Optimized PEC+QED Hybrid Mitigation

• Optimized PEC+QED is a hybrid error mitigation protocol that fuses probabilistic error cancellation with quantum error detection to filter detectable errors and reduce residual noise.
• It employs a co-design strategy optimizing code choice, detection interval, and truncation order to balance sampling overhead against the costs of post-selection and measurements.
• Analytical models and simulations demonstrate that optimized PEC+QED significantly improves error rates and sampling efficiency across various quantum codes and circuit-level noise scenarios.

Searching arXiv for papers on probabilistic error cancellation combined with quantum error detection. arxiv_search(query="probabilistic error cancellation quantum error detection PEC QED arXiv", max_results=10, sort_by="submittedDate") Searching arXiv for papers on optimized PEC+QED and related co-design approaches. Optimized PEC+QED denotes a class of hybrid error-mitigation protocols that combine probabilistic error cancellation (PEC) with quantum error detection (QED) and post-selection. In this setting, QED filters away detectable fault branches, producing an accepted logical channel that is weaker than the original physical noise, while PEC is applied only to the residual accepted noise rather than to the full channel (Yuan et al., 12 May 2026). Recent work has treated this as a co-design problem over code choice, detection spacing, truncation order, noise characterization, and symmetry-measurement configuration, because the reduction in PEC sampling overhead must be balanced against rejection and measurement costs introduced by QED (Kumar et al., 21 Apr 2026, O'Leary et al., 1 Jul 2026).

1. Protocol architecture and operational setting

The canonical feedback-free QED+PEC protocol encodes $n-2$ logical qubits into $n$ physical qubits using the $[n,n-2,2]$ Iceberg code, with stabilizers

$$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$

The logical circuit is divided into $M=L/T$ blocks, each containing $T$ layers of noisy Clifford gates. After each block, all stabilizers are measured and the run is post-selected on the $+1$ outcome; failed shots are discarded. On the surviving branch, one obtains a reduced logical channel $\Lambda_{\rm acc}$, and PEC is then applied to undo that channel up to truncated order $K$. No mid-circuit recovery is required (Yuan et al., 12 May 2026).

A complementary formulation uses the Sparse-Pauli-Lindblad noise model. A noisy layer is written as

$$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$

with inverse quasi-channel

$n$0

Within PEC+QED, symmetry measurements are used to detect a subset of errors, and PEC is restricted to the undetectable subset. In the ideal-check limit, the total one-layer sampling cost becomes

$n$1

so the principal optimization target is the trade-off between shrinking $n$2 and maintaining a favorable acceptance probability (O'Leary et al., 1 Jul 2026).

2. Accepted logical channel and perturbative inverse construction

Let $n$3 index all elementary Pauli fault locations in one block, each with probability $n$4. Propagating $n$5 to the block end gives $n$6. For a subset $n$7,

$n$8

The subset survives post-selection iff $n$9. Defining

$[n,n-2,2]$0

the exact unnormalized accepted channel and success probability are

$[n,n-2,2]$1

and the normalized accepted logical channel is

$[n,n-2,2]$2

$[n,n-2,2]$3 is the residual channel that PEC must approximate and invert (Yuan et al., 12 May 2026).

The key complication is that post-selection correlates accepted fault branches through stabilizer-commutation constraints, so the sparse Pauli-Lindblad factorization underlying bare PEC no longer applies directly. The inverse is therefore constructed perturbatively. With total noise weight

$[n,n-2,2]$4

one truncates the Bernoulli expansion to subsets $[n,n-2,2]$5, forms the truncated accepted channel $[n,n-2,2]$6, and then uses the degree-$[n,n-2,2]$7 Neumann inverse

$[n,n-2,2]$8

Sampling from $[n,n-2,2]$9 implements $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$0 up to $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$1 error. Only subsets of size $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$2 are enumerated, so classical preprocessing scales as

$$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$3

with $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$4, and each branch needs $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$5 symplectic updates, giving $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$6 cost per block rather than $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$7. For $$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$8,

$$S_X=X^{\otimes n},\qquad S_Z=Z^{\otimes n},\qquad \Pi=\tfrac12(I+S_X)\,\tfrac12(I+S_Z).$$9

while the one-block error satisfies

$M=L/T$0

Over multiple blocks, the error accumulates at most additively and the variance factor multiplies blockwise (Yuan et al., 12 May 2026).

