## Quantum Error Mitigation: “Nature Physics” Publ. by Jens Eisert’s Group

**A** recent publication in “Nature Physics”** by the research group of MATH+ member Jens Eisert shows that error mitigation does not work as a scalable scheme, contrary to how it is central to IBM’s corporate strategy. What challenges arise with the question of how quantum errors can be mitigated?**

Quantum computers are a new computer type that promise efficient solutions to classically hard computational problems. Although the concept of a quantum computer is not entirely new, it is only in recent years that significant strides have been made in building large-scale quantum computers. These machines are still noisy and not yet large, but the existence of quantum computers with over 1000 quantum bits, once inconceivable, has created an exciting state of affairs. In particular, recent months have seen announcements of architectures that have surprised the community with their level of sophistication.

To combat noise, one can think of quantum error correction which comes along with substantial overheads. Currently, the common practice is to address unwanted quantum noise more directly: Adding noise is more feasible than eliminating it, and in some cases, there is a clear understanding of the underlying noise mechanism. Methods of quantum error mitigation make use of these insights. Rather than a single solution, this involves a portfolio of methods that use classical computation to partially reverse quantum noise in post-processing. This innovative and elegant approach is now routinely applied in quantum experiments, effectively undoing unwanted noise. However, the question remains: How far can this approach take us?

To address the question of the potential and limitations of quantum error mitigation, this work explores the extent to which quantum error mitigation can work for all quantum circuits as a scalable scheme for large system sizes. For this purpose, a mathematical framework is developed that is general enough to encapsulate a large number of protocols that are being used in practice. The approach was motivated by the author’s previous work on the impact of noise on quantum computing, prompting a natural inquiry into its implications for quantum error mitigation.

The approach taken in this publication is radical, treating error mitigation as a statistical inference problem: If some of the noise can be canceled, it must be possible to discriminate certain input states and then relate the task to hypothesis testing. One can design reasonable circuits that mix so much that this task becomes impossible unless the number of samples scales exponentially with both the depth and the system size. This was an intricate step: The team had to find a circuit that is extremely mixing in a precise sense.

Ultimately, this means that even at log-log depth, a marginal increase beyond constant depth makes quantum error mitigation prohibitively costly. This bound is exponentially tighter than previously known bounds.

It doesn’t mean that quantum error mitigation does not work. It just means that it is not scalable in the most extreme sense. This aligns with MATH+ vision, as it tackles a rigorous mathematical problem closely tight to real-world technological application—specifically quantum technology.

However, like all no-go theorems, the result established here should also be seen as an invitation: Firstly, it’s a worst-case scenario. Long-ranged entanglement and quantum noise do not seem to work well together. By using more local architectures, we might achieve better scaling. Moreover, this result does not affect quantum error correction. Ultimately, this motivates the scientists to look for coherent instances of quantum error correction and mitigation that do not fall within the framework established in this publication. Eisert explained: “There are good reasons to believe that a positive interpretation of the result makes sense. In this way, this work, aligning with the MATH+ focus, can help to transform the world through mathematics.“

**Original publication in ***Nature Physics***:**

*Exponentially tighter bounds on limitations of quantum error mitigation*

Yihui Quek, Daniel Stilck França, Sumeet Khatri, Johannes Jakob Meyer, Jens Eisert

DOI: https://doi.org/10.1038/s41567-024-02536-7

In Nature Physics (July, 2024): https://www.nature.com/articles/s41567-024-02536-7

**LINKS:**

- Jens Eisert at FU Berlin, Dahlem Center for Complex Quantum Systems
- Research Group Jens Eisert (FU Berlin): Quantum many-body theory, quantum information theory, and quantum optics
- MATH+ Project EF1-11: Quantum Advantages in Machine Learning (EF: Emerging Field)
- MATH+ Project EF1-7: Quantum Machine Learning