In contrast to monolithic devices, modular, networked quantum architectures are based on interconnecting smaller quantum hardware nodes using quantum communication links and offer a promising approach to scalability. Virtual distillation (VD) is a technique that can, under ideal conditions, suppress errors exponentially as the number of quantum state copies increases. However, additional gate operations required for VD introduce further errors, which may limit its practical effectiveness. In this work, we analyze three practical implementations of VD that correspond to edge cases that maximize space-time trade-offs. Specifically, we consider an implementation that minimizes the number of qubits but introduces significantly deeper quantum circuits and contrast it with implementations that parallelize the preparation of copies using additional qubits, including a constant-depth implementation. We rigorously characterize their circuit depth and gate count requirements and develop explicit architectures for implementing them in networked quantum systems—while also detailing implementations in early fault-tolerant quantum architectures. We numerically compare the performance of the three implementations under realistic noise characteristics of networked ion trap systems and conclude the following. First, VD effectively suppresses errors even for very noisy states. Second, the constant-depth implementation consistently outperforms the implementation that minimizes the number of qubits. Finally, the approach is highly robust to errors in remote entangling operations, with noise in local gates being the main limiting factor to its performance.
In contrast to monolithic devices, modular, networked quantum architectures are based on interconnecting smaller quantum hardware nodes using quantum communication links and offer a promising approach to scalability. Virtual distillation (VD) is a technique that can, under ideal conditions, suppress errors exponentially as the number of quantum state copies increases. However, additional gate operations required for VD introduce further errors, which may limit its practical effectiveness. In this work, we analyze three practical implementations of VD that correspond to edge cases that maximize space-time trade-offs. Specifically, we consider an implementation that minimizes the number of qubits but introduces significantly deeper quantum circuits and contrast it with implementations that parallelize the preparation of copies using additional qubits, including a constant-depth implementation. We rigorously characterize their circuit depth and gate count requirements and develop explicit architectures for implementing them in networked quantum systems—while also detailing implementations in early fault-tolerant quantum architectures. We numerically compare the performance of the three implementations under realistic noise characteristics of networked ion trap systems and conclude the following. First, VD effectively suppresses errors even for very noisy states. Second, the constant-depth implementation consistently outperforms the implementation that minimizes the number of qubits. Finally, the approach is highly robust to errors in remote entangling operations, with noise in local gates being the main limiting factor to its performance.
Near-term and early fault-tolerant quantum computing architectures are expected to exhibit highly non-uniform error rates. In particular, local operations within a chip can be substantially more reliable than operations connecting different chips or dilution refrigerators. Such inter-module operations can therefore become a dominant bottleneck, even when quantum error correction is applied. Entanglement distillation provides a natural way to trade additional operations and qubits for higher-fidelity entanglement. Standard distillation protocols, however, are usually formulated in an LOCC resource model, in which several noisy Bell pairs are generated initially and all subsequent processing consists only of local operations and classical communication. Here, we consider a generalized model tailored to modular quantum computing hardware, in which the two modules have access to high-fidelity local operations and to repeated uses of the same noisy inter-module entangling operation during the protocol. We develop practical small-scale entanglement distillation protocols designed to minimize both space and time overhead. Remarkably, our main protocol requires only two qubits per module, yet achieves quadratic error suppression of inter-module errors, assuming local operations are much cleaner. Compared with existing small-scale protocols, our space-optimal protocol provides more space- and time-efficient quadratic error suppression and achieves the best performance in our simulations and experiments on noisy links of current superconducting quantum processors. These results suggest that inter-module-gate-assisted entanglement distillation can be a practical primitive for overcoming noisy links in modular quantum computing architectures.
Abstract Quantum error mitigation (QEM) is crucial for near-term quantum devices, as noise inherently exists in physical quantum systems and undermines the accuracy of quantum algorithms. A typical QEM method, called Virtual Distillation (VD), aims to mitigate errors for expectation values of observables and achieve effective exponential suppression using multiple copies of the noisy state. However, imperfect VD circuit implementation may yield negative mitigation outcomes, potentially more severe than those achieved without QEM. To address this, we introduce Circuit-Noise-Resilient Virtual Distillation (CNR-VD). This method, featuring a calibration procedure that utilizes easily-prepared input states, refines the outcomes of VD when its circuit is contaminated by noise, seeking to recover the results of an ideally conducted VD circuit. Simulation results demonstrate that the CNR-VD estimator effectively reduces deviations induced by noise in the VD circuit, showcasing improvements in accuracy by an order of magnitude at most compared to the original VD. Meanwhile, CNR-VD enables positive mitigation effects, even in the presence of higher noise levels where the original VD performs worse than the unmitigated estimation. The proposed CNR-VD significantly enhances the noise-resilience of VD, and thus is anticipated to elevate the performance of quantum algorithm implementations on near-term quantum devices.
Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits.Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then, it will be necessary to use alternative strategies to mitigate the impact of errors.We propose a near-term friendly strategy to mitigate errors by entangling and measuring M copies of a noisy state ρ.This enables us to estimate expectation values with respect to a state with dramatically reduced error ρ M =Trðρ M Þ without explicitly preparing it, hence the name "virtual distillation."As M increases, this state approaches the closest pure state to ρ exponentially quickly.We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of ρ).We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows.Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.
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.
Virtual distillation has been proposed as an error mitigation protocol for estimating the expectation values of observables in quantum algorithms. It proceeds by creating a cyclic permutation of M noisy copies of a quantum state using a sequence of controlled-swap gates. If the noise does not shift the dominant eigenvector of the density operator away from the ideal state, then the error in expectation-value estimation can be exponentially reduced with M . In practice, subsequent error mitigation techniques are required to suppress the effect of noise in the cyclic permutation circuit itself, leading to increased experimental complexity. Here, we perform a careful analysis of the effect of uncorrelated, identical noise in the cyclic permutation circuit and find that the estimation of expectation value of observables are robust against dephasing noise. We support the analytical result with numerical simulations and find that 67% of errors are reduced for M=2 , with physical dephasing error probabilities as high as 10% . Our results imply that a broad class of quantum algorithms can be implemented with higher accuracy in the near-term with qubit platforms where non-dephasing errors are suppressed, such as superconducting bosonic qubits and Rydberg atoms.
Distributed architectures have been proposed as a pathway to large-scale quantum computers. Combined with the need for fault-tolerance, such architectures require distributed quantum error correction (DQEC) and distributed logical gates. An important challenge is how to realize DQEC primitives in the setting where interaction between modules is restricted to shared Bell pairs that are significantly noisier than on-chip operations. We extend the framework known as fault tolerance by construction to this setting, deriving different strategies for handling the additional noise. Through fault-improvement we recover conventional entanglement distillation, and also find more dynamical protocols that enable space-time trade-offs. We show that integrated decoding halves the required distillation code distance compared to entanglement distillation implemented using separate decoding, thus requiring significantly fewer Bell pairs. As a main focus of the work, we synthesize efficient circuits for an important primitive in distributed fault tolerance: distributed stabilizer measurements. These circuits can be adapted to resource constraints, e.g. on the Bell pair generation rate or the space available for on-chip auxiliary qubits. Noting that full local fault-tolerance is not always needed to preserve the correct scaling of logical error rates, we further optimize the circuits depending on the surrounding context. We consider in particular the surface code and the color code, both as distributed memories and in the case of lattice surgery across separate modules. Here, robustness to certain hook and readout errors reduces the number of required Bell pairs even further, compared to the context-free setting. We numerically benchmark the resulting implementations under circuit level noise with additional interconnect noise.