**Because evolution is
unitary**.

## Making quantum error mitigation practical

Will Zeng and Nathan Shammah

Useful applications of quantum
computers require significant reductions
in logical error rates.^{1}^{2}^{3}^{4}^{5}^{6} 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.^{7}^{8}^{9}^{10}^{11}^{12}^{13}^{14}^{15}^{16}^{17}
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*
scales^{18} as
*N* ∝ *γ*^{2}/*δ*^{2},
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 as^{21}
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 noise^{22}
that could improve the performance of
error correction. Further, error
mitigation can be extended into the
fault-tolerant regime where it can
reduce overheads^{23}
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:

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.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.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.