Quantum error mitigation (QEM) is at the very heart of near-term quantum simulations and lattice gauge theories are no exceptions, rather their physical symmetries provide natural consistency checks on noise quantum states. In this work, we exploit the parity and fermion-number symmetries of a gauge theory, the (1+1)-dimensional Schwinger model, under depolarising noise and investigate symmetry verification under digital quantum simulation. We investigate two set-ups - symmetry- sector post-selection in adiabatic state preparation followed by real-time measurements of the chiral condensate and symmetry verification within a variational quantum eigensolver (VQE). In the first case, post-selection reduces the bias in the chiral condensate consistently removing up to 60% of the quantum noise induced error in our system. Motivated by the observed regularity of the residual bias (in the low-noise regime), we further introduce a global-noise calibration obtained from classically accessible smaller lattices and implemented on larger lattices recovering noiseless chiral condensate values within statistical uncertainty. However, in VQE, symmetry verification does not seem to generally improve the optimised parameters or the fidelity of the prepared states, although it reduces the bias in the estimated ground-state energy. This demonstates that improving a noisy cost-function estimator in variational algorithms does not necessarily improve the outcome of the algorithm. Our results show the strength of symmetry verification in different approaches while establishing that its usefulness critically depends on where it is applied in the computational workflow, and provide practical guidance for symmetry-assisted quantum error mitigation in quantum simulations of lattice gauge theories.
Quantum error mitigation (QEM) is at the very heart of near-term quantum simulations and lattice gauge theories are no exceptions, rather their physical symmetries provide natural consistency checks on noise quantum states. In this work, we exploit the parity and fermion-number symmetries of a gauge theory, the (1+1)-dimensional Schwinger model, under depolarising noise and investigate symmetry verification under digital quantum simulation. We investigate two set-ups - symmetry- sector post-selection in adiabatic state preparation followed by real-time measurements of the chiral condensate and symmetry verification within a variational quantum eigensolver (VQE). In the first case, post-selection reduces the bias in the chiral condensate consistently removing up to 60% of the quantum noise induced error in our system. Motivated by the observed regularity of the residual bias (in the low-noise regime), we further introduce a global-noise calibration obtained from classically accessible smaller lattices and implemented on larger lattices recovering noiseless chiral condensate values within statistical uncertainty. However, in VQE, symmetry verification does not seem to generally improve the optimised parameters or the fidelity of the prepared states, although it reduces the bias in the estimated ground-state energy. This demonstates that improving a noisy cost-function estimator in variational algorithms does not necessarily improve the outcome of the algorithm. Our results show the strength of symmetry verification in different approaches while establishing that its usefulness critically depends on where it is applied in the computational workflow, and provide practical guidance for symmetry-assisted quantum error mitigation in quantum simulations of lattice gauge theories.
Variational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by applying the recently proposed symmetry verification technique to the experimental estimation of the ground-state energy and ground state of the hydrogen molecule. A finely adjustable exchange interaction between two qubits in a circuit QED processor efficiently prepares variational ansatz states in the single-excitation subspace respecting the parity symmetry of the qubit-mapped Hamiltonian. Symmetry verification improves the energy and state estimates by mitigating the effects of qubit relaxation and residual qubit excitation, which violate the symmetry. A full-density-matrix simulation matching the experiment dissects the contribution of these mechanisms from other calibrated error sources. Enforcing positivity of the measured density matrix via scalable convex optimization correlates the energy and state estimate improvements when using symmetry verification, with interesting implications for determining system properties beyond the ground-state energy.
Quantum error mitigation (QEM) has emerged as a powerful tool for the extraction of useful quantum information from quantum devices. Here, we introduce the Subspace Noise Tailoring (SNT) algorithm, which efficiently combines the cheap cost of Symmetry Verification (SV) and low bias of Probabilistic Error Cancellation (PEC) QEM techniques. We study the performance of our method by simulating the Trotterized time evolution of the spin-1/2 Fermi-Hubbard model (FHM) using a variety of local fermion-to-qubit encodings, which define a computational subspace through a set of stabilizers, the measurement of which can be used to post-select noisy quantum data. We study different combinations of QEM and encodings and uncover a rich state diagram of optimal combinations, depending on the hardware performance, system size and available shot budget. We then demonstrate how SNT extends the reach of current noisy quantum computers in terms of the number of fermionic lattice sites and the number of Trotter steps, and quantify the required hardware performance beyond which a noisy device may compete with current state-of-the-art classical computational methods.
