Non-Gaussian Tensor Programs

Timo Felser, Simone Montangero

Ryutaro Nagai, Takao Tomono

Tensor networks have been eagerly developed in recent years as a simulation method for quantum computation on classical computers. The efficiency of tensor networks for simulating quantum computation has been shown by solving a task designed for demonstrating quantum computing supremacy. However, such a task is not useful for real application. Therefore, there is the room for research of finding specific applications suitable for exploiting tensor network simulation. As such an application, we propose optimization of circuits to generate non- Gaussian states. Non-Gaussian state is one of the crucial elements for achieving universal continuous variable quantum computation. We consider this problem as a variational optimization of optical circuits to generate desired non-Gaussian states. Tensor networks efficiently simulate the optical circuits, and their compatibility with automatic differentiation provides a heuristic way to optimize the circuits. We propose and report the result of our approach that combines tensor networks and automatic differentiation to perform variational optimization of circuits such that the desired non-Gaussian state is approximately generated. The complexity of contracting tensor networks is scaled by degree of entanglement, so we consider relatively shallow optical circuit. Such a restriction is beneficial for considering devices without error correction method, which are accessible in near-term.

Simone Montangero

Beatriz Dias, Robert König

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity.

Valentin Bonzom, Razvan Gurău, Matteo Smerlak

We introduce a new method, based on the recently developed random tensor theory, to study the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau's theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail how the critical temperature is renormalized by these correlations. We close with a speculation on possible applications of random tensor theory to finite-range spin glass models.

Simone Montangero