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Statistical Signal Processing for Quantum Error Mitigation
Kausthubh Chandramouli, Kelly Mae Allen, Christopher Mori, Dror Baron, Mário A. T. Figueiredo
TL;DR
The paper addresses reliable QEM under depolarizing and bit-flip noise by filtering uninformative observations and applying EM to estimate noiseless outputs. It reports successful recovery across small-qubit simulations and scalable accuracy on synthetic larger-qubit data, while noting hardware-noise and implementation limitations.
Problem
QEM needs to recover reliable noiseless outputs when deep-circuit measurements include heavy depolarizing noise and classical bit-flip errors.
Method
The method filters shots resembling uniform depolarizing noise, then applies EM to the remaining data to compute a maximum-likelihood output distribution.
Results
For K < 8 and D < 800, EM perfectly estimates all correct outputs with BER 0, while the total PKerror is 0.83% across parameter combinations.
Takeaways & Limitations
The method succeeds for n = 10, 12, 14 qubits and retains high accuracy and scalability on synthetic data consistent with depolarizing and bit-flip noise.
Takeaways & Limitations
The model assumes symmetric bit-flip probabilities, omits crosstalk, assumes finite K, and uses an EM implementation whose initialization and K estimation can fail occasionally.
Abstract
from arXiv · showhide
In the noisy intermediate-scale quantum (NISQ) era, quantum error mitigation (QEM) is essential for producing reliable outputs from quantum circuits. We present a statistical signal processing approach to QEM that estimates the most likely noiseless outputs from noisy quantum measurements. Our model assumes that circuit depth is sufficient for depolarizing noise, producing corrupted observations that resemble a uniform distribution alongside classical bit-flip errors from readout. Our method consists of two steps: a filtering stage that discards uninformative depolarizing noise and an expectation-maximization (EM) algorithm that computes a maximum likelihood (ML) estimate over the remaining data. We demonstrate the effectiveness of this approach on small-qubit systems using IBM circuit simulations in Qiskit and compare its performance to contemporary statistical QEM techniques. We also show that our method scales to larger qubit counts using synthetically generated data consistent with our noise model. These results suggest that principled statistical methods can offer scalable and interpretable solutions for quantum error mitigation in realistic NISQ settings.
I. INTRODUCTION
The paper frames QEM as classical post-processing for reliable NISQ outputs and proposes combining filtering with statistical estimation to recover likely noiseless results from noisy measurements.
- QEM post-processes noisy circuit outputs with classical computation to estimate ideal results.
- Depolarizing noise is difficult to estimate because it nearly uniformly distributes measurements and carries little exploitable information.
- The proposed method combines QEM, statistical signal processing, and classical filtering for binary outputs affected by depolarizing noise and bit-flip errors.
- The pipeline filters observations consistent with depolarizing noise and applies EM to estimate remaining binary output distributions.
- Deep circuits can produce output distributions in which 80–90% of shots appear random, motivating methods that handle heavy depolarizing noise.
A. Noise model
The noise model combines independent depolarizing noise on qubits with classical symmetric bit-flip readout errors, then formulates noiseless outputs as a finite mixture estimated by maximum likelihood.
- Depolarizing noise maps each qubit state to the maximally mixed state with probability p and otherwise leaves it unchanged.
- The readout model flips each classical bit with probability ε and assumes equal crossover probabilities for 0→1 and 1→0.
- After filtering depolarizing noise, the remaining bit-flip probabilities may vary independently across qubits as ε = {ε1, ε2, . . . , εn}.
- The observed dataset contains S noisy binary shots, generated from K unknown noiseless binary vectors with mixing weights αk.
- A latent one-hot variable identifies which noiseless vector generated each observation, yielding a finite-mixture likelihood and a maximum-likelihood estimation problem.
- Because the mixture optimization lacks a closed-form solution, the method applies EM, which converges to a local maximum of the likelihood.
C. Estimating K: Modified Likelihood Function
Because likelihood increases with the number of mixture components, the paper uses a modified minimum-message-length objective to estimate K while accounting for model complexity.
- When K is unknown, the ordinary likelihood cannot select the number of components because it increases with K.
- The modified objective is based on minimum message length and penalizes the number of nonzero components and their parameter dimensionality.
- For this model, each component has dimensionality d = n, the length of its binary output vector.
- The formulation should account for coding the K length-n solutions, but its MML characterization remains ongoing.
- The current formulation works sufficiently well for the reported experiments, while the authors focus on the core EM algorithm without major modification.
III. ALGORITHMIC PROCEDURE
The algorithm first removes shots resembling uniform depolarizing noise, then applies EM to the filtered data to estimate the remaining maximum-likelihood solution.
- The procedure has two stages: depolarization filtering followed by EM estimation on the filtered dataset.
- A. Filtering: Filtering removes uniformly spread shots so the retained data better match the bit-flip-only noise model.
- A. Filtering: The filter uses local Hamming neighborhoods and a tunable threshold η to identify observations with sufficient density.
- A. Filtering: Its implementation counts bit-string frequencies and 1-Hamming-distance neighbors, suppresses low-support entries, and reconstructs the dataset from retained entries.