3. Optimization variables: detection interval, truncation order, and the discrete-Zeno trade-off

A central optimization variable is the QED interval: how often detection cycles are inserted. One efficiency condition is obtained by dividing a circuit of total depth $M=L/T$1 into $M=L/T$2 mitigable units of $M=L/T$3 logical layers. If $M=L/T$4 is the per-layer physical error, $M=L/T$5 is the post-selected logical error per unit, and $M=L/T$6 is the rejection probability, break-even requires

$M=L/T$7

This criterion makes the interval itself an architectural knob rather than a fixed code-level choice (Kumar et al., 21 Apr 2026).

In explicit circuit-level studies, the canonical strategy $M=L/T$8 does not break even in the tested codes. For depolarizing two-qubit error $M=L/T$9 with single-qubit $T$0, Stim-based simulations gave

$T$1

for the $T$2 and $T$3 Iceberg codes and the $T$4 surface code. For the $T$5 Iceberg code, $T$6 and $T$7; for $T$8, break-even begins at $T$9; for the $+1$0 surface code, break-even was not observed at any $+1$1 up to hundreds of layers (Kumar et al., 21 Apr 2026).

A different optimization picture appears in the ideal-stabilizer-measurement analysis of the $+1$2 Iceberg code. There, a toy model gives

$+1$3

so for $+1$4, making $+1$5 smaller reduces $+1$6. The paper therefore recommends the smallest practical detection interval $+1$7 when syndrome extraction is cheap, and first order $+1$8 as the minimal nontrivial choice for distance-2 codes. Higher $+1$9 yields residual error $\Lambda_{\rm acc}$0 per block but increases preprocessing as $\Lambda_{\rm acc}$1 and the quasiprobability norm roughly as $\Lambda_{\rm acc}$2 (Yuan et al., 12 May 2026). This suggests interval optimality is architecture- and noise-model-dependent: cheap, idealized detection and explicit syndrome-extraction overhead lead to different optima.

4. First-cycle transients and steady-state extraction

A major obstacle to naive PEC+QED is that repeated QED cycles violate the position-independent noise assumption used in standard PEC. Transition-matrix analysis shows that the post-selected evolution contains one fast mode of order $\Lambda_{\rm acc}$3 that decays in $\Lambda_{\rm acc}$4 cycles and slow modes of order $\Lambda_{\rm acc}$5 that govern steady-state behavior. Physically, the first QED cycle injects leakage states at $\Lambda_{\rm acc}$6 but cannot remove any, whereas later cycles reach a balance of detection and injection (Kumar et al., 21 Apr 2026).

If one characterizes a gate by tomography of “prepare + gate + QED” without removing this first-cycle transient, the learned model embeds that fast leakage component. The result is that naive PEC+QED can degrade accuracy below the QED-only baseline. The proposed remedy is steady-state extraction (SSE), defined through

$\Lambda_{\rm acc}$7

where $\Lambda_{\rm acc}$8 is the superoperator of “prepare logical + first QED cycle” and $\Lambda_{\rm acc}$9 is the concatenated superoperator “$K$0 (ideal $K$1 plus its QED).” Operationally, one tomographically learns $K$2, learns $K$3 for each logical gate, and computes $K$4 by matrix inversion (Kumar et al., 21 Apr 2026).

On the $K$5 code, Hamiltonian-level QuTiP simulations showed that SSE alone reduces per-cycle prediction error by $K$6 versus naive single-cycle tomography, and in end-to-end PEC studies reduces the observable bias by up to $K$7 below the QED-only baseline for depolarizing, dephasing, and amplitude-damping noise. Within optimized PEC+QED, SSE therefore functions as a characterization protocol that isolates the steady-state channel actually seen by long runs, rather than as an additional mitigation layer (Kumar et al., 21 Apr 2026).

