Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications

Encoding quantum information into a set of harmonic oscillators is considered a hardware efficient approach to mitigate noise for reliable quantum information processing. Various codes have been proposed to encode a qubit into an oscillator -- including cat codes, binomial codes and Gottesman-Kitaev-Preskill (GKP) codes. These bosonic codes are among the first to reach a break-even point for quantum error correction. Furthermore, GKP states not only enable close-to-optimal quantum communication rates in bosonic channels, but also allow for error correction of an oscillator into many oscillators. This review focuses on the basic working mechanism, performance characterization, and the many applications of GKP codes, with emphasis on recent experimental progress in superconducting circuit architectures and theoretical progress in multimode GKP qubit codes and oscillators-to-oscillators (O2O) codes. We begin with a preliminary continuous-variable formalism needed for bosonic codes. We then proceed to the quantum engineering involved to physically realize GKP states. We take a deep dive into GKP stabilization and preparation in superconducting architectures and examine proposals for realizing GKP states in the optical domain (along with a concise review of GKP realization in trapped-ion platforms). Finally, we present multimode GKP qubits and GKP-O2O codes, examine code performance and discuss applications of GKP codes in quantum information processing tasks such as computing, communication, and sensing.Comment: 77+5 pages, 31 figures. Minor bugs fixed in v2. comments are welcome

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Transport and nonequilibrium dynamics in 1d many-body systems with bosonic degrees of freedom

In this thesis, different wave-function based numerical methods are used to study many-body systems with bosonic degrees of freedom in 1d. The main focus is on the Holstein model of spinless fermions, where ground-state phases, nonequilibrium dynamics, and thermalization are investigated. Using matrix-product-state methods for the investigation of such models with electron-phonon coupling poses special numerical challenges. Local basis optimization, projected purification, and subspace expansion are used to overcome these challenges.2021-12-1

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Technical Design Report EuroGammaS proposal for the ELI-NP Gamma beam System

The machine described in this document is an advanced Source of up to 20 MeV Gamma Rays based on Compton back-scattering, i.e. collision of an intense high power laser beam and a high brightness electron beam with maximum kinetic energy of about 720 MeV. Fully equipped with collimation and characterization systems, in order to generate, form and fully measure the physical characteristics of the produced Gamma Ray beam. The quality, i.e. phase space density, of the two colliding beams will be such that the emitted Gamma ray beam is characterized by energy tunability, spectral density, bandwidth, polarization, divergence and brilliance compatible with the requested performances of the ELI-NP user facility, to be built in Romania as the Nuclear Physics oriented Pillar of the European Extreme Light Infrastructure. This document illustrates the Technical Design finally produced by the EuroGammaS Collaboration, after a thorough investigation of the machine expected performances within the constraints imposed by the ELI-NP tender for the Gamma Beam System (ELI-NP-GBS), in terms of available budget, deadlines for machine completion and performance achievement, compatibility with lay-out and characteristics of the planned civil engineering

On The Dynamics and Control Strategy of Time-Delayed Vibro-Impact Oscillators

Being able to control nonlinear oscillators, which are ubiquitous, has significant engineering implications in process development and product sustainability design. The fundamental characteristics of a vibro-impact oscillator, a non-autonomous time-delayed feedback oscillator, and a time-delayed vibro-impact oscillator are studied. Their being stochastic, nonstationary, non-smooth, and dynamically complex render the mitigation of their behaviors in response to linear and stationary inputs very difficult if not entirely impossible. A novel nonlinear control concept featuring simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain is developed and subsequently incorporated to maintain dynamic stability in these nonlinear oscillators by denying bifurcation and route-to-chaos from coming to pass. Convergence of the controller is formulated to be inherently unconditional with the optimization step size being self-adaptive to system identification and control force input. Optimal initial filter weights are also derived to warrant fast convergence rate and short response time. These novel features impart adaptivity, intelligence, and universal applicability to the wavelet based nonlinear time-frequency control methodology. The validity of the controller design is demonstrated by evaluating its performance against PID and fuzzy logic controllers in controlling the aperiodic, broad bandwidth, discontinuous responses characteristic of the time-delayed, vibro-impact oscillator

Multi-parameter optimisation of quantum optical systems

Quantum optical systems are poised to become integral components of technologies of the future. While there is growing commercial interest in these systems---for applications in information processing, secure communication and precision metrology---there remain significant technical challenges to overcome before widespread adoption is possible. In this thesis we consider the general problem of optimising quantum optical systems, with a focus on sensing and information processing applications. We investigate four different classes of system with varying degrees of generality and complexity, and demonstrate four corresponding optimisation techniques. At the most specific end of the spectrum---where behaviour is best understood---we consider the problem of interferometric sensitivity enhancement, specifically in the context of long-baseline gravitational wave detectors. We investigate the use of an auxiliary optomechanical system to generate squeezed light exhibiting frequency-dependent quadrature rotation. Such rotation is necessary to evade the effect of quantum back action and achieve broadband sensitivity beyond the standard quantum limit. We find that a cavity optomechanical system is generally unsuitable for this purpose, since the quadrature rotation occurs in the opposite direction to that required for broadband sensitivity improvement. Next we introduce a general technique to engineer arbitrary optical spring potentials in cavity optomechanical systems. This technique has the potential to optimise many types of sensors relying on the optical spring effect. As an example, we show that this technique could yield an enhancement in sensitivity by a factor of 5 when applied to a certain gravitational sensor based on a levitated cavity mirror. We then consider a particular nanowire-based optomechanical system with potential applications in force sensing. We demonstrate a variety of ways to improve its sensitivity to transient forces. We first apply a non-stationary feedback cooling protocol to the system, and achieve an improvement in peak signal-to-noise ratio by a factor of 3, corresponding to a force resolution of 0.2fN. We then implement two non-stationary estimation schemes, which involve post-processing data taken in the absence of physical feedback cooling, to achieve a comparable enhancement in performance without the need for additional experimental complexity. Finally, to address the most complex of systems, we present a general-purpose machine learning algorithm capable of automatically modelling and optimising arbitrary physical systems without human input. To demonstrate the potential of the algorithm we apply it to a magneto-optical trap used for a quantum memory, and achieve an improvement in optical depth from 138 to 448. The four techniques presented differ significantly in their style and the types of systems to which they are applicable. Successfully harnessing the full range of such optimisation procedures will be vital in unlocking the potential of quantum optical systems in the technologies of the futur

Toward simulating Superstring/M-theory on a quantum computer

We present a novel framework for simulating matrix models on a quantum computer. Supersymmetric matrix models have natural applications to superstring/M-theory and gravitational physics, in an appropriate limit of parameters. Furthermore, for certain states in the Berenstein-Maldacena-Nastase (BMN) matrix model, several supersymmetric quantum field theories dual to superstring/M-theory can be realized on a quantum device. Our prescription consists of four steps: regularization of the Hilbert space, adiabatic state preparation, simulation of real-time dynamics, and measurements. Regularization is performed for the BMN matrix model with the introduction of energy cut-off via the truncation in the Fock space. We use the Wan-Kim algorithm for fast digital adiabatic state preparation to prepare the low-energy eigenstates of this model as well as thermofield double state. Then, we provide an explicit construction for simulating real-time dynamics utilizing techniques of block-encoding, qubitization, and quantum signal processing. Lastly, we present a set of measurements and experiments that can be carried out on a quantum computer to further our understanding of superstring/M-theory beyond analytic results.Comment: 68 pages; v2: minor correction