⚛️ quantum physics

HAMMR-L: Noise Reduction in Quantum Outcomes Using a Richardson-Lucy Deconvolution Algorithm for Quantum State Graphs

This paper introduces HAMMR-L, a circuit- and hardware-agnostic post-processing technique that applies Richardson-Lucy deconvolution to measurement results on a Hamming distance state graph to reduce noise and improve output distribution fidelity on NISQ-era quantum computers, outperforming existing methods like QBEEP.

Original authors: Jake Scally, Austin Myers, Ryan Carmichael, Phat Tran, Xiuwen Liu

Published 2026-04-01

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jake Scally, Austin Myers, Ryan Carmichael, Phat Tran, Xiuwen Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a friend's voice on a very noisy phone call. The signal is there, but static, crackles, and background chatter are drowning out the words. You know your friend is saying something specific, but the noise makes it sound like gibberish.

This is exactly the problem scientists face with Quantum Computers today. These machines are incredibly powerful, but they are also extremely "noisy." When they try to solve a problem, the answer they give is often a jumbled mess of the correct answer and many wrong answers, all mixed together by errors.

This paper introduces a new tool called HAMMR-L to clean up that noise. Here is how it works, explained simply:

1. The Problem: A Blurry Photo

Think of a quantum computer's output not as a list of numbers, but as a blurry photograph.

The correct answer is the sharp, clear subject of the photo (like a person's face).
The noise is like a smudge or a blur that spreads the light from that face onto the surrounding pixels, making it look like the person is standing next to a bunch of other people who aren't really there.

In the quantum world, this "blur" happens because of something called Hamming Distance. If the correct answer is 111, the computer might accidentally give you 110 or 101 (where just one bit flipped). These wrong answers are "neighbors" to the right answer, and the noise spreads the probability of the right answer out to these neighbors.

2. The Old Way: Guessing the Neighborhood

Previous methods (like one called QBEEP) tried to fix this by looking at the "neighborhood." They would say, "Hey, 110 is close to 111, so maybe 111 is the real answer." They used a fixed rule (like a Poisson distribution) to guess how the noise spreads. It worked okay, but it was like trying to clean a photo using a generic, one-size-fits-all filter.

3. The New Way: HAMMR-L (The Smart De-blur)

The authors of this paper realized that the noise pattern looks a lot like a blurred image. So, they borrowed a tool from photography called Richardson-Lucy Deconvolution.

The Analogy: Imagine you have a blurry photo of a star. You know that stars are bright points of light, but the camera lens smeared the light out. If you know exactly how the lens smears light (the "Point Spread Function"), you can mathematically reverse the process to make the star sharp again.
The Innovation: HAMMR-L treats the quantum computer's messy list of answers as that blurry photo. It assumes the "smear" is caused by bit-flips (changing a 0 to a 1 or vice versa). It then uses a mathematical algorithm to "un-smear" the data, pulling the probability back from the wrong neighbors and concentrating it on the correct answer.

4. How They Tested It

To test this, they used a standard quantum puzzle called the Bernstein-Vazirani algorithm.

Imagine a secret code (like 111111111).
They ran this code on real IBM quantum computers. Because the computers are noisy, the result was a mess. The correct code might have been ranked 4th or 5th, buried under wrong answers.
They ran the HAMMR-L algorithm on the messy results.

The Result:
In many cases, HAMMR-L took the correct answer from being buried deep in the list (e.g., rank 4) and boosted it to the very top (rank 1). It did this better than the previous best method (QBEEP), especially when the noise was very heavy.

5. Why This Matters

It's Agnostic: Unlike some other tools that need to know the specific details of the computer's hardware to work, HAMMR-L is like a universal lens cleaner. It works on different types of quantum computers without needing a manual for each one.
It's a Framework: The authors admit their current "lens" (the math they use to describe the blur) is a bit simple. They suggest that in the future, we could make it even smarter by using AI or Blind Deconvolution (where the computer figures out the blur pattern itself without being told).

The Bottom Line

HAMMR-L is a clever way of using old-school image processing math to fix new-school quantum computer errors. It takes the "fuzzy" results we get from today's noisy machines and mathematically sharpens them, helping us get the right answer more often without needing to wait for perfect, error-free quantum computers to be built.

1. Problem Statement

Current quantum computers operate in the Noisy Intermediate-Scale Quantum (NISQ) era, characterized by significant noise that degrades output fidelity as circuit depth and qubit counts increase. While Quantum Error Correction (QEC) is the long-term goal, it is not yet feasible for large-scale systems. Consequently, Quantum Error Mitigation (QEM) is essential for improving the utility of current hardware.

Existing QEM techniques, particularly those based on Hamming space (the metric space of binary strings where distance represents the number of bit-flip errors), have shown promise. Notable predecessors include:

HAMMER: Uses a state graph connected by Hamming distance to assign "neighborhood scores" to outcomes.
QBEEP: An improvement over HAMMER that uses a Poisson distribution for edge weighting, tailored to specific circuit and hardware error rates.

