Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. This basic task has applications to quantum communication over noisy channels and quantum computation with imperfect devices, but has only been studied previously for the case of qubits. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. Treating the initial error parameter and the dimension as constants, we show that our procedure has sample complexity asymptotically optimal in the final error parameter. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.
Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. This basic task has applications to quantum communication over noisy channels and quantum computation with imperfect devices, but has only been studied previously for the case of qubits. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. Treating the initial error parameter and the dimension as constants, we show that our procedure has sample complexity asymptotically optimal in the final error parameter. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.
Given multiple copies of a mixed quantum state with an unknown, nondegenerate principal eigenspace, quantum state purification is the task of recovering a quantum state that is closer to the principal eigenstate. A streaming protocol relying on recursive swap tests has been proposed and analysed for noisy depolarized states with arbitrary dimension and noise level. Here, we show that the same algorithm applies much more broadly, enabling the purification of arbitrary mixed states with a nondegenerate principal eigenvalue. We demonstrate this through two approaches. In the first approach, we show that, given the largest two eigenvalues, the depolarized noise is the most difficult noise to purify for the recursive swap tests, thus the desirable bounds on performance and cost follow from prior work. In the second approach, we provide a new and direct analysis for the performance of purification using recursive swap tests for the more general noise. We also derive simple lower bounds on the sample complexity, showing that the recursive swap test algorithm attains optimal sample complexity (up to a constant factor) in the low-noise regime.
Abstract Quantum state purification is crucial in quantum communication and computation, aiming to recover a purified state from multiple copies of an unknown noisy state. This work introduces a general state purification framework designed to achieve the highest fidelity with a specified probability and characterize the associated trade-offs. For i.i.d. quantum states under depolarizing noise, our framework can replicate the purification protocol proposed by Barenco et al (1997 SIAM J. Comput. 26 1541–57) and further provide exact formulas for the purification fidelity and probability with explicit trade-offs. We prove the protocols’ optimality for two copies of noisy states with any dimension and confirm its optimality for higher numbers of copies and dimensions through numerical analysis. Our methodological approach paves the way for proving the protocol’s optimality in more general scenarios and leads to optimal protocols for other noise models. Furthermore, we present a systematic implementation method via block encoding and parameterized quantum circuits, providing explicit circuits for purifying three-copy and four-copy states under depolarizing noise. Finally, we estimate the sample complexity and generalize the protocol to a recursive form, demonstrating its practicality for quantum computers with limited memory.
Quantum state purification protocols, which mitigate noise by converting multiple copies of noisy quantum states into fewer copies with a lower noise level, have applications in quantum communication and computation with imperfect devices. Here, we systematically study the task of state purification in distributed quantum systems, demanding that purification be achieved by local operations and classical communication (LOCC). We prove that, in the presence of depolarizing noise, no LOCC purification protocol starting from two copies can work blindly for all the states in three important sets: the set of all pure two-qubit states, the set of all two-qubit maximally entangled states, and the Bell basis. In stark contrast, we show that a targeted, single-state purification is always achievable in the presence of depolarizing noise, and we provide an explicit analytical LOCC protocol for every given two-qubit state. For arbitrary finite sets of pure states and arbitrary noise profiles, we develop an optimization-based algorithm that systematically designs LOCC purification protocols, and we demonstrate it through concrete examples. Overall, our results identify both fundamental limitations and practical noise-reduction strategies for distributed quantum information processing.
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.
Quantum state purification is the functionality that, given multiple copies of an unknown state, outputs a state with increased purity. This will be an essential building block for near- and middle-term quantum ecosystems before the availability of full fault tolerance, where one may want to suppress errors not only in expectation values but also in quantum states. We propose an effective state purification gadget with a moderate quantum overhead by projecting M noisy quantum inputs to their symmetric subspace defined by a set of projectors forming a symmetric subgroup with order M. Our method, applied in every short evolution over M redundant copies of noisy states, can suppress both coherent and stochastic errors by a factor of 1/M, respectively. This reduces the circuit implementation cost M times smaller than the state projection to the full symmetric subspace proposed by Barenco et al. more than two decades ago. We also show that our gadget purifies the depolarised inputs with probability p to asymptotically O(p2) with an optimal choice of M when p is small. The sampling cost scales O(p−1) for small p, which is also shown to be asymptotically optimal. Our method provides flexible choices of state purification depending on the hardware restrictions before fully fault-tolerant computation is available.
We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-r qudit state ρ=∑i=1rλi∣φi⟩⟨φi∣, prepare a copy of the state proportional to Πantir(∣φ1⟩⊗…⊗∣φr⟩), with Πantir the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses O(r/λmin) copies of ρ, where λmin is the minimum non-zero eigenvalue of ρ. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses O(r/λmin) copies of ρ, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies n and local dimension d, as well as in an algorithm for optimal purity amplification for qudit states.