We investigate the quantum Zeno effect as a framework for designing and analyzing quantum algorithms for Hamiltonian simulation. We show that frequent projective measurements of an ancilla qubit register can be used to simulate quantum dynamics on a target qubit register with a circuit complexity similar to randomized approaches. The classical sampling overhead in the latter approaches is traded for ancilla qubit overhead in Zeno-based approaches. We show that the quantum circuits over the combined register can be identified as a subroutine commonly used in post-Trotter Hamiltonian simulation methods. We build on this observation to reveal connections between different Hamiltonian simulation algorithms. Finally, we develop higher order Zeno formulas to improve scaling and discuss implementations through unitary kicks. 

According to Quantum Mechanics, the (pure) states of a quantum system are described by normalised elements of a complex separable Hilbert space, while their evolution is given by a t-dependent Hamiltonian, H(t), through the Schrödinger equation. In the general case, the Hamiltonian is an unbounded t-dependent self-adjoint operator. Guided by the example of a free particle confined in an interval, we will present a framework to study quantum control systems based on the relationship between self-adjoint operators and quadratic forms: control by form perturbations. This framework will allow us to explore some unconventional quantum control problems such as controlling the quantum state of particle in a box by moving its walls, by manipulating its boundary conditions or even by means of delta-like interactions.

In the physics literature it is common to use “in cascade” the rotating wave approximation and the adiabatic approximation for chirped pulses of two-level quantum systems driven by one external field, in particular when the resonance frequencies are not known precisely. Both approximations need relatively long time and are based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. We study how the cascade of the two approximations can be justified, while preserving the robustness of the adiabatic strategy. As a by-product, we obtain results of ensemble controllability for n-level systems driven by a single real-valued control providing an extension of a celebrated result by Khaneja and Li (for spin systems driven by two controls).

The quantum speed limit (QSL) is the upper bound on the rate of change of a quantum system subjected to an external disturbance. This conceptis of great practical interest for many time-critical applications where the external disturbance is a controlled electromagnetic field. However, a precise definition of QSL is not straightforward.Intuitively, the QSL may arise from: (1) a limitation on the admissible amplitude of the external control (for instance, to avoid the destruction of the system), or (2) an intrinsic limitation of the dynamics that prevents instantaneous transitions, even under non-physical, extremely large controls. This talk aims to present various estimates of the QSL for bilinear conservative quantum systems in these two scenarios, both in the finite- and infinite-dimensional frameworks. It may serve as an introduction to the talk by E. Pozzoli on the (im)possibility of achieving arbitrarily fast transitions for infinite-dimensional bilinear quantum systems.

We will investigate bilinear control systems of the form

Ẋ(t, θ) = (θA + u1(t)B1 + u2(t)B2) X(t, θ), (1)

where A, B1 and B2 are elements of the Lie algebra g of a given (compact) Lie group G (e.g. g = su(n) and G = SU(n)) and θ ∈ [a,b] is an additional unknown system parameter (e.g. some unknown Lamor frequence). Our aim is to provide sufficient conditions for approximate accessibility and simultaneous controllability of the above class of systems. To this end we consider Eq. (1) as a bilinear system on the infinite dimensional Lie group of all continuous maps from [a,b] to G.

Joint work with F. vom Ende (FU Berlin), E. Malvetti (TU Munich), and T. Schulte-Herbrüggen (TU Munich)

We propose a platform for implementing a universal, globally driven quantum computer based on a 2D ladder and 1D arrays geometry, hosting  different species of superconducting qubits. In stark contrast with the existing literature, our scheme exploits the always-on longitudinal ZZ coupling. The latter, combined with specific driving frequencies, enables the reach of a blockade regime, which plays a pivotal role in the computing scheme. Based on arXiv:2407.01182.

The talk provides an introduction to dynamical decoupling, an error mitigation procedure in quantum information theory that aims to stabilise open quantum systems undergoing decoherence. We look at some of the mathematical challenges, provide conditions as to when and how dynamical decoupling does or does not work in given quantum systems and what the resulting dynamics looks like. The talk is based on a series of joint papers with C. Arenz, D. Burgarth and P. Facchi. 