However, these methods often lack generality (being hardware/circuit dependent) or rely on heuristic weighting. The authors identify a need for a principled, hardware-agnostic approach that can mathematically model the "blurring" of quantum probability distributions caused by noise.

2. Methodology: HAMMR-L

The authors propose HAMMR-L, a post-processing technique that reframes quantum error mitigation as an image deconvolution problem.

Core Concept: Deconvolution in Hamming Space

The authors model the noisy output distribution of a quantum circuit as a blurred version of the true (error-free) distribution.

The "Image": The state graph of observed measurement outcomes, where nodes represent bitstrings and their "brightness" represents observed probability.
The "Blur": The Point Spread Function (PSF), which describes how a correct bitstring "spreads" into incorrect strings due to bit-flip errors.
The Distance Metric: The blur is governed by Hamming distance ( $h(i, j)$ ), the number of bit flips between two strings.

The Algorithm

HAMMR-L adapts the Richardson-Lucy (R-L) deconvolution algorithm, traditionally used for image restoration, to discrete quantum state graphs.

Graph Construction: A state graph is built where nodes are observed strings connected if their Hamming distance is 1.
PSF Definition: The authors define the PSF, $p_{ij}$ , based on the Hamming distance between strings $i$ and $j$ :
$p_{ij} = \frac{1}{h(i, j) + 1}$
This function assumes that the probability of an error decreases as the Hamming distance from the correct string increases.
Iterative Deconvolution:
- Let $d_i$ be the observed (noisy) probability and $u_i$ be the true (error-free) probability.
- The relationship is modeled as a discrete convolution: $d = u * P$ .
- The algorithm iteratively estimates the true distribution $\hat{u}$ using the R-L update rule:
  $\hat{u}^{(n+1)}_j = \hat{u}^{(n)}_j \sum_i \frac{d_i}{c_i} p_{ij}$
  where $c_i$ is a normalization factor derived from the current estimate.
Convergence: The process repeats until convergence or a maximum iteration count is reached, followed by normalization to ensure probabilities sum to 1.

3. Key Contributions

Novel Framework: Introduces the first application of Richardson-Lucy deconvolution to quantum state graphs, treating noise mitigation as a mathematical deblurring problem rather than a heuristic re-weighting.
Hardware and Circuit Agnosticism: Unlike QBEEP, which requires tailoring the error distribution model to specific hardware and circuit layouts, HAMMR-L uses a universal PSF based on Hamming distance, making it applicable across different quantum processors without re-calibration.
Superior Performance: Demonstrates that a principled deconvolution approach outperforms current state-of-the-art Hamming-based methods (specifically QBEEP) in recovering correct outcomes.

4. Experimental Results

The authors evaluated HAMMR-L using the Bernstein-Vazirani (BV) algorithm on IBM quantum hardware. They generated datasets for 9-qubit and 10-qubit circuits with varying "ones counts" (number of 1s in the secret string), which correlates to the number of CNOT gates and thus the error rate.

Key Metrics:

Rank Improvement: Measuring how much the rank of the correct secret string improves (e.g., moving from 4th to 1st).
Mean Rank Change: The average shift in rank across all permutations.

Findings:

Overall Superiority: HAMMR-L outperformed QBEEP in 7 out of 10 datasets.
Mean Rank Change:
- HAMMR-L: Average rank increase of +3.081.
- QBEEP: Average rank change of -0.328 (indicating QBEEP often worsened the ranking on average).
High-Error Regimes: HAMMR-L showed exceptional performance in high-noise scenarios (e.g., 8-ones datasets). In one 9-qubit case with a secret string of all 1s, HAMMR-L recovered the correct string from the 4th rank to the 1st rank, increasing its probability from <1% to nearly 8%.
Consistency: While HAMMR-L showed some run-to-run variance (likely due to hardware fluctuations), it consistently provided positive rank improvements in high-error cases where QBEEP often failed or degraded results.

5. Significance and Future Work

Significance:
HAMMR-L provides a principled, generalizable framework for error mitigation. By leveraging well-established image processing mathematics, it offers a robust method to "deblur" quantum outputs without needing detailed knowledge of the underlying hardware noise models. This makes it a powerful tool for immediate NISQ-era applications.

Future Directions:
The authors identify three paths for improvement:

Hardware-Aware PSF: Tailoring the PSF to specific hardware noise models (similar to QBEEP) to increase accuracy, though this sacrifices some agnosticism.
Blind Deconvolution: Estimating the PSF directly from the blurred data structure without prior knowledge, a technique well-developed in image processing.
Deep Learning Integration: Combining R-L deconvolution with deep learning to learn the PSF and accelerate convergence, potentially offering further gains in speed and accuracy.

In conclusion, HAMMR-L represents a significant step forward in making NISQ-era quantum computers more useful by effectively mitigating noise through a mathematically rigorous deconvolution approach.