# Error mitigation for variational quantum algorithms through mid-circuit measurements

**Authors:** Ludmila Botelho, Adam Glos, Akash Kundu, Jarosław Adam Miszczak, Özlem Salehi, Zoltán Zimborás  
**arXiv:** [2108.10927](https://arxiv.org/abs/2108.10927)  
**Date:** 2021-08-24  
**Summary page:** https://www.summarizepaper.com/en/paper/2108.10927v1/

## Key points

- Authors address challenges of Noisy Intermediate-Scale Quantum (NISQ) algorithms in error mitigation
- Strategy involves equipping logical qubits with hundreds/thousands of physical qubits for noise robustness
- Focus on problem instances with distinct quantum state structures like Fock states and W-states
- Utilize prior knowledge to detect errors during circuit evolution through post-selection mechanisms
- Introduce mid-circuit post-selection schemes for common encodings like one-hot, binary, gray, and domain-wall encoding
- Novel method proposed for compressing Hilbert space into more manageable subspace without ancilla qubits
- Efficacy demonstrated within Quantum Alternating Operator Ansatz algorithm
- Method suitable for existing quantum hardware setups lacking classical control conditional operators

## Summary

In their paper titled "Error mitigation for variational quantum algorithms through mid-circuit measurements," authors Ludmila Botelho, Adam Glos, Akash Kundu, Jarosław Adam Miszczak, Özlem Salehi, and Zoltán Zimborás address the challenges posed by Noisy Intermediate-Scale Quantum (NISQ) algorithms in the realm of error mitigation. These algorithms necessitate innovative approaches to counteract noise and ensure the reliability of quantum computers. One key strategy involves equipping each logical qubit with hundreds or thousands of physical qubits to enhance noise robustness. However, due to current limitations in quantum hardware capabilities, employing memory-intensive techniques for devices with limited physical qubits is not feasible. The authors focus on specific problem instances where valid quantum states exhibit a distinct structure, such as Fock states and W-states characterized by fixed Hamming weights. In these cases, the evolution occurs within a smaller subspace of the full Hilbert space. Leveraging this prior knowledge enables the detection of errors during circuit evolution by filtering out states that do not conform to the expected pattern through post-selection mechanisms. The study introduces mid-circuit post-selection schemes tailored for commonly used encodings like one-hot, binary, gray, and domain-wall encoding. For the subset of one-hot states in particular, a novel method is proposed that involves compressing the expansive Hilbert space into a more manageable subspace. This compression facilitates projection onto the desired subspace without necessitating ancilla qubits. The efficacy of this approach is demonstrated within the context of the Quantum Alternating Operator Ansatz algorithm. Notably, the proposed method is well-suited for existing quantum hardware setups where measurement and resetting operations are viable but classical control conditional operators are not readily available. By offering a practical solution for error mitigation in variational quantum algorithms through mid-circuit measurements, this research contributes valuable insights towards enhancing the performance and reliability of NISQ-era quantum computing systems.