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.