99191 - DISPOSITIVI, CIRCUITI ED ALGORITMI PER IL CALCOLO QUANTISTICO M

Learning outcomes

The course introduces the fundamental concepts for understanding the hardware and software systems underlying quantum computing. Upon completion, the student masters the principles underlying quantum computation, an understanding of the quantum circuit model, the main algorithms, the physical concepts underlying the implementation of hardware systems, and the main electronic architectures.

Course contents

The course is divided into two modules. In the first module the concepts of qubits and quantum circuit are introduced, together with the operations with which quantum algorithms are defined starting from the composition of fundamental gates. The module also introduces the problems related to hardware and software emulation of quantum circuits on classical systems and currently possible approaches. Computer exercises are planned to illustrate examples of realization of elementary quantum gates and the implementation of some algorithms.

In the second module, the physical principles underlying the realization and operation of qubits are introduced and the physical behaviour of fundamental quantum gates is analysed, with reference to solutions based on superconducting circuits and electron/hole spins in semiconductor quantum dots. The module is complemented with computer exercises aimed at illustrating the main concepts and simulating the behaviour of some quantum gates.

First module

• Some brief historical notes. The computer as a physical system. The limits of classical computers.
• What is a quantum computer. The quantum representation of information. The concept of qubits and its peculiarities: superposition of states, entanglement, interference.
• Reversibility, unit transformations and quantum gates (I, X, Y, Z, S). No-cloning theorem. Qubit systems. The initial and final states of a calculation: preparation, measurement and Born rules. Collapse of the state.
• Logic gates and quantum circuits (the three universal gates: H, T, CNOT). Construction of arbitrary 1- and 2-qubit states. Universality.
• The Quantum Fourier Transform. Some important algorithm: Deutsch-Jozsa, Simon, Shor-Steane and Grover. HHL Algorithm. Quantum Wavelet Transform, Q-Machine Learning.
• Some error correction techniques.
• The quantum representation of digital signals and its preparation. Realization of arithmetic operations.
• Simulation of quantum circuits: the fundamental problems and the possible software and hardware solutions (GPUs and FPGAs). Different and computationally efficient approaches for the description of quantum state evolution.

Second module

• The formalism of Quantum Mechanics: ket/bra vectors, linear operators, Hermitian operators, the eigenvalue problem, observables, completeness, operator functions, matrix representation of vectors and operators.
• The postulates of Quantum Mechanics: quantization, commutator of conjugate operators, Schroedinger equation. Measures of observables and probabilistic interpretation. Heisenberg's uncertainty principle. Dynamic behavior of a quantum system: time-dependent Schroedinger equation, evolution operator. Schroedinger, Heisenberg and interaction picture.
• Elementary quantum systems: free particle, electron in a rectangular potential well. Harmonic oscillator. Creation and annihilation operators and their properties.
• Brief phenomenological introduction to superconductivity. Notes on the Ginzburg-Landau theory. Josephson junction: fundamental relationships between current, voltage and phase.
• Superconductor qubits. The quantum LC oscillator. The transmon qubit: structure, Hamiltonian and analysis of the energy levels. Flux-tuning transmon. Outline of other types of superconducting qubits.
• Noise and decoherence in qubits: models (outline) and characterization, common sources of noise, techniques to mitigate their impact.
• Single qubit gates: qubit control and dynamics, Rabi frequency. Virtual Z-gate. Advanced qubit control techniques to reduce leakage and phase errors (outline).
• Two qubit/oscillator systems: Hamiltonian analysis and related simplifications, dispersive regime. Coupling between two transmons.
• 2-qubit gates: iSWAP gate, efficient implementation of the CPHASE gate using higher energy levels, cross-resonance gates, parametric gates (outline).
• Qubit readout: general problems related to detection and amplification of electromagnetic signals in quantum regime, dispersive qubit readout, criteria to increase the measurement fidelity.
• Notes on spin qubits based on semiconductor quantum dots: operating principles for the manipulation and readout of single and double qubits.

 

Readings/Bibliography

1. Daniel D. Stancil, Gregory T. Bird, “Principles of Superconducting Quantum Computers”, Wiley, 2022
2. Ray LaPierre, “Introduction to Quantum Computing”, Springer Cham, 2021
3. Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press, 2010

Teaching methods

Lectures supplemented by computer exercises.

Assessment methods

The exam consists of an oral interview on the topics covered in both teaching modules. Not only the student's critical ability in connecting different parts of the program and to justify reasoning will be evaluated, but also the possible presence of training gaps or the use of inappropriate language. The final grade expresses the evaluation of the mastery of the concepts and the critical capacity shown by the student.

Teaching tools

Slides, documentation provided by teachers on specific topics, multimedia material.

Office hours

See the website of Antonio Gnudi

See the website of Nicolò Attilio Speciale