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.
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.
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum 'magic' or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
Claims of quantum advantage rest on the classical hardness of simulating quantum circuits. Magic, operator scrambling, anticoncentration, and non-Gaussianity for fermionic circuits are standard diagnostics of complex quantum dynamics. For pure states, some of these have been rigorously connected to classical simulability. Whether these diagnostics reliably track the limits of efficient classical simulation under noise remains unclear. Here, we show that in noisy Clifford+T and nearest-neighbour matchgate+SWAP circuits, dynamical diagnostics and classical simulability can separate in both directions. In particular, the diagnostics can remain nontrivial after classical simulation becomes efficient, or become trivial before known efficient classical algorithms apply. We trace this mismatch to the different statistical properties they probe: magic and scrambling depend on fourth-order statistics of the Pauli spectrum, whereas the simulation algorithms depend primarily on second-order moments, which local noise suppresses at different rates. Thus, dynamical diagnostics measured on a noisy quantum device do not by themselves constitute evidence for classical hardness.
We construct a polynomial-time classical algorithm that samples from the output distribution of noisy geometrically local Clifford circuits with any product-state input and single-qubit measurements in any basis. Our results apply to circuits with nearest-neighbor gates on an O(1)-D architecture with depolarizing noise after each gate. Importantly, we assume that the circuit does not contain qubit resets or mid-circuit measurements. This class of circuits includes Clifford-magic circuits and Conjugated-Clifford circuits, which are important candidates for demonstrating quantum advantage using non-universal gates. Additionally, our results can be extended to the case of IQP circuits augmented with CNOT gates, which is another class of non-universal circuits that are relevant to current experiments. Importantly, these results do not require randomness assumptions over the circuit families considered (such as anticoncentration properties) and instead hold for every circuit in each class as long as the depth is above a constant threshold. This allows us to rule out the possibility of fault-tolerance in these circuit models. As a key technical step, we prove that interspersed noise causes a decay of long-range entanglement at depths beyond a critical threshold. To prove our results, we merge techniques from percolation theory and Pauli path analysis.
Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
Determining whether a quantum evolution can be efficiently simulated classically is central to understanding the boundary between classical and quantum computation. However, polynomial-time simulability is an asymptotic statement, and does not by itself determine whether the (quantum-inspired) classical simulation is actually practical. Indeed, different polynomial scalings can lead to vastly different computational costs, particularly when expensive preprocessing or quantum data acquisition is required. In this work, we ask whether classically simulable quantum dynamics are in practice more resource-efficient to simulate classically than to execute directly on quantum hardware. We analyze this question using three resource metrics, quantum sample, quantum time, and classical time complexity, for several widely studied classically simulable circuit families. Using representative hardware-level estimates, we identify regimes in which quantum simulation can be faster despite the existence of a polynomial-time classical algorithm, as well as regimes in which classical simulation remains more efficient. At the same time, the large quantum sampling cost needed to characterize unknown input states can make this polynomial-time classical simulation prohibitively expensive with current cloud-based hardware access prices. Ultimately, our work indicates that guarantees of classical simulability with polynomial resources alone are insufficient to determine the preferred implementation.