Quantum simulation is a powerful tool for exploring complex quantum many-body systems such as condensed matter physics and gauge theories. Trotterization, which approximates the ideal time evolution operator by decomposing it into a sequence of local gate operations, is one of the most widely used quantum simulation algorithms. However, such Trotterized implementations generally fail to preserve the symmetries of the target Hamiltonian during compilation. As a result, they can drive quantum states out of symmetrically allowed subspaces, leading to unphysical dynamics and symmetry-violating algorithmic errors. In this work, we propose a symmetry-based Trotter error mitigation protocol using classical post-processing. By applying symmetry transformations to the initial state or interleaving them between discrete Trotter layers, and then averaging an ensemble of the resulting measurement outcomes via classical post-processing, our method systematically projects out the symmetry-violating components of the Trotter error while leaving the ideal dynamics unchanged. Importantly, this framework naturally accommodates non-local spatial symmetries and anti-unitary operations such as time reversal, which are difficult or impossible to implement directly with hardware-native quantum gates. We benchmark our protocol on the one-dimensional XY model and the one-dimensional Schwinger model. In the XY model, enforcing reflection symmetry suppresses the leading-order Trotter error, whereas in the Schwinger model, interleaving gauge transformations between Trotter layers enables gauge-twirling effectively to reduce unphysical violations of local Gauss's law. These results demonstrate that symmetry-based post-processing provides a depth-preserving route to substantially improving the fidelity of Trotterized quantum simulations on near-term devices.
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 estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 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 random 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.
We present a new quantum error mitigation technique (QEM), called GUiding Extrapolations from Symmetry decayS (GUESS), which exploits Hamiltonian symmetries to improve accuracy of noisy quantum computations. This method is explicitly designed for quantum algorithms that estimate expectation values of observables and consists in learning the extrapolation coefficients from a symmetry observable of the system to then estimate the value of a target observable. Furthermore, we propose a Hamiltonian impurity technique to enforce symmetries allowing the mitigation of local observables of interest. We employ the IBM Heron r2 quantum processing unit '\texttt{ibm\_basquecountry}' to simulate the time evolution of average magnetization and nearest-neighbor correlator observables for transverse field Ising and XZ Heisenberg models in 1D with open boundary conditions. We benchmark the accuracy of our method against baseline Zero Noise Extrapolation (ZNE) and tensor network simulations for systems of 100 qubits. Remarkably, GUESS achieves a relative error around 10% for circuits containing up to 8000 CZ gates, while showcasing lower variance than ZNE on average across 20 observables and requiring only twice the number of shots per observable compared to baseline ZNE. Furthermore, we demonstrate that GUESS enables statistical post-selection based on the outcomes of the symmetry observable, which provides critical information about the quality of the target qubits by means of its mean and variance. These results indicate that GUESS is a powerful QEM technique capable of mitigating utility-scale circuit outcomes, delivering high accuracy and reduced variance for large-scale circuits with minimal quantum overhead.
Non-Abelian gauge theories underlie our understanding of fundamental forces of modern physics. Simulating them on quantum hardware is an outstanding challenge in the rapidly evolving field of quantum simulation. A key prerequisite is the protection of local gauge symmetries against errors that, if unchecked, would lead to unphysical results. While an extensive toolkit devoted to identifying, mitigating, and ultimately correcting such errors has been developed for Abelian groups, non-commuting symmetry operators complicate the implementation of similar schemes in non-Abelian theories. Here, we discuss two techniques for error mitigation through symmetry verification, tailored for non-Abelian lattice gauge theories implemented in noisy qudit hardware: dynamical post-selection (DPS), based on mid-circuit measurements without active feedback, and post-processed symmetry verification (PSV), which combines measurements of correlations between target observables and gauge transformations. We illustrate both approaches for the discrete non-Abelian group D 3 in 2+1 dimensions, explaining their usefulness for current NISQ devices even in the presence of fast fluctuating noise. Our results open new avenues for robust quantum simulation of non-Abelian gauge theories, for further development of error-mitigation techniques, and for measurement-based control methods in qudit platforms.