluations (e.g., $E(0), E(1), E(2)$) suffice to parameterize this model, allowing closed-form determination of $E_\infty$ and $\alpha$. Extrapolation to $n \rightarrow n_{\max}$, or infinity, yields the bias-suppressed estimator $E_{\text{exp}}$ (Langfitt et al., 20 Jun 2024).

3. Statistical and Theoretical Properties

The residual systematic bias for $n$ inserted check pairs in the exponential ansatz decays as $$|E(n) - E_\infty| = |E(0) - E_\infty| e^{-\alpha n}$$. Achieving target bias $\varepsilon$ requires $$n \gtrsim \frac{1}{\alpha} \ln\bigl(|E(0) - E_\infty|/\varepsilon\bigr)$$. The leading order sample complexity to achieve mean-squared error $\varepsilon^2$ with $n_\text{pts}$ fit points is $$M_{\text{tot}} = O(n_\text{pts}\, \varepsilon^{-2})$$. PCE distinguishes itself from ZNE by avoiding circuit-depth overheads that amplify variance and by employing a noise dial—namely, the number of inserted Pauli checks—that typically preserves order-unity acceptance for moderate $n$ (Langfitt et al., 20 Jun 2024).

4. Applications to Classical Shadows and Expectation Tasks

PCE integrates seamlessly into classical shadow estimation. Each snapshot $\hat\sigma^{(n)}_i$ is collected after preparing the circuit with $n$ checks, and for observable $O$, the estimator

$\widehat{E}(n) = \frac{1}{M_n} \sum_{i=1}^{M_n} \Tr[O\, \hat{\sigma}_i^{(n)}]$

is computed. The fit (either linear or exponential) is applied to $\widehat{E}(n)$. In practice, variance inflation due to extrapolation is limited (10–20% relative increase over a single-point estimate at equivalent shot-count), ensuring scalability for shadow tomography and expectation evaluation in the presence of circuit-level noise (Langfitt et al., 20 Jun 2024).

5. Numerical and Experimental Validation

Simulations for VQE-ground states (5 qubits, 4-layer ansatz) show that PCE-exponential achieves a target fidelity $F>0.95$ with approximately 30% fewer samples than PCS—while PCE-linear approaches this within 10% for moderate noise. On IBM Quantum hardware (7-qubit Heisenberg model, 6-layer ansatz), error suppression follows:

• Raw error: $\sim$0.12
• PCS(2): $\sim$0.06
• PCE-linear: $\sim$0.03
• PCE-exponential: $\sim$0.025

All methods used $M_{\text{tot}} = 5 \times 10^4$ shots. Empirical convergence tracks the predicted exponential decay with added check pairs (Langfitt et al., 20 Jun 2024).

6. Pauli Correlation Encoding and Warm-Start Extensions

The second use of "PCE" refers to Pauli Correlation Encoding, enabling encoding of $m$ classical variables $x_i \in \{-1,1\}$ (or $\{0,1\}$ after shifting) as the sign of expectation values of weight-$k$ Pauli correlators $\Pi_{(S, P)}$ on an $n$-qubit variational state $\ket{\Psi(\theta)}$. For $k=2$,

$m = \tfrac{3n(n-1)}{2}, \qquad n = O(\sqrt{m})$

yielding a polynomial reduction compared to one-hot mappings. The smooth assignment $s_i(\theta) = \tanh[\alpha \langle \Pi_i \rangle]$ with hyperparameter $\alpha>0$ enables gradient-based optimization. Typical objective for MaxCut (or QUBO) is

$$\mathcal{L}_{\text{PCE}}(\theta) = \sum_{(i,j)\in E} W_{ij} s_i(\theta) s_j(\theta) + \mathcal{L}^{(\text{reg})}$$

where a regularizer controls correlator magnitudes (Carmo et al., 17 Sep 2025).

Warm-Start PCE incorporates Goemans–Williamson (GW) randomized rounding as a classical bias, scaling edge weights according to GW cut preferences:

$$\mathcal{L}_{\text{Warm-PCE}}(\theta) = \sum_{(i,j)\in E} W_{ij}[1 + |\widehat{c}_i^\star - \widehat{c}_j^\star|]s_i(\theta)s_j(\theta) + \mathcal{L}^{(\text{reg})}$$

where $\widehat{c}_i^\star$ is the clamped GW bit. This modification preserves problem-agnostic applicability and is parameterized by the clamping threshold $\varepsilon$ (Carmo et al., 17 Sep 2025). Variational optimization employs TwoLocal hardware-efficient ansätze, with gradient-free routines (e.g., COBYLA).

In benchmark tasks for the 5-city TSP (converted to MaxCut), Warm-PCE achieves the optimum in $28$–$64\%$ of instances, against $4$–$26\%$ for standard PCE, and demonstrates monotonic improvement in the approximation ratio with increasing circuit depth $p$, particularly for $p \geq 3$.

7. Practical Recommendations and Limitations

For error mitigation PCE, inserting $n=1$ or $2$ check pairs is typically optimal for superconducting-qubit noise rates $\alpha \approx 0.3$–$0.6$ per check, as this already suppresses bias by $50$–$75\%$ with manageable sampling and gate overhead. The exponential fit is advisable with three or more $n$-values and visible nonlinear convergence as a function of $n$; the linear fit is appropriate for shallow circuits or limited shot budgets. Shot allocation can be balanced for exponential fits or biased toward $n=0$ in linear fits to reduce variance. The variance cost of extrapolation is mild, and sample complexity is favorable compared to ZNE-type protocols (Langfitt et al., 20 Jun 2024).

For Pauli Correlation Encoding, using weight-2 correlators efficiently encodes $m$-bit combinatorial problems with $O(\sqrt{m})$ qubits. Warm-Start PCE (incorporating GW bias) provides substantial quality gains at shallow circuit depths and requires minimal additional computation. Warm-PCE accelerates convergence to high-quality approximations, reducing hardware requirements for similar solution quality (Carmo et al., 17 Sep 2025).

Notable limitations for the error mitigation use-case include the circuit gate overhead per Pauli check and possible mis-modeling under strong non-Markovian noise. For the encoding context, future directions include exploring higher-order correlations ($k>2$), scaling to larger combinatorial problems, and benchmarking against advanced quantum and classical heuristics.

8. Summary Table: PCE Variants

Variant	Core Mechanism	Primary Application
Pauli Check Extrapolation	Extrapolation along Pauli check axis; error mitigation	Shadow estimation, VQE, qubit devices (Langfitt et al., 20 Jun 2024)
Pauli Correlation Encoding	Qubit-efficient mapping of classical variables via correlators	MaxCut, TSP, QUBO optimization (Carmo et al., 17 Sep 2025)
Warm-Start PCE	Incorporates GW bias in PCE loss	Improved combinatorial optimization (Carmo et al., 17 Sep 2025)

Both error mitigation and encoding-centric forms of PCE leverage Pauli structure to parameterize and suppress the impacts of noise, sample complexity, or encoding overhead in near-term quantum algorithms. The unifying thread is the exploitation of Pauli operators' combinatorial and algebraic properties for scalable and efficient quantum computation and mitigation strategies.