  Even with the recent rapid developments in quantum hardware, noise remains
the biggest challenge for the practical applications of any near-term quantum
devices. Full quantum error correction cannot be implemented in these devices
due to their limited scale. Therefore instead of relying on engineered code
symmetry, symmetry verification was developed which uses the inherent symmetry
within the physical problem we try to solve. In this article, we develop a
general framework named symmetry expansion which provides a wide spectrum of
symmetry-based error mitigation schemes beyond symmetry verification, enabling
us to achieve different balances between the errors in the result and the
sampling cost of the scheme. We show a way to identify a near-optimal symmetry
expansion scheme that could achieve much lower errors than symmetry
verification. By numerically simulating the Fermi-Hubbard model for energy
estimation, the near-optimal symmetry expansion can achieve estimation errors
$6$ to $9$ times below what is achievable by symmetry verification when the
average number of circuit errors is between $1$ to $2$. The corresponding
sampling cost for shot noise reduction is just $2$ to $6$ times higher than
symmetry verification. Beyond symmetries inherent to the physical problem, our
formalism is also applicable to engineered symmetries. For example, the recent
scheme for exponential error suppression using multiple noisy copies of the
quantum device is just a special case of symmetry expansion using the
permutation symmetry among the copies.