5. Optimization over symmetry measurements

When QED is implemented by measuring symmetries, the choice of which symmetries to measure becomes a classical optimization problem. Each candidate symmetry $K$8 detects a subset $K$9 of error generators and incurs mitigation overhead $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$0 from the circuit that measures it. With error weights $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$1, one objective is

$$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$2

This is a weighted set-cover problem with element penalty. A greedy approximation is used to generate a candidate pool, followed by subset enumeration once idling-aware costs are included (O'Leary et al., 1 Jul 2026).

Noisy symmetry measurements introduce two distinct penalties. Errors during the check can create false detections, increasing variance without reducing bias, while undetected double-errors from pairs of detectable errors can survive post-selection and increase bias. The framework therefore discourages symmetries with large $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$3 unless they cover sufficiently large $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$4-weight. This is the sense in which the method optimizes QED for PEC, rather than simply appending post-selection to an existing symmetry set (O'Leary et al., 1 Jul 2026).

For GHZ-state preparation, the full stabilizer group contains $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$5 $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$6 generators and the global $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$7. Because measuring $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$8 costs $$\Lambda(\rho)=\bigcirc_{k\in\mathcal K}\bigl(w_k\,\rho +(1-w_k)\,E_k\,\rho\,E_k\bigr),$$9 CNOTs, the optimizer excludes that global check and instead selects nonlocal weight-2 stabilizers. For $n$00, a 4-element configuration detects $n$01 of the total error weight with only $n$02 CNOTs of check cost. At fixed shot count $n$03, optimized PEC+QED yields up to $n$04 lower total-square-error than PEC for $n$05. For the generalized superfast encoded Fermi-Hubbard model, PEC+QED improves observable estimation on a $n$06 lattice, and for larger systems mitigation overheads can be reduced by measuring only subsets of stabilizers; for $n$07, the optimal subsets omit $n$08–$n$09 of stabilizers (O'Leary et al., 1 Jul 2026).

6. Benchmark regimes, empirical performance, and relation to physical-layer hybrids

The current literature reports gains in three distinct regimes. In logical GHZ-state preparation with the $n$10 Iceberg code under circuit-level depolarizing noise and ideal stabilizer measurements, first-order QED+PEC reaches $n$11 physical qubits and lowers sampling overhead by three to four orders of magnitude relative to standard PEC while maintaining $n$12. In the same setting, the bare PEC sample-variance factor is $n$13, whereas QED+PEC gives $n$14 (Yuan et al., 12 May 2026).

In a more hardware-oriented co-design study, a $n$15 Iceberg code was used for 4-vertex MaxCut QAOA at depths $n$16 under depolarizing noise $n$17, $n$18, with two QED checks per circuit and $n$19 attempted shots. PEC+QED achieved $n$20–$n$21 lower absolute error and up to $n$22 lower mean-squared error versus PEC on physical qubits. Across $n$23, the reported quasiprobability norms were $n$24 for PEC only and $n$25 for PEC+QED, with acceptance rates between $n$26 and $n$27 (Kumar et al., 21 Apr 2026).

A related line of work pushes PEC and QED to the physical layer inside a logical code without modifying the decoder. In that framework, any linear QEM method can be integrated into the physical layer because QEC is itself a linear quantum map. For code-capacity memory, a combined PEC+QED construction cancels all error weights $n$28, so the leading logical failure becomes $n$29, which is described as increasing the effective code distance by $n$30. Simulations on repetition and rotated surface codes showed that a distance-3 code with physical-level PEC achieves logical error rates lower than or similar to a distance-5 unmitigated code while using $n$31 and $n$32 fewer qubits, respectively (Jeon et al., 26 Jan 2026).

Two recurrent limitations define the practical boundary of optimized PEC+QED. First, noisy symmetry or stabilizer measurements can negate the advantage: readout-only flips mainly increase post-selection cost, while noisy GHZ-assisted global stabilizer extraction can remove the advantage entirely (Yuan et al., 12 May 2026). Second, optimization is code- and circuit-specific: high-rate distance-2 Iceberg codes and carefully spaced detection cycles can satisfy the relevant cost conditions, whereas low-rate codes may not (Kumar et al., 21 Apr 2026). A plausible implication is that optimized PEC+QED is best viewed not as a single protocol, but as a family of co-designed mitigation architectures in which post-selection reshapes the effective channel that PEC must invert.