Abstract

Quantum simulators aim at gaining a deeper understanding of complex physical quantum systems. Within this perspective discrete time quantum walks offer a possible algorithmic description for relativistic quantum fields. The Dirac equation, ruling the kinematics of free massive Fermions, is settled as the small wave-vector approximation of a quantum walk evolution. We present an interacting Fermionic quantum walk that provides the quantum circuit counterpart of the Fermionic Hubbard model. The Hubbard Fermionic quantum, which consists of two Dirac quantum walks with the addition of an on-site interaction, is solved analytically via the Bethe-ansatz. The class of solutions contains both scattering states and bound states, the last ones corresponding to the two particles joined together to form a localized molecular state.

Discrete-time quantum walks (DTQWs) model chiral transport on a lattice. They are universal building blocks of quantum algorithms, and can mimic condensed-matter phenomenology. They have been implemented with several objects and setups, such as photons in optical networks or fibers, or cold atoms in optical lattices. When DTQWs decohere, they tend to classical random walks. In the continuous limit, i.e. in the limit of weak spin-components coupling, for modes having large wavelengths and time periods with respect to the space and time steps, respectively, DTQWs reproduce a substantial branch of classical field theory (first quantization), including couplings with (synthetic) Abelian [1, 2] and non-Abelian [3] Yang-Mills gauge fields, and/or relativistic gravitational fields [4]. Several gauge invariances on the lattice have been exhibited, showing that the connexions between DTQWs and gauge theories are not a mere emergent property of the continuous limit, but exist at the discrete level. This 15-min talk will deal with the quantum simulation, with DTQWs, of the propagation of Dirac fermions in a (1+2)-dimensional curved spacetime [5]. The construction of such DTQWs will be discussed, and an application to gravitational waves will then be studied, beyond the continuum limit. Eventually, the aforementioned ongoing DTQW-based construction of lattice gauge theories will be quickly reviewed.

Quantum computers have been shown to be useful for many applications, among them the ecient simulation of quantum systems. It may, however, be some time before a large scale universal quantum computer is built. In the meantime, several intermediate, non-universal models of quantum computation, have been developed and may prove easier to implement. The Instantaneous Quantum Poly-time (IQP) machine is one such non- universal model with signicant practical advantages. In spite of the fact that IQP uses only commuting gates, it is believed to remain hard to classically simulate. Hence, providing evidence that a machine can perform hard IQP computations would be a proof of its quantum supremacy. A hypothesis test that can be passed only by devices capable of eciently simulating IQP machines provides the aforementioned evidence. While previous work builds such a hypothesis test assuming some conjectures on computational complexity, in the present work we are able to use tools from blind quantum computing to develop an information-theoretically secure hypothesis test. To do so, we develop an implementation of an IQP computation in Measure- ment Based Quantum Computing (MBQC). This allows us to derive a blind delegated protocol for IQP computations that keeps the details of the computa- tion hidden from the device performing it. We can prove information-theoretic security of this protocol in a composable framework. This requires that we empower the client with minimal quantum capabilities such as those required in standard Quantum Key Distribution schemes. Finally, we develop our own hypothesis test for quantum supremacy, which a limited quantum client can run on an untrusted Server.

The Ising model has proven ubiquitous in combinatorial/multiparticle problems. Its study connects to issues as diverse as (anti-) ferromagnetism, satisfiability problems, or the construction of non-trivial link invariants. In this talk, we will introduce a scheme for the detection of partition functions of the Ising model at complex temperature. In the context of ordered qubit registers, this scheme finds a natural translation in terms of global operations and single measurements on the edge of the arrays. Interestingly, the kind of state preparations and measurements involved can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Through appropriate Wick rotations, estimates for real temperature partition functions can be deduced. Bounds on the estimation error, valid with high confidence are provided through a central-limit theorem. We will also show how the scheme allows to evaluate some link invariants.

Analog quantum simulation schemes allow to implement spin-boson models where two-photon couplings are the dominating terms of light-matter interaction. In this case, when the coupling strength becomes comparable with the characteristic frequencies, a spectral collapse can take place, i.e. the discrete system spectrum can collapse into a continuous band. We analyzed the many-body limit of the two-photon Dicke model, which describes the interaction of a chain of qubits with a single bosonic mode. We find that there exists a parameter regime where two-photon interactions induce a superradiant phase transition, before the spectral collapse occurs. We characterize the transition by analyzing the low-energy spectrum of the system, and we compare the critical behavior with the one-photon case.

We analyze the properties of a two and three dimensional quantum walk that are inspired by the idea of a brane-world model put forward by Rubakov and Shaposhnikov. In that model, particles are dynamically confined on the brane due to the interaction with a scalar field. We translated this model into an alternate quantum walk with a coin that depends on the external field, with a dependence which mimics a domain wall solution. As in the original model, fermions (in our case, the walker), become localized in one of the dimensions, not from the action of a random noise on the lattice (as in the case of Anderson localization), but from a regular dependence in space. On the other hand, the resulting quantum walk can move freely along the “ordinary” dimensions.

