Mathematical modelling of quantum computers

Mazyar Mirrahimi

This course is destined to applied mathematics and physics students with interest in quantum information and computation. The course subject is the mathematical modelling of devices that enable us to manipulate the quantum state of light and matter in the aim of realizing a quantum processor. While the developed methodology is applicable to various hardware problems in recently developed quantum technologies, the quantum software problems (e.g. quantum algorithms) are not initiated in this course.

One of the most needed requirements for any operating quantum machine is to prepare various non-classical states with high fidelities, manipulate them reliably and protect them over arbitrary long times. In manipulating such quantum systems, one needs to take into account various experimental uncertainties and the destructive effects of the coupling to an uncontrolled environment leading to the loss of coherence in quantum information. On the other hand, in order to protect (against decoherence) the quantum information carried by such a system, one needs to intentionally couple the system to an environment either to measure some physical observables (in an active measurement-based feedback scheme) or to evacuate the entropy (in a passive dissipation engineering scheme).

This course focuses on dynamical models (ordinary and partial differential equations, discrete and continuous-time stochastic systems) behind such quantum systems and control tools allowing the manipulation and stabilization of some interesting non-classical states. By focusing on the particular case of a qubit (two-level quantum system) coupled to a harmonic oscillator, we will present open-loop control tools (pulse shaping techniques) that are robust to experimental systematic errors and stochastic time-dependent noise. We will also introduce and analyze various dynamical models for open quantum systems undergoing repetitive or continuous measurements, and will study their stabilization through closed-loop control techniques (feedback). The topics covered by the course include:

  1. Introduction to spin-spring systems, Jaynes-Cummings hamiltonian and propagators, laser manipulation of trapped ions.
  2. Resonant control and rotating wave approximation, multi-frequency averaging of first and second orders, Rabi oscillations and Bloch-Siegert shift, lambda systems and Raman transition, single trapped ion and Law-Eberly method, single and two qubit logical gates.
  3. Robust adiabatic control, time-adiabatic approximation without gap conditions, chirped pulses for a 2-level system, Stimulated Raman Adiabatic Passage (STIRAP).
  4. Optimal control, Pontryagin maximum principle and first-order stationary condition, monotonic numerical scheme, Gradient Ascent Pulse Engineering (GRAPE) algorithm.
  5. Quantum measurement, projective and Positive Operator Valued Measure (POVM), Quantum Non-Demolition (QND) measurement, stochastic process attached to a POVM, repeated QND measurements for non-deterministic state preparation, measurement-based feedback, Measurement uncertainties and Bayesian filtering, structure of discrete-time open quantum systems: Markov models, Kraus and unital maps, quantum filtering.
  6. Relaxation as unread measurement, spontaneous emission and Lindblad master equation, slow/fast dissipative systems and model reduction by adiabatic elimination, dissipation as a control tool: optical pumping, coherent population trapping, dynamical cooling, quantum reservoir engineering.
  7. Continuous measurement, quantum stochastic master equation and model reduction, feedback stabilization.
  8. Introduction to quantum error correction.

Pre-requisites: The course «  Physique quantique avancée » is very helpful to progress rapidly throughout the course but it is not mandatory. The course is self-contained and all required quantum physics concepts will be reminded.