Quantum computing holds the promise of dramatically accelerating computations for specific problems compared to classical methods. However, the significant challenge lies in the inherent noise of current quantum systems, which severely limits their performance. The prevailing solution is the development of fault-tolerant quantum computers, which are currently beyond our technological reach. This paper directly addresses the crucial question of whether current, noisy quantum processors can provide a computational advantage before fault tolerance is achieved. The prevailing belief is that even simple quantum circuits capable of outperforming classical methods will have to wait for more advanced, error-corrected processors. This is supported by fidelity calculations which suggest very low fidelities even for relatively simple circuits. Nevertheless, the potential for extracting accurate information from these noisy devices remains an open and critical question. This research focuses on demonstrating the first step towards establishing quantum advantage: showing that current devices can perform accurate computations at a scale beyond classical simulation. The researchers do not aim to implement algorithms with proven speedups but rather to demonstrate the foundational capability of accurate computation at scale.

The paper references several key works in quantum computing and error mitigation. It cites research on probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE) as error mitigation techniques. It also mentions the use of tensor network methods (matrix product states (MPS) and isometric tensor network states (isoTNS)) for classical simulation of quantum systems and their limitations in handling high entanglement. The literature review implicitly acknowledges the ongoing debate regarding the potential for near-term quantum advantage and highlights the need for experimental evidence to address this debate. The paper builds upon previous work on sparse Pauli-Lindblad noise models, which are used to characterize and control the noise in the quantum processor. Previous work on quantum error mitigation techniques, specifically PEC and ZNE, and their respective strengths and weaknesses are discussed and compared within this paper. The limitations of classical simulation methods, particularly tensor network methods like MPS and isoTNS, for high-entanglement quantum states are also cited and form the basis for the experimental comparisons.

The experiment utilizes an IBM Eagle processor, a 127-qubit superconducting quantum processor with improved coherence times (T1 and T2). The researchers employ a Trotterized time evolution of a 2D transverse-field Ising model, a well-known model in physics, to benchmark the quantum computer's performance. This choice is motivated by the Ising model's ability to exhibit strong entanglement growth, presenting a challenge for classical approximation methods. The time evolution is simulated using a first-order Trotter decomposition, and the resulting circuits are executed on the quantum processor. Noise is introduced using a sparse Pauli-Lindblad noise model, allowing for controlled noise amplification in ZNE. To improve accuracy, probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE) techniques are implemented. PEC involves learning a noise model and effectively inverting it through sampling, while ZNE extrapolates from noisy expectation values at different noise levels. The study begins with Clifford circuits, which are classically verifiable, to calibrate the error mitigation techniques. Then, the researchers move to non-Clifford circuits to assess the performance in more complex and practically relevant scenarios. The classical simulation is performed using exact methods where possible and tensor network methods (MPS and isoTNS) when exact methods are computationally intractable. For higher-weight observables or deeper circuits, light-cone and depth-reduced (LCDR) circuits are used, which allow for exact classical simulation of a subset of qubits, providing another point of comparison. The experimental data is analyzed by performing statistical analysis and comparing the results obtained from the quantum computer to the exact results from classical calculations or approximations.

The researchers successfully demonstrated accurate measurement of expectation values for quantum circuits with up to 60 layers of two-qubit gates (2,880 CNOT gates) on a 127-qubit processor, a scale beyond brute-force classical simulation. The accuracy of the measurements was verified through comparisons with exactly verifiable Clifford circuits and approximate classical calculations. Zero-noise extrapolation (ZNE) significantly improved the accuracy of the expectation values, especially for deeper circuits, outperforming linear extrapolation in many cases. In regimes of strong entanglement, the quantum computer provided correct results whereas leading classical approximations such as MPS and isoTNS methods broke down. This breakdown of classical methods was observed even with relatively small bond dimensions (χ), highlighting the limitations of classical approximation techniques. The results for both Clifford and non-Clifford circuits showed that error mitigation techniques significantly improved the accuracy of the results obtained from the noisy quantum computer. In cases where classical simulation was intractable, even with light-cone and depth-reduction techniques, the quantum results showed agreement with the predictions made using tensor network simulations. Comparisons were made between mitigated experimental results and exact and approximate classical simulations across various circuit depths, observables, and entanglement strengths. The study showed that even with noise, the quantum computer consistently produces reliable results in cases where classical methods struggle. The scalability of the error mitigation techniques is also demonstrated, showing their effectiveness even for larger circuit volumes and non-trivial observables.

The findings demonstrate that current, noisy intermediate-scale quantum (NISQ) devices can perform accurate computations beyond the reach of classical methods, even before the advent of fault-tolerant quantum computation. The successful use of error mitigation techniques like ZNE highlights their potential for expanding the capabilities of NISQ devices. The superior performance of the quantum computer in strongly entangling regimes suggests the potential for quantum advantage in specific applications. The observed limitations of classical approximation methods, such as MPS and isoTNS, in handling high entanglement provide further support for the utility of quantum computation. The results of the experiments motivate the exploration of other quantum algorithms to exploit the demonstrated accuracy and reliability for practically relevant computations. This work encourages future research on quantum algorithms and error mitigation techniques as well as the development of better classical approximation techniques. The results challenge the conventional expectation that quantum advantage must wait for fault-tolerant machines, urging further research into near-term quantum applications.

This research provides strong experimental evidence for the potential utility of noisy quantum computers in a pre-fault-tolerant era. The ability to accurately measure expectation values for large-scale, complex quantum circuits using error mitigation techniques marks a significant step towards realizing practical quantum applications. Future work should focus on developing and testing novel quantum algorithms designed to exploit the demonstrated accuracy and scalability, and continue to improve error mitigation strategies and explore more advanced classical approximation methods. The continuing improvement in hardware performance will further increase the potential for achieving quantum advantage in various areas.

The study primarily focuses on the Trotterized time evolution of the 2D transverse-field Ising model, which might not fully represent the complexity of other quantum algorithms. While the error mitigation techniques were effective, they still introduce a degree of bias (ZNE) or require high sampling overhead (PEC). The classical simulations using tensor network methods are limited by computational resources, particularly the bond dimension, which limits their ability to handle high entanglement accurately. The noise model used in the simulations might not capture all aspects of the real noise in the quantum processor, potentially affecting the accuracy of the results. The study focuses on a specific quantum processor (IBM Eagle), and the results may not generalize directly to other quantum computing architectures.