Understanding the boundary between classical simulatability and the power of quantum computation is a fascinating topic. Direct simulation of noisy quantum computation requires solving an open quantum many-body system, which is very costly. Here, we develop a tensor network formalism to simulate the time-dynamics and the Fourier spectrum of noisy quantum circuits. We prove that under general conditions most of the quantum circuits at any constant level of noise per gate can be efficiently simulated classically with the cost increasing only polynomially with the size of the circuits. The result holds even if we have perfect noiseless quantum gates for some subsets of operations, such as all the gates in the Clifford group. This surprising result reveals the subtle relations between classical simulatability, quantum supremacy, and fault-tolerant quantum computation. The developed simulation tools may also be useful for solving other open quantum many-body systems.
Understanding the boundary between classical simulatability and the power of quantum computation is a fascinating topic. Direct simulation of noisy quantum computation requires solving an open quantum many-body system, which is very costly. Here, we develop a tensor network formalism to simulate the time-dynamics and the Fourier spectrum of noisy quantum circuits. We prove that under general conditions most of the quantum circuits at any constant level of noise per gate can be efficiently simulated classically with the cost increasing only polynomially with the size of the circuits. The result holds even if we have perfect noiseless quantum gates for some subsets of operations, such as all the gates in the Clifford group. This surprising result reveals the subtle relations between classical simulatability, quantum supremacy, and fault-tolerant quantum computation. The developed simulation tools may also be useful for solving other open quantum many-body systems.
Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[iθX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[iaX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
As quantum devices continue to improve in scale and precision, a central challenge is how to effectively utilize noisy hardware for meaningful computation. Most existing approaches aim to recover noiseless circuit outputs from noisy ones through error mitigation or correction. Here, we show that noisy quantum devices can be directly leveraged as computational resources to enhance the classical simulation of quantum circuits. We introduce the Noisy-device-enhanced Classical Simulation (NDE-CS) protocol, which improves stabilizer-based classical Monte Carlo simulation methods by incorporating data obtained from noisy quantum hardware. Specifically, NDE-CS uses noisy executions of a target circuit together with noisy Clifford circuits to learn how the target circuit can be expressed in terms of Clifford circuits under realistic noise. The same learned relation can then be reused in the noiseless Clifford limit, enabling accurate estimation of ideal expectation values with substantially reduced sampling cost. Numerical simulations on Trotterized Ising circuits demonstrate that NDE-CS achieves orders-of-magnitude reductions in sampling cost compared to the underlying purely classical Monte Carlo approaches from which it is derived, while maintaining the same accuracy. We also compare NDE-CS with Sparse Pauli Dynamics (SPD), a powerful classical framework capable of simulating quantum circuits at previously inaccessible scales, and provide an example where the cost of SPD scales exponentially with system size, while NDE-CS scales much more favorably. These results establish NDE-CS as a scalable hybrid simulation approach for quantum circuits, where noise can be harnessed as a computational asset.
Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.
The classical simulation of quantum dynamics plays an important role in our understanding of quantum complexity and in the development of quantum technologies. Efficient techniques such as those based on the Gottesman-Knill theorem for Clifford circuits, tensor networks for low entanglement-generating circuits, or Wick's theorem for fermionic Gaussian states have become central tools in quantum computing. In this work, we contribute to this body of knowledge by presenting a framework for classical simulations, dubbed “ g -sim ”, which is based on the underlying Lie algebraic structure of the dynamical process. When the dimension of the algebra grows at most polynomially in the system size, there exist observables for which the simulation is efficient. Indeed, we show that g -sim enables new regimes for classical simulations, is able to deal with certain forms of noise in the evolution, as well as can be used to tackle several paradigmatic variational and nonvariational quantum computing tasks. For the former, we perform Lie-algebraic simulations to train and optimize parametrized quantum circuits (thus effectively showing that some variational models can be dequantized), design enhanced parameter initialization strategies, solve tasks of quantum circuit synthesis, and train a quantum-phase classifier. For the latter, we report large-scale noiseless and noisy simulations on benchmark problems. By comparing the limitations of g -sim and certain Wick's theorem-based simulations, we find that the two methods become inefficient for different types of states or observables, hinting at the existence of distinct, nonequivalent resources for classical simulation.