Quantum adversarial machine learning is an emergent interdisciplinary research frontier that studies the vulnerability of quantum learning systems in adversarial scenarios and the development of potential countermeasures to enhance their robustness against adversarial perturbations. In this talk, I will first make a brief introduction to this field and review some recent progresses. I will show, through concrete examples, that typical quantum classifiers are extremely vulnerable to adversarial perturbations: adding a tiny amount of carefully crafted noises into the original legitimate samples may lead the classifiers to make incorrect predictions at a high confidence level. I will talk about possible defense strategies against adversarial attacks.

I will also talk about a recent experimental demonstration of quantum adversarial learning with programmable superconducting qubits.

Ref:
[1] S.-R. Lu, L. M. Duan, and D.-L. Deng, Phys. Rev. Research 2, 033212 (2020)
[2] W.-Y. Gong and D.-L. Deng, National Science Review 9, nwab130 (2022)
[3] W.-H. Ren et al., Nature Computational Science 2, 711 (2022)
[4] H.-L. Zhang et al., Nature Communications 13, 4993 (2022)

We provide a unified approach with which we may explain several important notions arising from current quantum information theory. They include separability/entanglement and Schmidt numbers of bipartite states as well as various kinds of positive maps like k-superpositive maps, k-positive maps and completely positive maps. Recall that a linear map is 1-superpositive if and only if it is entanglement breaking if and only if its Choi matrices are separable.

Main tools for our approach are bi-linear pairings and Choi matrices. After we explain the above notions with the usual bi-linear pairings and Choi matrices, we find all bilinear pairings between mapping spaces and tensor products of matrices, which retains the duality between k-positivity and Schmidt number k. We also characterize all linear isomorphisms of Choi type from mapping spaces onto tensor products, which retains the correspondences between k-superpositivity and Schmidt number k, as well as Choi’s original correspondence between completely positive maps and positive (semi-definite) matrices.

Finally, we consider infinite dimensional analogues of Choi matrices. For this purpose, we assign two objects, bounded linear operators and trace class operators for a class of linear maps between von Neumann factors. With this approach, we extend the above mentioned notions for the type I factor acting on the infinite dimensional separable Hilbert space.

We propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems. LCHS does not rely on converting the problem into a dilated linear system problem, or on the spectral mapping theorem. The latter is the mathematical foundation of many quantum algorithms for solving a wide variety of tasks involving non-unitary processes, such as the quantum singular value transformation (QSVT). The LCHS method can achieve optimal cost in terms of state preparation. We also demonstrate an application for open quantum dynamics simulation using the complex absorbing potential method with near-optimal dependence on all parameters.

Ref:  [1] Dong An, Jin-Peng Liu, Lin Lin. Physical Review Letters, 131(15):150603, 2023.

Quantum codes achieving approximate quantum error correction (AQEC) are useful, often fundamentally important, from both practical and physical perspectives but lack a systematic understanding. In this work, we establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) properties, covering both all-to-all and geometric scenarios including lattice systems. To this end, we introduce a type of code parameter that we call “subsystem variance”, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) critical threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial “phases” of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. Our theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems.

In addition to showcasing applications to a wide variety of AQEC codes arising from diverse contexts spanning computer science and physics, we also demonstrated how our theory offers new physical insights through the lens of both gapped and gapless systems. Specifically, it enables a long-sought rigorous understanding of the gap between strict definitions of gapped topological order and the widely-used long-range entanglement and topological entanglement entropy (TEE) signatures, as well as a discussion of how AQEC codes exhibiting physically significant power-law-error behavior emerge at low energies of conformal field theory (CFT), potentially advancing the understanding of quantum gravity through holography. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant “scaling threshold” of subsystem variance for features associated with nontrivial quantum order.

Quantum walks are the quantum mechanical counterpart of classical random walks and provide a general and flexible tool to describe the quantum propagation of a particle or an excitation in a network, i.e. a discrete structure modeled as a graph. Besides the fundamental interest, quantum walks represent a valuable tool to describe, design and optimize energy transport in artificial or biological systems. Continuous-time quantum walks (CTQW) on graphs are traditionally defined as the quantum analogue of classical random walks (RW), by promoting the classical transfer matrix, i.e. the RW graph Laplacian, to an Hamiltonian. However, this association does not encompass all the possible quantum evolutions of a walker on a graph, thus strongly limiting the set of exploitable quantum Hamiltonians to achieve quantum enhancement in specific tasks. A question thus arises on whether it is possible to define more general quantum walks on a graph, by considering all Hermitian Hamiltonians compatible with a given graph topology, and how do these generalized QWs compare with their classical analogues.

In this talk, I present a systematic approach to chiral quantum walks, introducing a full characterization of all the possible Hamiltonians describing the time evolution over a given topology. As applications, we address the problem of optimally routing classical and quantum information encoded in the position of a quantum walker on the simplest graphs apt to this task, and prove that chirality may be exploited to achieve the quantum speed limit in search algorithms on graphs.

We give a new complete description theorem of the intermediate operator algebras, which unifies the discrete Galois correspondence results and the crossed product splitting results. As an application, we obtain a Galois’s type result for Bisch—Haagerup type inclusions arising from isometrically shift-absorbing actions of compact-by-discrete groups.

Based on my preprint arxiv:2406.00304