To harness the rapidly increasing complexity of quantum platforms, such as scalable quantum computing devices, machine learning is establishing itself as a valuable toolbox. In this talk I will give an overview of how machine learning can be used to discover novel quantum error correction strategies and quantum control procedures, also including feedback. One prominent approach relies on reinforcement learning, a set of techniques used to discover strategies from scratch. I will introduce our recent extension of the popular GRAPE approach to cases with feedback. Beyond our numerical simulations, I will also describe a project where we worked with experimentalists to implement for the first time a real-time neural-network based agent controlling a superconducting qubit.

I will present a stability result of the non-autonomous Schrödinger equation for Hamiltonians with constant form domain. The sharp estimates obtained improve previous results in the literature. As an application I will show how one can prove some approximate controllability results for infinite dimensional quantum control problems. This is joint work with A. Balmaseda and D. Lonigro.

We show that infinite-dimensional closed quantum systems can be approximately controllable in arbitrarily small times, when arbitrarily large controls are allowed. We illustrate this property on two paradigmatic examples of bilinear Schrödinger PDEs, namely, quantum rotors and harmonic oscillators. Such results in particular imply that there is no intrinsic limitation to instantaneous transitions in such systems. The control technique we develop to prove such results can be seen as an infinite-dimensional extension of the geometric approach ("pulse-drift-pulse") used in the celebrated work on time-optimal control of finite-dimensional spin systems by Khaneja, Brockett, and Glaser. Interestingly, drastically different phenomena appear in the infinite-dimensional setting. This is a joint work with Karine Beauchard.

We connect quantum control theory with quantum thermodynamics within the framework of Lie-semigroup theory. In particular, we sketch a Markovianity Filter, i.e. how to construct the Markovian counterparts of several types of quantum Thermal Operations via their respective Lie wedge. In an explicit qubit example,we parameterise the Markovian subset of maps within the set of all the Thermal Operations. As an application, we give inclusions in terms of d-majorisation for reachable sets of bilinear control systems, where coherent controls are complemented by switchable couplings to a thermal bath as additional thermal resource.

Joint work with Gunther Dirr, Frederik vom Ende, and Emanuel Malvetti.

I will present techniques for deriving state-dependent convergence rates for dynamical limit problems such as quantum speed limits or Trotter products. These techniques apply to finite and infinite dimensional systems with open or closed system dynamics. The core idea of the approach is to pick a suitable reference Hamiltonian and to prove convergence rates for the energy-constrained diamond and operator norms introduced by Shirokov and Winter. This approach builds two new ingredients: (1) a submultiplicativity inequality for the energy-constrained norms and (2) methods to estimate the output energy as a function of the input energy of quantum dynamics.

I will present applications of this toolbox in the form of explicit convergence rates for the Trotter product formula in continuous variable systems. The latter are joint work with Simon Becker, Niklas Galke, and Robert Salzmann.

We present a universal nonperturbative bound on the distance between unitary evolutions generated by time-dependent Hamiltonians in terms of the difference of their integral actions. This allows us to obtain an explicit error bound for the rotating-wave approximation. We also show that our universal bound can be used to prove and to generalize other known theorems such as the adiabatic theorem, product formulas, and Zeno limits.

Reference: D. Burgarth, P. Facchi, G. Gramegna, and K. Yuasa, "One bound to rule them all: from Adiabatic to Zeno". Quantum 6, 737 (2022).

The Unruh effect is a quantum field prediction according to which a uniformly accelerated observer in the vacuum detects thermal particles with a temperature directly proportional to its acceleration. In other words, there is no difference in the detection results between an accelerated observer in the vacuum and an observer at rest immersed in a thermal bath. In order to prove this phenomenon, we consider a detector, specifically a two-level atom, uniformly accelerated in a 1+1 spacetime and coupled to a boson massless scalar field in the vacuum. We consider two possible interaction models: Rabi and Jaynes–Cummings. We compute the response function for the accelerated detector in both cases, obtaining zero for the Jaynes–Cummings interaction and a response function equal to the thermal one for the Rabi interaction. 

The asymptotic dynamics of open quantum systems has been a hot topic since the seventies because of its fundamental interest. Moreover, with the advent of quantum technologies, it turns out to be the theoretical framework of a variety of applications, such as quantum information processing within noiseless/decoherence-free subsystems, quantum reservoir engineering, and quantum associative memories. In this poster we will show sharp upper bounds on the number of steady and asymptotic states of quantum evolutions. The bounds are universal as they are only related to the dimension of the system.

