Importance sampling based on quasi-probability decomposition is the backbone of many widely used techniques, such as error mitigation, circuit knitting, and, more generally, virtual quantum resource distillation, as it allows one to simulate operations that are not accessible in a given setting. However, this class of protocols faces a fundamental problem -- it only allows to estimate expectation values. Here, we provide a general framework that lifts any quasi-probability-based protocol from expectation value estimation to a weak simulator, realizing sampling from the desired distribution only using a restricted class of quantum resources. Our method runs with the sampling cost proportional to the negativity of the quasi-probability, in stark contrast to the naive estimation-based approach that requires a large number of samples even in the case of small negativity. We show that our method requires significantly fewer samples in a number of relevant scenarios, such as error mitigation, entanglement distillation and magic state distillation. Our framework realizes the weak simulation of quantum resources without actually distilling the state, introducing a new notion of quantum resource distillation.
Importance sampling based on quasi-probability decomposition is the backbone of many widely used techniques, such as error mitigation, circuit knitting, and, more generally, virtual quantum resource distillation, as it allows one to simulate operations that are not accessible in a given setting. However, this class of protocols faces a fundamental problem -- it only allows to estimate expectation values. Here, we provide a general framework that lifts any quasi-probability-based protocol from expectation value estimation to a weak simulator, realizing sampling from the desired distribution only using a restricted class of quantum resources. Our method runs with the sampling cost proportional to the negativity of the quasi-probability, in stark contrast to the naive estimation-based approach that requires a large number of samples even in the case of small negativity. We show that our method requires significantly fewer samples in a number of relevant scenarios, such as error mitigation, entanglement distillation and magic state distillation. Our framework realizes the weak simulation of quantum resources without actually distilling the state, introducing a new notion of quantum resource distillation.
Randomized protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted "free" operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call "virtual resource distillation," provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.
Virtual distillation is a technique that aims to mitigate errors in noisy quantum computers. It works by preparing multiple copies of a noisy quantum state, bridging them through a circuit, and conducting measurements. As the number of copies increases, this process allows for the estimation of the expectation value with respect to a state that approaches the ideal pure state rapidly. However, virtual distillation faces a challenge in realistic scenarios: preparing multiple copies of a quantum state and bridging them through a circuit in a noisy quantum computer will significantly increase the circuit size and introduce excessive noise, which will degrade the performance of virtual distillation. To overcome this challenge, we propose an error mitigation strategy that uses circuit-cutting technology to cut the entire circuit into fragments. With this approach, the fragments responsible for generating the noisy quantum state can be executed on a noisy quantum device, while the remaining fragments are efficiently simulated on a noiseless classical simulator. By running each fragment circuit separately on quantum and classical devices and recombining their results, we can reduce the noise accumulation and enhance the effectiveness of the virtual distillation technique. Our strategy has good scalability in terms of both runtime and computational resources. We demonstrate our strategy’s effectiveness through noisy simulation and experiments on a real quantum device.
A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation.We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analogue of the Wigner function for quantum systems of odd dimension.This connection allows us to resolve an open question on the existence of bound states for magic state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations.We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.
Quantum resource distillation converts imperfect resource states into highly resourceful targets and is a central task in quantum information processing. When distillation is probabilistic, unfavorable outcomes may be discarded and the quality of the output is evaluated conditioned on success, leading to a notion of reliability that differs from the deterministic setting. Here we study the conditional error exponents of probabilistic quantum resource distillation under approximately resource-nongenerating instruments and derive its general connection to the postselected composite quantum hypothesis testing against the corresponding set of free states. This reduction leads naturally to scale-invariant projective distinguishability measures and provides finite-blocklength bounds for a broad class of resource theories. We apply the framework to entanglement, coherence, and magic. For entanglement and magic, we obtain exact zero-rate asymptotic characterizations of the conditional error exponent in terms of regularized projective distinguishability, while for coherence we derive general finite-blocklength and asymptotic bounds. For representative entangled and magic states, the resulting quantities can be evaluated analytically and reveal regimes in which postselection yields a strictly larger conditional error exponent than deterministic distillation. Furthermore, we also show that the conditional error exponents for probabilistic entanglement distillation under a strictly positive success probability constraint are smaller than those with no constraints on the success probability.
Virtual distillation is a promising error-mitigation technique that exploits multiple copies of a noisy quantum state to estimate observables as if measured on a purified state. Although originally introduced in the context of bosonic many-body systems under the name of virtual cooling, its development and applications have largely focused on qubit-based quantum computation. Here, we establish a framework for virtual distillation in bosonic quantum information processing and continuous-variable quantum computing. Building on a diagonalization of cyclic shift operators implemented with passive linear-optical interferometers, we derive experimentally accessible protocols for estimating virtually distilled expectation values of observables relevant to bosonic architectures. In particular, we show how to recover noise-mitigated expectation values of number operators, phase-shift operators, and arbitrary quadratures from multi-copy measurements. For number operators, we further demonstrate the estimation of virtually distilled correlators of arbitrary order through the characteristic function of the photon-number distribution. We apply the framework to states affected by photon loss and dephasing, two of the dominant noise mechanisms in bosonic quantum computation, and quantify the resulting suppression of noise contributions. Our results extend virtual distillation beyond its original setting and provide a practical route toward error-mitigated measurements in bosonic quantum processors using experimentally available linear-optical resources.