Probabilistic Readout Error Mitigation (PROM)

• Probabilistic Readout Error Mitigation (PROM) is a framework that treats quantum measurements as noisy classical channels and applies inversion techniques to recover ideal statistics.
• It encompasses diverse methods such as matrix inversion, iterative Bayesian unfolding, quasi-probabilistic cancellation, and tensor-network modeling, each suited for different experimental regimes.
• PROM is applied for correcting full outcome distributions, local observables, and mid-circuit measurements, though careful calibration is required to manage SPAM entanglement and increased variance.

Probabilistic Readout Error Mitigation (PROM) denotes a family of techniques that treat quantum measurement as a noisy classical channel and then use classical post-processing, probabilistic unfolding, or quasi-probabilistic inversion to infer ideal statistics from noisy readout. The acronym is not used uniformly across the literature: closely related methods appear as assignment-matrix inversion, iterative Bayesian unfolding, quasi-probabilistic readout correction, BFA-based PROM, and tensor-network readout mitigation. What unifies them is a calibrated map such as $$\mathbf{p}_{\mathrm{noisy}}=\mathbf{\Lambda}\mathbf{p}_{\mathrm{ideal}}$$, $q=A p$, or $\bar p=\mathcal E^T p$, together with an approximate or exact inverse applied either to full distributions, to marginals, or directly to observables (Maciejewski et al., 2019, Nguyen, 2023, Hashim et al., 2023, Guo et al., 24 Jun 2026).

1. Measurement-channel formulation

PROM is grounded in the observation that readout noise is frequently well modeled as a classical channel acting on ideal measurement outcomes. In the detector-tomography formulation, an experimental POVM $M^{\mathrm{exp}}$ is related to an ideal POVM $M^{\mathrm{ideal}}$ by an invertible left-stochastic matrix $\Lambda$,

$$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$

so that correction is formally $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$ (Maciejewski et al., 2019). In computational-basis language, the same object appears as a confusion matrix $\mathcal M$ with entries

$\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$

or as a product model $q=A p$0 under independent single-qubit assignment errors (Hashim et al., 2023, Nguyen, 2023).

This classical-channel picture has two immediate consequences. First, readout mitigation is fundamentally an inverse problem on probabilities rather than a modification of the quantum state dynamics. Second, the inverse channel is generally non-stochastic, so corrected histograms can contain negative entries or values exceeding one; in that sense, PROM often works with quasi-probabilities even when the forward noise model is purely classical (Maciejewski et al., 2019, Guo et al., 9 Oct 2025).

Operator-space formulations are equivalent. Under classical bit-flip readout noise, one may correct expectation values of $q=A p$1-type observables directly, rather than inverting the full outcome distribution. For single- and two-qubit observables, explicit correction formulas can be written in terms of calibrated flip probabilities, and extensions including $q=A p$2-qubit readout correlations can be expressed as inversion of a $q=A p$3 operator map $q=A p$4 built from the joint conditional probabilities $q=A p$5 (Alexandrou et al., 2021, Alexandrou et al., 2021). This operator-space view is especially natural when the target quantities are Pauli expectations rather than full histograms.

2. Main methodological families

PROM is not a single algorithm but a methodological class. The principal variants differ in how they parameterize the readout channel, how they compute or approximate its inverse, and whether they act on full distributions, sparse supports, or observables.

Family	Core object	Representative papers
Matrix inversion / detector tomography	$q=A p$6, $q=A p$7, $q=A p$8	(Maciejewski et al., 2019, Lee, 4 Jun 2025)
Iterative Bayesian unfolding	EM/KL update for $q=A p$9	(Nguyen, 2023)
Sparse / perturbative inversion	truncated support or Hamming-distance expansion	(Yang et al., 2022, Peters et al., 2021)
Randomized-measurement PROM	scalar suppression factors under randomized bases	(Arrasmith et al., 2023)
Quasi-probabilistic cancellation	signed distribution over bit-flip corrections	(Hashim et al., 2023, Koh et al., 2024)
Tensor-network PROM	MPO/PEPO model of $\bar p=\mathcal E^T p$0	(Guo et al., 24 Jun 2026)

Iterative Bayesian unfolding (IBU) is the clearest example of PROM in a strictly probabilistic sense. It maintains a valid distribution at every iteration and updates

