---
title: Pauli Check Extrapolation (PCE)
url: https://www.emergentmind.com/topics/pauli-check-extrapolation-pce
type: topic
---

# Pauli Check Extrapolation (PCE)

Pauli Check Extrapolation (PCE) encompasses two distinct frameworks in current quantum information literature. The first, introduced in "Pauli Check Extrapolation for Quantum Error Mitigation" [2406.14759], 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" [2509.14414], 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 $n$ [2406.14759].

## 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 [2406.14759].

### 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 $E_{\text{exp}}$ [2406.14759].

## 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 $n$ [2406.14759].

## 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 [2406.14759].

## 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 [2406.14759].

## 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 [2509.14414].

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 $\varepsilon$ [2509.14414]. 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 [2406.14759].

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 [2509.14414].

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 [2406.14759] |
| Pauli Correlation Encoding| Qubit-efficient mapping of classical variables via correlators | MaxCut, TSP, QUBO optimization [2509.14414] |
| Warm-Start PCE            | Incorporates GW bias in PCE loss                          | Improved combinatorial optimization [2509.14414] |

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.

Source: https://www.emergentmind.com/topics/pauli-check-extrapolation-pce