Antonio Anna Mele

Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster et al.

Variational quantum machine learning is an extensively studied application of near-term quantum computers. The success of variational quantum learning models crucially depends on finding a suitable parametrization of the model that encodes an inductive bias relevant to the learning task. However, precious little is known about guiding principles for the construction of suitable parametrizations. In this work, we holistically explore when and how symmetries of the learning problem can be exploited to construct quantum learning models with outcomes invariant under the symmetry of the learning task. Building on tools from representation theory, we show how a standard gateset can be transformed into an equivariant gateset that respects the symmetries of the problem at hand through a process of gate symmetrization. We benchmark the proposed methods on two toy problems that feature a non-trivial symmetry and observe a substantial increase in generalization performance. As our tools can also be applied in a straightforward way to other variational problems with symmetric structure, we show how equivariant gatesets can be used in variational quantum eigensolvers.

Junyu Liu, Frederik Wilde, Antonio Anna Mele et al.

Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided for example by means of stochastic gradient descent methods. In this work, we provide evidence that the saddle points problem can be naturally avoided in variational quantum algorithms by exploiting the presence of stochasticity. We prove convergence guarantees and present practical examples in numerical simulations and on quantum hardware. We argue that the natural stochasticity of variational algorithms can be beneficial for avoiding strict saddle points, i.e., those saddle points with at least one negative Hessian eigenvalue. This insight that some levels of shot noise could help is expected to add a new perspective to notions of near-term variational quantum algorithms.

Antonio Anna Mele, Lennart Bittel

Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions $d_{\mathrm{in}},d_{\mathrm{out}}$ and Kraus rank at most $k$ can be learned to accuracy $\varepsilon$ using $O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})$ channel uses. Conversely, we prove that $Ω(d_{\mathrm{in}}d_{\mathrm{out}}k)$ uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate $Ω(1/\varepsilon^{2})$ contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.

Marco Fanizza, Vishnu Iyer, Junseo Lee et al.

Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only $2m+2$ queries, where $m$ is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.