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

Jake Scally, Austin Myers, Ryan Carmichael, Phat Tran, Xiuwen Liu

Abstract

Current quantum computers present significant noise, especially as circuit depth and qubit count increase. Prior work has demonstrated that erroneous outcomes exhibit some behavior in Hamming space, enabling improvements in the output distributions of NISQ-era computers. We present HAMMR-L: a principled post-processing technique for improving the fidelity of output distributions by applying Richardson-Lucy image deconvolution on a state graph of measurement results connected by Hamming distance. We show that this preliminary implementation of HAMMR-L outperforms existing cutting-edge Hamming-based post-processors such as QBEEP while being circuit and hardware agnostic, which QBEEP is not. HAMMR-L also demonstrates clear potential for future improvements and we discuss how such improvements might be realized while highlighting the strengths, limitations, and generality of the underlying concept.

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

Abstract

Current quantum computers present significant noise, especially as circuit depth and qubit count increase. Prior work has demonstrated that erroneous outcomes exhibit some behavior in Hamming space, enabling improvements in the output distributions of NISQ-era computers. We present HAMMR-L: a principled post-processing technique for improving the fidelity of output distributions by applying Richardson-Lucy image deconvolution on a state graph of measurement results connected by Hamming distance. We show that this preliminary implementation of HAMMR-L outperforms existing cutting-edge Hamming-based post-processors such as QBEEP while being circuit and hardware agnostic, which QBEEP is not. HAMMR-L also demonstrates clear potential for future improvements and we discuss how such improvements might be realized while highlighting the strengths, limitations, and generality of the underlying concept.

Paper Structure

This paper contains 12 sections, 12 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Bernstein-Vazirani Algorithm for QEM Evaluation
  4. Hamming Space
  5. Previous Hamming-Based QEM Implementations
  6. HAMMR-L Implementation
  7. Graph Construction
  8. Richardson-Lucy Deconvolution
  9. Experimental Techniques
  10. Results
  11. Discussion and Future Work
  12. Conclusion

Figures (4)

  • Figure 1: BV example circuit for the secret string "110." The classical register $c$ should hold exactly "110" with certainty on a fault-tolerant quantum computer. However, this is not the case in practice.
  • Figure 2: Example state graph based on experimental results for the 3-qubit Bernstein-Vazirani circuit for a secret string "111." Nodes have their observed string and observed probabilities based on the number of times that the string was observed divided by the total shots. Nodes are connected to their neighbors in Hamming space, meaning that nodes one edge away are a single bit flip from the source node, two edges away are two bit flips from the source node, and so on.
  • Figure 3: Example showing HAMMR-L on the top ten most likely strings for the nine-qubit Bernstein-Vazirani circuit for the secret string "111111111." As this is a circuit with high entanglement and many CNOTs, the output is hardly usable. HAMMR-L is able to increase the rank from 4th to 1st in 100 convolutions and increase the probability from less than 1% to nearly 8%.
  • Figure 4: 9-qubit Bernstein-Vazirani circuit for the secret string "111111111." This circuit exhibits extremely high error rates on IBM's quantum computers due to the depth, qubit count, number of two-qubit gates, and high entanglement.