Tag - Quantum computing

Strong interactions between the components of a quantum system are critical for leveraging quantum effects for quantum technologies. In this talk, I present a protocol to enhance such interactions through local controls, thereby speeding up the system’s evolution. I first show that although this is impossible for finite dimensional systems, interactions mediated through infinite dimensional systems, such as quantum harmonic oscillators, can be enhanced through local parametric controls of high frequency, creating squeezing of different quadratures. I discuss phase insensitive amplification and speeding up Rabi oscillations as two potential applications of the protocol and report on an experimental demonstration in an ion trapped system. Finally, I show that the developed protocol can be generalized to non-linear interactions that are critical for universal quantum computing in photonic systems. From this observation I argue that quantum algorithm implementations can be arbitrarily sped up in these latter systems as long as strong and fast squeezing is available.

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letier by Tarnopolsky-Kruchkov-Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators, and analytic hypoellipticity. Recent mathematical progress also includes the proof of existence of infinitely many generalized magic angles, of classically forbidden regions for eigenstates and computer assisted proofs of existence of real ones (Luskin-Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems.

It is well-known that bilinear control systems of the form

Ẋ(t) = (u₁(t)B₁ + u₂(t)B₂)X(t),     X(0) = I_n     (1)

are generically controllable on semi-simple Lie groups like SL_n(ℝ) or SL_n(ℂ). Loosely speaking, this means that for a randomly chosen tuple (B₁,B₂) system (1) will be controllable. This readily implies generic controllability for finite-dimensional closed quantum systems of the form

U̇(t) = −i(H + u₁(t)H₁)X(t),     U(0) = I_n.     (2)

But what about infinite-dimensional quantum systems? Of course, solving this question in full generality is currently out of (our) reach. Therefore, in this talk we will focus on a certain 'natural' and rather large class of systems which often occurs in applications. Within this subclass we prove that the answer to the above raised question is 'yes'.

We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence and the result is proved by a novel pertubation series approach. In addition, we establish, for the first time, the existence of a quantum Zeno limit for the case of unbounded generators. Moreover, using the mentioned perturbation series approach, we prove the first strong damping result for infinite-dimensional open quantum systems with unbounded generators.

We recently proved a KAM-like theorem for finite-dimensional quantum systems, showing that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations. For fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. The long-term stability of the robust symmetries is based on the 'eternal' adiabaticity of perturbed evolutions of finite-dimensional quantum systems. In this talk, I give an overview of this result, to stimulate discussions on possible generalizations to infinite- dimensional systems.

In this talk I will discuss the problem of estimating dynamical parameters of a quantum Markov chain by means of sequential measurements. The key tool will be the use of a coherent quantum absorber which transforms the problem into a simpler one pertaining to a system with a pure stationary state. I will then define certain translationally invariant modes of the output and show that the output state reduces to a coherent state of these modes. This provides a concrete representation of the local asymptotic normality phenomenon for Markov dynamics. I will then show how to optimally estimate the unknown parameter by using the recently developed technique of displaced-null measurements.