- A. Filtering: The threshold is calibrated from the expected uniform frequency λ = S/2^n and the neighborhood count expectation λ(n+1).
B. Expectation maximization (EM) algorithm
The paper applies EM to estimate parameters in a mixture model with unobserved component assignments. Its E-step computes expected latent assignments, while the M-step maximizes the resulting expected complete log-likelihood.
- EM estimates model parameters when latent variables identify which mixture component produced each observation.
- E-step: The E-step computes the expected complete log-likelihood over latent assignments using current parameter estimates.
- M-step: The M-step updates parameters by maximizing the expected complete log-likelihood.
- E-step: The conditional expectation W represents latent-variable assignments given the observations and current parameter estimates.
2) M-step:
The M-step maximizes a modified likelihood with respect to mixture probabilities and component parameters. The procedure can remove components whose estimated probabilities become zero and uses decoupled updates where possible.
- M-step: Parameter updates maximize LMML(θ; Y) with the Q-function replacing the original likelihood.
- M-step: Maximization over mixture probabilities is independent of the other parameters and can reduce the number of active components.
- M-step: The component-parameter maximization is decoupled across candidate solutions and across bit dimensions.
- Algorithm: Algorithm 2 initializes bounds, parameter estimates, and a stopping threshold before iterating E- and M-step updates.
- Algorithm: Components with zero estimated mixture probability are removed, decreasing the current count of nonzero components.
IV. EXPERIMENTAL RESULTS
The experiments evaluate binary-output recovery using BER, incorrect-estimation rates for the number of solutions, and Hellinger fidelity comparisons. They use simulated IBM hardware, IBM Brisbane data, and synthetic data for larger systems.
- Experiments use simulated IBM hardware in Qiskit, IBM Brisbane QPU data, and synthetic data when larger systems exceed simulation or hardware limits.
- BER measures the average proportion of differing bits between estimated and true binary-vector sets.
- Table I reports BER for the EM algorithm at K = 8 and circuit depth 800.
- The procedure sometimes incorrectly estimates the number of valid solutions, and this error rate is computed as incorrect estimates divided by total executions.
- Hellinger fidelity is adopted to compare the method with existing statistical QEM schemes by measuring similarity between discrete probability distributions.
1) Methodology:
The methodology simulates random circuits with finite sets of noiseless binary outputs and applies IBM-derived noise. Filtering removes depolarization noise before EM estimates the remaining solution mixture without knowing the true number of outputs or circuit depth.
- Simulation setup: Simulations use K = {2, 4, 6, 8} valid outputs, n = {10, 12, 14} qubits, and depths D = {500, 600, 700, 800}.
- Simulation setup: Each target state randomly selects K solution vectors from 2^n possible bit vectors and assigns them equal amplitudes.
- Simulation setup: Random entangling circuits and their inverses are constructed in Qiskit, with noise applied to every gate using an IBM Brisbane-derived noise model.
- Processing pipeline: Filtering removes most depolarization noise before EM receives the data and generates a maximum-likelihood solution.
- Processing pipeline: The EM algorithm does not know the number of valid outputs K or the circuit depth D after filtering.
2) Numerical results:
The EM algorithm accurately recovers noiseless outputs across most tested configurations, while estimation reliability depends on the number of shots and the number of valid outputs K.
- 0.83% total PKerror was recorded across all parameter combinations.
- BER was 0 for all configurations with K < 8 and circuit depth D < 800.At K = 8 and D = 800, Table I evaluates BER as the number of qubits n varies.
- PKerror converges to 0 as the number of shots increases, while EM performance generally degrades as K increases.For the selected problem sizes, 10,000 shots appeared sufficient for perfect K estimation across configurations.
- Table II compares quantum error mitigation methods.
B. Comparison with other QEM approaches on IBM hardware
On IBM hardware and synthetic large-qubit data, the EM-based method improves mitigation fidelity and maintains accurate recovery, while remaining subject to modeling and implementation limitations.
- IBM hardware comparison: Marked fidelity improvements were demonstrated on IBM Brisbane circuits ranging from 3 to 14 qubits.The comparison included HAMMER, M3, and Q-BEEP on Bernstein-Vazirani, GHZ, W, cat, and adder circuits.
- Scaling with more qubits: BER was 0 for every binary vector in synthetic n = 128 experiments with K = 2, 4, 8 solutions.The experiments used 20,000 shots with depolarizing noise p = 0.9 and per-bit flip probabilities selected from [0.05, 0.15].
- Conclusion: The method succeeds for n = 10, 12, 14 qubits with K = 2, 4, 6, 8 valid solutions under calibrated machine noise models.In this small-qubit regime, it outperforms HAMMER, M3, and Q-BEEP and retains high accuracy on synthetic larger-n data.
- Limitations: The approach assumes symmetric bit-flip probabilities, excludes complex readout errors such as crosstalk, and relies on a finite-K formulation.Extension to variational algorithms is left as future work.
- Limitations: Initialization strongly influences ML estimation, K estimation can occasionally fail, and the depolarizing-noise filter is heuristic.The authors identify these issues as directions for future work.
- Conclusion: The EM algorithm is presented as a promising and scalable framework for quantum error mitigation.