Joint work with Paolo Facchi (UNIBA & INFN Bari). References: D. A., and P. Facchi, Number of steady states of quantum evolutions, Sci. Rep. 14, 14366 (2024). 

We propose a method for describing coherent control of electronic quantum states in a nanowire subject to a spatially varying magnetic (Zeeman) field. Such manipulation is realized by a holonomic transformation that arises from a (non-Abelian) generalization of the Berry phase mechanism. We show that, under specific conditions, the electron is fully transmitted through the wire and the electronic quantum state undergoes a pure holonomic transformation. Notably, our approach ensures stability against external perturbations by leveraging an intrinsic topological protection: the transformation depends solely on specific boundary conditions of the field at the edges of the wire, without imposing restrictions on the field’s variation or its rate of variation along the wire. This approach enables control over both the spin and the momentum degrees of freedom of the injected electron. For example, we demonstrated the capability to induce a perfect spin-flip and to accelerate or decelerate the electron through the wire, thereby changing its momentum. Additionally, we are able to induce an entangled state between the electron’s spin and momentum degrees of freedom. In conclusion, our approach is applicable to nanowires in one, two, and three dimensions. 

Controlling a quantum system should be most effective in resonance. To examine and use this, we derive for any control system with a bounded Hamiltonian a criterium to rule out controllability within a given time using the available frequency range of the time dependent control with respect to the transition energies of the uncontrolled system. We derive another one specialised for high frequency controls using that the controls average out. For a qubit system a third inequality specialised for low frequency controls is derived, which uses that the qubit will approximately evolve adiabatically. As a by-product we examine how much a qubit can be controlled adiabatically and we derive for any system with a bounded Hamiltonian H(t) a new error bound on the adiabatic approximation that in contrast to existing bounds does not need H(t) to be twice differentiable but Ḣ(t) to have a quickly decaying Fourier transformation.

We study the influence of single-atom dissipation and dephasing noise on the performance of Dicke and Tavis-Cummings quantum batteries, where the electromagnetic field of the cavity hosting the system acts as a charger. For these models a genuine charging process can only occur in the transient regime. Indeed, unless the interaction with the environment is cut off, the asymptotic energy of the battery is solely determined by the environment and does not depend on the initial energy of the electromagnetic field. We numerically estimate the fundamental figures of merit for the model, including the time at which the battery reaches its maximum ergotropy, the average energy, and the energy that needs to be used to switch the battery-charger interaction on and off.Depending on the scaling of the coupling between the battery and the charger, we show that the model can still exhibit a subextensive charging time. However, for the Dicke battery, this effect comes with a higher cost when switching the battery-charger interaction on and off. We also show that as the number of battery constituents increases, both the Dicke and Tavis-Cummings models become asymptotically free, meaning the amount of energy that is not unitarily extractable becomes negligible. We obtain this result numerically and demonstrate analytically that it is a consequence of the symmetry under permutation of the model. Finally, we perform simulations for different values of the detuning, showing that the optimal regime for the Dicke battery is off-resonance, in contrast to what is observed in the Tavis-Cummings case.

Periodically driven quantum systems exhibit a diverse set of phenomena but are more challenging to simulate than their equilibrium counterparts. Here, we introduce the Quantum High-Frequency Floquet Simulation (QHiFFS) algorithm as a method to simulate fast-driven quantum systems on quantum hardware. Central to QHiFFS is the concept of a kick operator which transforms the system into a basis where the dynamics is governed by a time-independent effective Hamiltonian. This allows prior methods for time-independent simulation to be lifted to simulate Floquet systems. We use the periodically driven biaxial next-nearest neighbor Ising (BNNNI) model, a natural test bed for quantum frustrated magnetism and criticality, as a case study to illustrate our algorithm. We implemented a 20-qubit simulation of the driven two-dimensional BNNNI model on Quantinuum's trapped ion quantum computer. Our error analysis shows that QHiFFS exhibits not only a cubic advantage in driving frequency ω but also a linear advantage in simulation time t compared to Trotterization. 

