Useful applications of quantum computers require significant reductions in logical error rates.¹²³⁴⁵⁶ One direction to achieve this is to implement quantum error correcting codes. Another direction, complementing quantum error correction, are new techniques for quantum error mitigation.^{7891011121314151617} These are algorithmic methods that are designed to be less experimentally demanding than full quantum error correction. However, this benefit comes at the cost of being less general and more heuristic.

Challenges in quantum error mitigation

There are several key challenges in making error mitigation practical.

Reducing error mitigation overhead. For example, in some techniques, the number of samples N required to approximate the expectation value output from an ideal quantum computer to within an error δ scales¹⁸ as N ∝ γ²/δ², where γ is a constant that becomes larger as the quantum program becomes larger and the quantum computer becomes noisier. The γ values of approximately 1.02 have been measured in IBM processors.¹⁹ This exponential dependence emphasizes how important it is to improve performance of different error mitigating techniques and to study their fundamental limits.²⁰

Calibrating optimal techniques. While there are a growing number of options available, this means the programmer must choose what techniques to use and with what parameters. Making this choice well depends on the hardware target and on having a good model of the noise. Further, there is a tradeoff between spending valuable quantum computer time further calibrating the error migitation vs. exploiting the model that is currently available. Additionally, while there have been shown benefits to composing error mitigating techniques—such as²¹ where generalizing PEC and ZNE produces a more robust method—there are open research questions about how best to do this composition. These calibration and composition choices need to be made scalable so that they apply to larger QPUs whose output cannot be simulated and to problems where we cannot train on a previously known answer. Finally, several error mitigating techniques require lower level access to control electronics that is not always available from vendors. More abstract techniques and the integration of error mitigation at lower levels of the stack are needed to improve performance.

Error mitigation and fault-tolerance. How can error mitigation be applied to accelerate the deployment of error correcting codes? For example, Pauli twirling can convert coherent errors into stochastic noise²² that could improve the performance of error correction. Further, error mitigation can be extended into the fault-tolerant regime where it can reduce overheads²³ and, in some examples, improve the number of logical operations that can be applied by a factor of 1000X.²⁴

Opportunities for quantum error mitigation

These challenges are opportunities to both improve the performance of today’s quantum computers and also accelerate roadmaps across hardware modalities, including quantum sensors and networks. If properly seized, then error mitigation can provide a smooth ramp up towards quantum advantage,²⁵ making it easier for the quantum technology industry to cross the chasm to valuable applications. We describe three key categories of opportunity:

1. There is an opportunity to use open source software, such as the cross platform error-mitigating compiler Mitiq,²⁶ to study and automate the calibrations needed for optimal error mitigation. Open source error mitigation implementations are accretive, allowing researchers and programmers to take advantage of the state of the art without needing to implement everything from scratch themselves. The community using this software can study and fine tune these techniques across hardware platforms and upstream their learning.

2. Integrating these error mitigating techniques with hardware design offers an opportunity for hardware-software co-design. Here, error mitigating techniques can be considered in both NISQ and fault-tolerant quantum computer architectures. One could, for example, tailor the noise channels towards ones that are easy for mitigating techniques to calibrate and counter.

3. Research at the intersection of error mitigation and error correction. As error correction becomes more practical, it is likely that there are new error mitigating techniques that can be discovered that integrate well with error correction.

Assessment and Timeline

Progress on error mitigation can be assessed using benchmarks of performance such as effective quantum volume,²⁷ improved performance of application level benchmarks, or improvements in logical gate fidelity or coherence. It is important that the assessed performance takes into account the cost and time of classical post- and pre- computations used in the error mitigation. Ideally these assessments of mitigation performance will occur in the supremacy regime where it is non-trivial (or impossible) to classically simulate the results directly. A final assessment for software tools, such as error mitigating compilers, is their usage by the community with metrics like downloads, GitHub stars, citations, etc.

Now is a good time to focus on these error mitigation challenges since (1) we have a stable pool of techniques that are ready to be reduced to practice and (2) we have a need from applications and fault-tolerant design to reduce error rates as quickly as possible. Success on these challenges can meaningfully affect the timeline to useful quantum computing across the whole field.

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