$\bar p=\mathcal E^T p$1

This update is equivalent to an EM algorithm minimizing a KL-based objective. The same paper distinguishes structural mitigation, where the ideal distribution is concentrated on a few bitstrings, from unstructural mitigation, where only local marginals needed for observables are mitigated; on 127-qubit GHZ data, structural EM with $\bar p=\mathcal E^T p$2 latent bitstrings recovered nearly perfect correlations below about 80 qubits with only $\bar p=\mathcal E^T p$3 shots, while local IBU and local least-squares performed almost identically for $\bar p=\mathcal E^T p$4 and $\bar p=\mathcal E^T p$5 marginals (Nguyen, 2023).

Sparse and perturbative variants exploit the fact that many near-term distributions are sharply peaked and that high-Hamming-weight readout events are suppressed. One line of work gives two QREM methods with $\bar p=\mathcal E^T p$6 time for $\bar p=\mathcal E^T p$7-qubit distributions from $\bar p=\mathcal E^T p$8 shots, enabling mitigation of a 65-qubit GHZ state in a few seconds and observation of a 29-qubit GHZ fidelity exceeding $\bar p=\mathcal E^T p$9 (Yang et al., 2022). Another develops perturbative inversion with error $M^{\mathrm{exp}}$0 when Hamming-weight-$M^{\mathrm{exp}}$1 readout events are $M^{\mathrm{exp}}$2, yielding a relative speedup of $M^{\mathrm{exp}}$3 for estimating the all-zeros probability and a generalized full-distribution scheme based on a truncated Neumann series (Peters et al., 2021).

Randomized-measurement PROM replaces a large confusion matrix by a small set of scalar suppression factors. In approximate-state estimation and classical shadows, random single-qubit measurement bases symmetrize the effective readout channel so that the noisy expectation of a Pauli string is multiplicatively suppressed, and the ideal value is estimated as a ratio of two empirical means. For local observables, the required number of calibration shots does not grow with system size $M^{\mathrm{exp}}$4, and an implementation on Rigetti hardware collected and processed $M^{\mathrm{exp}}$5 samples in less than $M^{\mathrm{exp}}$6 minutes (Arrasmith et al., 2023).

3. Quasi-probabilistic PROM and measurement randomization

A particularly explicit realization of PROM is the quasi-probabilistic readout correction built on measurement randomized compiling (MRC). The starting point is a general noisy measurement map $M^{\mathrm{exp}}$7. By choosing a random Pauli string $M^{\mathrm{exp}}$8, compiling it into the final single-qubit layer, measuring in the computational basis, and classically undoing the $M^{\mathrm{exp}}$9 or $M^{\mathrm{ideal}}$0 components, MRC implements a Pauli twirl,

$M^{\mathrm{ideal}}$1

For computational-basis readout, the twirled channel becomes a classical stochastic bit-flip model,

$M^{\mathrm{ideal}}$2

so arbitrary, non-unital, context-dependent, and coherent readout noise is replaced by a distribution $M^{\mathrm{ideal}}$3 over classical bit-flip patterns (Hashim et al., 2023).

Once the effective channel is a bit-flip convolution $M^{\mathrm{ideal}}$4, PROM amounts to constructing a signed inverse distribution $M^{\mathrm{ideal}}$5 such that $M^{\mathrm{ideal}}$6. The first-order inverse is

$M^{\mathrm{ideal}}$7

with higher-order inverses $M^{\mathrm{ideal}}$8 pushing the residual bias to order $M^{\mathrm{ideal}}$9 at the expense of larger variance. Experimentally this inverse is applied by deterministic histogram reweighting rather than per-shot sampling. Because MRC makes the bit-flip distribution state-independent, the full $\Lambda$0 confusion matrix is avoided: one can estimate $\Lambda$1 from a single prepared basis state, typically $\Lambda$2, at cost $\Lambda$3 rather than $\Lambda$4 (Hashim et al., 2023).

The eight-qubit experimental demonstration is the strongest evidence for this formulation. On a ring of eight superconducting transmon qubits with multiplexed resonator readout and noticeable crosstalk, the method was tested on 200 different 8-qubit circuits—100 structured one-layer circuits over $\Lambda$5 and 100 random local $\Lambda$6 circuits. Using second-order quasi-probabilistic readout correction after MRC, the total variation distance to the ideal distribution was lower than both local readout correction and full confusion-matrix inversion in over $\Lambda$7 of circuits, and a single inverse calibrated from $\Lambda$8 corrected all rows of an 8-qubit confusion matrix measured under MRC (Hashim et al., 2023).

