State preparation that initializes quantum systems in a fiducial state and measurements to read outcomes after the evolution of quantum states, both essential elements in quantum information processing in general, may contain noise from which errors, in particular, referred to as static errors, may appear even with noise-free evolution. In this work, we consider noisy resources such as faulty manufacturing and improper maintenance of systems by which static errors in state preparation and measurement are inevitable. We show how to suppress static errors and purify noiseless SPAM by repeatedly applying noisy resources. We present the purification protocol for noisy initialization and noisy measurements and verify that a few qubits are immediately cost-effective to suppress error rates up to $10^{-3}$. We also demonstrate the purification protocol in a realistic scenario. The results are readily feasible with current quantum technologies.
State preparation that initializes quantum systems in a fiducial state and measurements to read outcomes after the evolution of quantum states, both essential elements in quantum information processing in general, may contain noise from which errors, in particular, referred to as static errors, may appear even with noise-free evolution. In this work, we consider noisy resources such as faulty manufacturing and improper maintenance of systems by which static errors in state preparation and measurement are inevitable. We show how to suppress static errors and purify noiseless SPAM by repeatedly applying noisy resources. We present the purification protocol for noisy initialization and noisy measurements and verify that a few qubits are immediately cost-effective to suppress error rates up to 10−3. We also demonstrate the purification protocol in a realistic scenario. The results are readily feasible with current quantum technologies.
To achieve the practical applications of near-term noisy quantum devices, low-cost ways to mitigate the noise damages in the devices are essential. In many applications, the noiseless state we want to prepare is often a pure state, which has recently inspired a range of purification-based quantum error mitigation proposals. The existing proposals either are limited to the suppressions of only the leading-order state preparation errors, or require a large number of long-range gates that might be challenging to implement depending on the qubit architecture. This article will provide an overview of the different purification-based quantum error mitigation schemes and propose a resource-efficient scheme that can correct state preparation errors up to any order while requiring only half of the qubits and less than half of the long-range gates compared to before.
The manipulation of quantum "resources" such as entanglement, coherence, and magic states lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to "purify" the quantum resources since they are inevitably contaminated by noise and thus often lose their power or become unreliable for direct usage. Here we prove fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind of quantum resource. More explicitly, we derive nontrivial lower bounds on the error of converting any full-rank noisy state to any target pure resource state by any free protocol (including probabilistic ones)-it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely used type of resource purification routine that underpins many key applications of quantum information science. In particular, this general result induces the first explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation. Implications for the standard error-correction-based methods are specifically discussed.
Quantum error correction physically removes errors from a quantum state, while quantum error mitigation improves observable estimates by processing noisy measurement data. We introduce \emph{purification quantum error suppression} (PQES), a hybrid approach that uses multiple noisy copies of an unknown state to combine these two ideas. The protocol uses SWAP tests to physically reduce errors by purification, while the full outcome record is used to combine all branches without postselection. In this way, PQES avoids the fixed-success-outcome requirement of standard SWAP-test purification while still accessing the power-purified state ρN. The SWAP identities allow purification steps to be interleaved with unitary circuit blocks, so errors can be suppressed during a computation rather than only at the final measurement. We provide both a parallel binary-tree implementation and a more compact register-recycled implementation using O(Mℓ) coherent data qubits for an M-qubit register and N=2ℓ input copies. We analyze the resulting error thresholds under representative noise models. For local depolarizing noise on the product-state family studied here, the threshold is pth=3/4 for any register size, while local dephasing of ∣+⟩⊗M has a threshold of pth=1/2. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.
Current noisy quantum computers have multiple types of errors, which can occur in the state preparation, measurement/readout, and gate operation, as well as intrinsic decoherence and relaxation. Partly motivated by the booming of intermediate-scale quantum processors, measurement and gate errors have been recently extensively studied, and several methods of mitigating them have been proposed and formulated in software packages (e.g., in IBM Qiskit). Despite this, the state preparation error and the procedure to quantify it have not yet been standardized, as state preparation and measurement errors are usually considered not directly separable. Inspired by a recent work of Laflamme, Lin, and Mor \cite{laflamme2022algorithmic}, we propose a simple and resource-efficient approach to quantify separately the state preparation and readout error rates. With these two errors separately quantified, we also propose methods to mitigate them separately, especially mitigating state preparation errors with linear (with the number of qubits) complexity. As a result of the separate mitigation, we show that the fidelity of the outcome can be improved by an order of magnitude compared to the standard measurement error mitigation scheme. We also show that the quantification and mitigation scheme is resilient against gate noise and can be immediately applied to current noisy quantum computers. To demonstrate this, we present results from cloud experiments on IBM's superconducting quantum computers. The results indicate that the state preparation error rate is also an important metric for qubit metrology that can be efficiently obtained.
Quantum error mitigation is essential for many computing tasks on the noisy quantum computer with a limited number of qubits. In this paper, we propose a practical protocol of error mitigation by virtually purifying the quantum state without qubit overhead or requiring only one ancilla qubit. In dual-state purification, we effectively generate a purified state with increased fidelity using the erroneous state and its dual state, respectively, prepared with the noisy quantum circuit and the dual map of its inverse circuit. Combined with tomography purification, we can make sure that the final estimate of an observable is obtained from a pure state. While the absolute error increases, the numerical result, based on a depolarizing model, suggests that our protocol reduces the error compared to the case without the protocol by a rescaling factor decreasing with the qubit number and circuit depth; i.e., the relative performance of purification is better for larger circuits. On a cloud quantum computer, we successfully demonstrate the reduced error with a quantum variational eigensolver circuit.
Noise remains the primary obstacle to realizing quantum advantage, continuously degrading the resources that enable quantum technologies. Purification aims to reverse this degradation by extracting high-fidelity resources from noisy ensembles, yet its conventional formulation is intrinsically static, acting only after noise has taken effect. Here we instead recast purification as a dynamical task, introducing a spatiotemporal framework that distributes interventions across the noise process. This formulation reveals operational capabilities inaccessible to existing approaches and gives rise to forward-assisted purifications that extend achievable performance. In certain regimes, a single-copy protocol already exceeds what can be achieved with up to 50 copies under conventional purification, demonstrating a significant overhead in required resources. Beyond these gains, our framework circumvents no-purification theorems within conventional protocols, including for Bell-state ensembles, thereby enabling purification previously considered impossible and pointing toward an efficient route to mitigating noise in quantum systems.