The theoretical and experimental progress in quantum simulation with ultracold atoms of the last decay have pushed the realm of simulable models well deep into condensed matter and has started touching high-energy and gravitational physics. In this talk, I will focus on the latter. In particular, I will present the simulation of (special) background spacetimes with ultracold atoms in optical lattices and I will discuss possible physical applications like the simulation of the Unruh effect. By drawing a parallelism with the most advanced program of quantum simulating gauge theories in optical lattices, I will conclude commenting on my general view and on the prospects of the quantum simulation of gravitational physics.

Quantization: Classical random walks (CRW) are considered form the quantization point of view. Modes of quantization of CRW are formulated and criteria for QW manifesting quantum features are provided. Simulations : 1o) The role of classical noise in quantum walks (QW) is investigated in the form of discrete telegraphic noise affecting the reshuffling matrices of QW on integers. A classical noise induced transition from the classical regime with diffusion rate O(sqrt(t)), to the quantum regime with ballistic diffusion rate O(t), is proven. 2o) The maximum likelihood parameter estimation problem (MLE) is addressed in the context of quantum walk theory. The estimation of an angular parameter theta is formulated as the equivalent problem of estimating the rotation angle determining the reshuffling matrix operating in 2D coin space of a general QW. A quantum channel formulation of the original QW provides the ground for an operational approach to the MLE problem. The operational approach is applied to show cases such as QW on integers, period lattices. 3o) By applying space conditional quantization rules for CRW on the set Zn, resulting in marking areas of fast and slow diffusion, it is shown that the resulting QW dynamics is able to read topology (revealing connectivity of hidden subsets on ring lattices). Ramifications: Extending the quantum framework, algebraic random walks (ARW) are introduced via Hopf algebras and interrelations between CRW, QW and ARW are investigated. By means of (positive) operator valued probability measures (P)OVM defined on several measurable sets (plane, line, cycle), it is shown how QWs lead in their asymptotic limit to master equations describing open quantum systems.

We investigate a discrete quantum walk in a square lattice having an interface separating two regions with distinct topology. The presence of spatial disorder breaks the particle-hole symmetry of the original clean system, making it of the same class as the integer quantum Hall system: edge states at the interface should in principle be preserved. We verify that this is the case for weak disorder, however strong disorder drive the walk to a diffusion or localization regime. We next introduce a nonlinear dependence of the walk on its own state. In the continuum limit it tends to a nonlinear Dirac equation in $2+1$ dimensions. The effect of the nonlinearity restore, even for strong noise, the ballistic transport along the edge channels, which in addition, become an attractor.

I will report on quantum walk experiments employing ultracold Caesium atoms trapped in polarisation-synthetized (PS) optical lattices. Polarisation-synthetized optical lattices are a conceptually novel realization of spin-dependent optical lattices, which enable a wide range of quantum walk experiments. Atoms in spin-up and spin-down states are trapped in two distinct optical standing waves, whose position and depth can be individually controlled in time. This allows us to perform arbitrary shift operations of atoms in a fully independent manner for spin-up and spin-down components, on the timescale of microseconds and with a spatial precision of about 1Å. The next frontier for our ultracold-atom experiments is the realization of 2D discrete-time quantum walks. The implementation of a novel scheme for spin-dependent transport in the x-y plane with polarization-synthesized optical lattices is currently underway. We plan to use this setup to study Floquet topological phases in one and two dimensions. I will also present our most recent technological advances about the construction of the 2D experimental setup.

Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the evolution, and may be driven to be in a quantum superposition of graphs---in accordance to the superposition principle. We show that these unitary causal operators must decompose as a finite-depth circuit of local unitary gates. This unifies a result on Quantum Cellular Automata with another on Reversible Causal Graph Dynamics. Along the way we formalize a notion of causality which is valid in the context of quantum superpositions of time-varying graphs, and has a number of good properties. Keywords: Quantum Lattice Gas Automata, Block-representation, Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks, Graph Rewriting.

The pioneering work of Couder and Fort on bouncing oil droplets established that de Broglie-Bohm trajectories have a macroscopical counterpart. For instance, in a two slit experiment, the droplet passes through one of the slits but gets influenced by the second aperture, similar to what is called one particle non-locality in a Bohmian approach. Several models have been developed to explain these features, mostly of hydrodynamical inspiration, but we privilegged a new approach, inspired by de Broglie double solution program, in which non-linearity plays an essential role, together with Brownian motion. From this point of view, droplets are a great tool for simulating de Broglie-Bohm trajectories, and provide a concrete illustration of the hydrodynamical formulation of quantum dynamics. They also open new fields of research and investigation, one of them being the quantitative understanding of their mutual, pseudo-gravitational, interaction...