4. Mid-circuit measurements and feedforward

PROM becomes qualitatively different when measurements occur mid-circuit and control later operations. In that setting, a readout error does not merely corrupt the recorded bitstring; it can cause the wrong branch of the quantum program to execute. Standard terminal readout correction cannot repair that branch mismatch after the fact. The general PROM protocol for feedforward circuits therefore inserts a random bitmask $\Lambda$9 into the classical control path, applies feedforward as $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$0 instead of $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$1, and reweights terminal observables by a sign and normalization derived from a quasiprobability vector $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$2 satisfying $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$3, where $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$4 is the symmetrized confusion matrix after bit-flip averaging. The mitigated estimator is

$$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$5

For a target precision and confidence, the required number of shots increases by a factor $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$6, with the bound $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$7 when the total mid-circuit readout error rate is $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$8. On superconducting processors, this protocol reduced error by up to $$M^{\mathrm{exp}}_i=\sum_j \Lambda_{i,j} M^{\mathrm{ideal}}_j, \qquad \mathbf p_{\mathrm{exp}}=\Lambda\,\mathbf p_{\mathrm{ideal}},$$9 on dynamic qubit resets, shallow-depth GHZ preparation, and multi-stage teleportation, without increasing circuit depth or two-qubit gate counts (Koh et al., 2024).

The quasi-probabilistic viewpoint had already been demonstrated for mid-circuit ancilla measurements using artificial $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$0 insertions before the ancilla readout. In that scheme, runs with and without the inserted bit-flip are combined with signed weights to cancel the classical control error in expectation. For a single mid-circuit measurement with error probability $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$1, the effective number of useful shots is $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$2. In an eight-qubit superconducting memory-protection experiment, memory error grew from $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$3 at $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$4 to $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$5 at $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$6 without mitigation, to $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$7 with MRC alone, and was reduced by an additional $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$8 by quasi-probabilistic correction of the ancilla measurement, consistent with a $$\mathbf p_{\mathrm{ideal}}=\Lambda^{-1}\mathbf p_{\mathrm{exp}}$$9 ancilla bit-flip probability (Hashim et al., 2023).

A later branch-resolved study of dynamic teleportation sharpened the comparison between PROM and simpler post-processing. Using BFA-based PROM inside branch Choi-shadow tomography, two layouts on IBM Fez were compared. In a layout with higher mid-circuit readout error, the operational feed-forward penalty was approximately $\mathcal M$0–$\mathcal M$1, and PROM produced higher branch qualities than post-processing for every branch. In a second layout with lower readout error, the operational feed-forward penalty increased to roughly $\mathcal M$2, and post-processing exceeded PROM for all branch qualities. The same study reported symmetrized syndrome distributions $\mathcal M$3 in the higher-error layout and $\mathcal M$4 in the lower-error layout, making explicit that PROM is most advantageous when mid-circuit readout noise is sufficiently large to justify the quasiprobability overhead (Edwards et al., 30 Apr 2026).

5. Scalability, variance, and modeling assumptions

PROM’s practical value is determined by how it trades calibration cost, classical complexity, and variance. In matrix-inversion schemes, the naive object is a $\mathcal M$5 matrix, so scalable variants rely on structure. Sparse-outcome QREM exploits the fact that only a small set of bitstrings appears in many NISQ workloads and achieves $\mathcal M$6 time for $\mathcal M$7 qubits and $\mathcal M$8 shots (Yang et al., 2022). Perturbative inversion exploits the suppression of high-Hamming-weight readout events and yields $\mathcal M$9 residual error when the inverse is truncated at order $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$0 (Peters et al., 2021). Randomized-measurement PROM avoids exponential calibration by collapsing readout effects into scalar suppression factors for local observables (Arrasmith et al., 2023). Tensor-network PROM models the full correlated channel $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$1 as an MPO or PEPO learned by likelihood optimization, and numerical experiments up to 20 qubits show sample cost growing only near-linearly with system size while capturing correlations that product models miss (Guo et al., 24 Jun 2026).

