Zhenhuan Liu

Estimating nonlinear properties of quantum states, such as $\mathrm{tr}(\rho^{k} O)$, is fundamental in quantum information but challenging due to the intrinsic linearity of quantum mechanics. Typical approaches such as generalized swap test rely on joint access to $k$ replicas to convert nonlinear functions in $\rho$ into linear functions in $\rho^{\otimes k}$. In this work, we prove that this conversion is not only sufficient but also necessary: any protocol that can only perform $(k-1)$-replica joint measurements require exponentially many samples to estimate $\mathrm{tr}(\rho^{k}O)$ with nonzero $\tr(O)$, while $k$-replica joint measurements allow efficient estimation. This establishes, for the first time, an exponential separation between $(k-1)$- and $k$-replica protocols for general $k$, thereby defining a fine-grained hierarchy for replica quantum advantage and solving an open question in the literature. The workhorse of our proofs is a general indistinguishability principle showing that any ensemble assembled from Haar-random states is hard to distinguish from its average. We also leverage this principle in spectrum testing, proving that $k$-replica joint measurements are also necessary to efficiently distinguish two spectra that match on all moments up to degree $k-1$. Our work draws sharp boundaries on the power of joint measurements, shedding light on resource-complexity tradeoffs in quantum learning theory.

Quantum networks, which integrate multiple quantum computers and the channels connecting them, are crucial for distributed quantum information processing but remain inherently susceptible to channel noise. The channel purification protocol emerges as a promising technique for directly suppressing noise in quantum channels without complex encoding and decoding operations, making it particularly suitable for remote quantum information transmission in optical systems. In this work, leveraging the spatial and polarization degrees of freedom of photons, we propose a novel experimental configuration that efficiently implements the channel purification protocol, utilizing two Fredkin gates to coherently interfere independent noise channels. Based on this configuration, we experimentally demonstrate that the protocol can suppress the noise with various noise levels and forms. Furthermore, we apply our protocol in a practical application of entanglement distribution, showing that the channel purification can effectively protect the distributed entanglement from channel noise.

Quantum error mitigation is a key approach for extracting target state properties on state-of-the-art noisy machines and early fault-tolerant devices. Using the ideas from flag fault tolerance and virtual state purification, we develop the virtual channel purification (VCP) protocol, which consumes similar qubit and gate resources as virtual state purification but offers up to exponentially stronger error suppression with increased system size and more noisy operation copies. Furthermore, VCP removes most of the assumptions required in virtual state purification. Essentially, VCP is the first quantum error mitigation protocol that does not require specific knowledge about the noise models, the target quantum state, and the target problem while still offering rigorous performance guarantees for practical noise regimes. Further connections are made between VCP and quantum error correction to produce one of the first protocols that combine quantum error correction and quantum error mitigation beyond concatenation. We can remove all noise in the channel while paying only the same sampling cost as low-order purification, reaching beyond the standard bias-variance trade-off in quantum error mitigation. Our protocol can also be adapted to key tasks in quantum networks like channel capacity activation and entanglement distribution.

We prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental limitation of quantum information processing. In particular, no quantum operation can transform a finite number of copies of an unknown quantum state or channel into a pure state or channel that depends on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Relaxing the requirement of exact pure output, we further extend our result to establish quantitative sample complexity bounds for approximate purification, independent of any task details or operational constraints. To illustrate the practical consequences of this principle, we examine the task of approximately preparing pure dilation and, for the first time, prove an exponential lower bound on the required sample complexity.

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products, in sharp contrast to the PPT criterion. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

Quantum learning encounters fundamental challenges when estimating non-linear properties, owing to the inherent linearity of quantum mechanics. Although recent advances in single-copy randomized measurement protocols have achieved optimal sample complexity for specific tasks like state purity estimation, generalizing these protocols to estimate broader classes of non-linear properties without sacrificing optimality remains an open problem. In this work, we introduce the observable-driven randomized measurement (ORM) protocol enabling the estimation of Tr(Oρ^2) for an arbitrary observable O---an essential quantity in quantum computing and many-body physics. We establish an upper bound for ORM's sample complexity and show its optimality for observables with a large trace-norm, including Pauli and local observables, closing a gap in the literature. For these observables, ORM admits an efficient implementation with Clifford circuits. Numerical experiments validate that ORM requires substantially fewer state samples to achieve the same precision compared to classical shadows. Additionally, we introduce a braiding randomized measurement protocol for multiple low-rank non-linear observables, reducing circuit complexities in practical applications.