In this work we show that the time evolutions given by finite-dimensional approximations of a Hamiltonian H do not necessarily converge to the original time evolution: instead, they converge to the dynamics induced by the Friedrichs extension of the Hamiltonian restricted on the approximation basis. As an example, we discretize the particle in an arbitrary box with a boundary-blind basis, and show that the time evolution converges to the Dirichlet time evolution—independently of the original boundary conditions. 

Joint work with Daniel Burgarth and Davide Lonigro.

The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm designed for Combinatorial Optimization Problem (COP). We show in the asymptotic limit that if a spin glass type COP with an underlying Erdos--Renyi hypergraph exhibits the Overlap Gap Property (OGP), then a random regular hypergraph exhibits it as well. As such, we re-derived the fact that the average-case value obtained by QAOA for the Max-qXORSAT is bounded away from optimality even when the algorithm runs indefinitely if optimized using the so-called tree parameters. While this result was proven before, the proof is rather technical compared to ours. In addition, we show that their result implicitly also implies limitation at logarithmic depth. Lastly, the results suggests that even when sub-optimised, the performance of QAOA on spin glass is equal in performance to classical algorithms in solving the mean field spin glass problem providing further evidence that the conjecture of getting the exact solution under limit swapping for the Sherrington--Kirkpatrick model to be true. 

An important result in the theory of quantum control is the "universality" of 2-local unitary gates, i.e. the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.

Poster presenting the work done for the paper arXiv:2409.11407.

The processing unit of a solid-state quantum computer consists in an array of coupled qubits, each locally driven with on-chip microwave lines that route carefully-engineered control signals to the qubits in order to perform logical operations. This approach to quantum computing comes with two major problems. On the one hand, it greatly hampers scalability towards fault-tolerant quantum computers, which are estimated to need a number of qubits -- and, therefore driving lines -- on the order of 10^6. On the other hand, these lines are a source of electromagnetic noise, exacerbating frequency crowding and crosstalk,  while also contributing to power dissipation inside the dilution fridge. We here tackle these two overwhelming challenges by presenting a novel quantum processing unit (QPU) for a universal quantum computer which is globally (rather than locally) driven. Our QPU relies on a string of superconducting qubits with always-on ZZ interactions, enclosed into a closed geometry, which we dub "conveyor belt''. Strikingly, this architecture requires only O(N) physical qubits to run a  computation on N logical qubits, in contrast to previous O(N^2) proposals for global quantum computation. Additionally, universality is achieved via the implementation of single-qubit gates and a one-shot Toffoli gate. The ability to perform multi-qubit operations in a single step could vastly improve the fidelity and execution time of many algorithms.

We analytically find quantitative, non-perturbative bounds to the validity of the rotating-wave approximation (RWA) for the multi-atom generalization of the quantum Rabi model: the Dicke model. Precisely, we bound the norm of the difference between the evolutions of states generated by the Dicke model and its rotating-wave approximated counterpart, that is, the Tavis–Cummings model. The intricate role of the parameters of the model in determining the bounds is discussed and compared with numerical results. Our bounds are intrinsically state-dependent and, in particular, are significantly different in the cases of entangled and non-entangled states; this behaviour also seems to be confirmed by the numerical results.

Joint work with Daniel Burgarth and Davide Lonigro. Reference: L. Richter, D. Burgarth and D. Lonigro, arXiv:2410.18694.

We show that it is possible to refine the classification of a quantum symmetry in terms of its robustness with respect to small perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust with respect to a single fixed perturbation and we use such result to characterize stability with respect to larger sets of perturbations for gapped unbounded Hamiltonians.

Quantum control theory provides the foundation upon which modern quantum technologies are being built. Previous research has unveiled a deep connection between symmetries, controllability of quantum systems and quantum speed limits. This talk explores how this relationship can be used to further our understanding of quantum control theory.Firstly, a new kind of quantum speed limit is established. While previously known speed limits are only applicable to very specialized time evolutions, we derive a quantum speed limit for any arbitrary unitary target gate or effective target Hamiltonian.Secondly, the symmetries of the quadratic Hamiltonians of a single bosonic mode will be presented. Even though the quadratic Hamiltonians are not sufficient to control the mode, they do not admit a shared second order symmetry. This is only possible in infinite dimensional systems. However, there still exists a shared third order symmetry. The conserved quantities of the symmetries relate to geometric properties of the Wigner function.