Variance amplification is intrinsic. In classical bit-flip correction, the variance of mitigated Pauli operators can be expressed explicitly in terms of the corrected expectation values and the noisy variances, and hardware experiments on IBM and Rigetti show good agreement with the theory. The empirical shot overhead needed to keep the same variance was about $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$2 on Rigetti Aspen-9 for a two-qubit observable and about $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$3 on ibmq_16_melbourne for a three-qubit observable, indicating that the increase of the variance due to the mitigation procedure is only moderate in those settings (Alexandrou et al., 2021). In quasi-probabilistic schemes, the same effect appears through $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$4 or the quasi-norm of the inverse channel (Koh et al., 2024, Hashim et al., 2023).

These gains depend on assumptions. Local IBU on marginals assumes that cross-talk can be ignored (Nguyen, 2023). Tensor-product matrix models assume independent readout noise or at least weak correlations (Yang et al., 2022, Lee, 4 Jun 2025). Randomized compiling analyses presume Markovian noise and stationarity during calibration (Hashim et al., 2023). Detector-tomography studies show that readout on IBM and Rigetti transmon devices is dominated by classical noise, but coherent detector errors and pairwise correlations are not strictly zero; modeling them improves some two-qubit tasks, though on the tested IBM Falcon hardware such correlations did not have a strong effect on the final $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$5 observable once dominant single-qubit bias was removed (Maciejewski et al., 2019, Alexandrou et al., 2021).

6. Limitations, controversies, and current directions

A central limitation is the entanglement of readout calibration with SPAM. Conventional mitigation based on a calibration matrix cannot, in general, distinguish state-preparation errors from measurement errors. One analysis shows that the effective mitigation matrix actually learned in practice is contaminated by preparation errors $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$6, so that corrected expectations acquire factors $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$7. For many-qubit observables this leads to relative errors of the form

$\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$8

which grow exponentially with qubit number. The same work argues that large-scale entangled-state fidelities can be significantly overestimated in the presence of state-preparation error and gives the approximate safety bound $\mathcal M_{ij}=\operatorname{Tr}[E_i\rho_j],$9 for tolerable per-qubit preparation error at target relative error $q=A p$00 (Guo et al., 9 Oct 2025). This is not a criticism of a specific PROM implementation so much as a warning that inversion-based readout mitigation is only as reliable as the separation between preparation and measurement errors.

A second limitation is that PROM is not universally superior to simpler correction strategies. In broad-output settings, local least-squares and local IBU can behave almost identically on the observables of interest (Nguyen, 2023). In high-fidelity mid-circuit readout regimes, branch-resolved teleportation shows that post-processing can outperform PROM because the latter pays quasiprobability variance and additional control complexity for a small residual bias reduction (Edwards et al., 30 Apr 2026). In dynamic circuits more generally, PROM corrects only the mid-circuit measurement channel; gate errors, decoherence, leakage, and drift remain and can dominate once readout bias is removed (Koh et al., 2024, Hashim et al., 2023).

Current directions focus on richer channel models and data-driven refinement. One direction represents the readout channel itself as an MPO or PEPO, trains it by likelihood optimization, and then uses it for nonlocal observable correction, random circuit sampling, classical shadows, learning-based tomography, and tensor-network QEC decoding; in 2D, the same framework enables joint inference over data and readout errors (Guo et al., 24 Jun 2026). Another direction refines a standard assignment matrix $q=A p$01 using low-depth circuits and linear regression, producing a personalized matrix $q=A p$02; on a seven-qubit simulated IBM Perth backend at depth four, this gave median improvements of $q=A p$03 in fidelity, $q=A p$04 in mean-squared error, and $q=A p$05 in Hellinger distance over the standard mitigation approach (Lee, 4 Jun 2025). A plausible implication is that PROM is evolving away from static calibration matrices toward structured, workload-aware probabilistic models.

Taken together, the literature presents PROM as a broad probabilistic framework for measurement-noise inversion rather than a single standardized protocol. Its mature forms range from confusion-matrix inversion and EM unfolding to quasi-probabilistic channel cancellation, randomized-measurement suppression-factor estimation, and tensor-network inference. The common strengths are hardware compatibility and a precise classical statistical interpretation; the common weaknesses are calibration dependence, variance amplification, and sensitivity to model mismatch.