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Quantum state purification

Honghao Fu

Published Aug 29, 2016• 1 citation
OpenAlex

Abstract

Quantum state purification is a process in which decoherence is partially reversed by using multiple copies of the input states that have been subject to the same decoherence effect. This thesis focuses on purifying the decoherence caused by the depolarizing channel. In the first half of the thesis, the purification problem is formally introduced and one efficient purification procedure featuring the swap-test is presented and analyzed. The rest of the thesis formulates the optimal purification problem as an optimization problem and applies it to qubit and qutrit purification.

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Quantum state purification is a process in which decoherence is partially reversed by using multiple copies of the input states that have been subject to the same decoherence effect. This thesis focuses on purifying the decoherence caused by the depolarizing channel. In the first half of the thesis, the purification problem is formally introduced and one efficient purification procedure featuring the swap-test is presented and analyzed. The rest of the thesis formulates the optimal purification problem as an optimization problem and applies it to qubit and qutrit purification.
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.
Abstract We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on N noisy copies, requires minimally O(Mlog 2 N) data qubits to process the M-qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be 75% for any register size. For local dephasing, the error threshold is reduced to 50% but may be boosted using twirling.
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.
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.
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.
We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete U(2)\mathrm{U}(2)U(2)-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as O(1/n)O(1/n)O(1/n) with nnn channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.