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arXiv:2503.22644

Streaming quantum state purification for general mixed states

Daniel Grier, Debbie Leung, Zhi Li, Hakop Pashayan, Luke Schaeffer

Published Mar 28, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2503.22644

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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.
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.
In quantum purity amplification, one is given nnn copies of a noisy quantum state ρ∈Cd×dρ\in \mathbb{C}^{d \times d}ρ∈Cd×d and asked to prepare kkk copies of its principal eigenstate ∣vd⟩|v_d\rangle∣vd​⟩. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples nnn tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if ρρρ's eigenvalues are sorted p1≤⋯≤pdp_1 \leq \cdots \leq p_dp1​≤⋯≤pd​ and pd−1<pdp_{d-1} < p_dpd−1​<pd​, then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least 1−δ1-δ1−δ with ∣vd⊗k⟩|v_d^{\otimes k}\rangle∣vd⊗k​⟩. Our bound holds for arbitrary spectra, and is independent of the dimension ddd. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.
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 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 ρNρ^NρN. 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)O(M\ell)O(Mℓ) coherent data qubits for an MMM-qubit register and N=2ℓN=2^\ellN=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=3/4p_{\mathrm{th}}=3/4pth​=3/4 for any register size, while local dephasing of ∣+⟩⊗M|+\rangle^{\otimes M}∣+⟩⊗M has a threshold of pth=1/2p_{\mathrm{th}}=1/2pth​=1/2. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.
Purification protocols take several copies of a mixed state and output a smaller number of copies with higher purity. Recently, a protocol based on the Swap test was shown to have optimal resource consumption, with the number of samples scaling proportional to the inverse of the error. In this work, we consider a more realistic scenario in which all the prepared states are noisy, including the auxiliary qubits used by the protocol. Here, we show that this generalization does not compromise convergence, with the protocol still converging for all non-extreme noise values. Moreover, we estimate the number of iterations and resources needed in the generalized scenario. Such an estimation allows us to address the optimality of the protocol with noisy state preparation and show that the protocol is no longer optimal.
We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-rrr qudit state ρ=∑i=1rλi∣φi⟩ ⁣⟨φi∣ρ=\sum_{i=1}^r λ_i |\varphi_i\rangle\!\langle\varphi_i|ρ=∑i=1r​λi​∣φi​⟩⟨φi​∣, prepare a copy of the state proportional to Πantir(∣φ1⟩⊗…⊗∣φr⟩)Π_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)Πantir​(∣φ1​⟩⊗…⊗∣φr​⟩), with ΠantirΠ_{\mathrm{anti}}^rΠantir​ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses O~(r/λmin⁡)\widetilde{O}(r / λ_{\min})O(r/λmin​) copies of ρρρ, where λmin⁡λ_{\min}λmin​ is the minimum non-zero eigenvalue of ρρρ. 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⁡)O(r / λ_{\min})O(r/λmin​) copies of ρρρ, 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 nnn and local dimension ddd, as well as in an algorithm for optimal purity amplification for qudit states.