PHY 265: Introduction to Quantum Computing and Quantum Information

(Spring 2024)



Lecture notes

Organizational meeting

On Thursday Jan 18, 2024 we will have an organizational meeting as part of the first lecture.

Lecture Time: Tuesdays+Thursdays at 4:50-6:05pm
Lecture Location: Meliora 224
Instructor: Alice Quillen, Office: Bausch and Lomb Hall 424.
Email:alice.quillen x*x
Class Website:
Course materials are usually posted on the website.
Textbook: Quantum Computing: A Gentle Introduction, Reifel & Polak, MIT press @2011
This is available on-line through our library: permalink
I will share some pdfs of literature that I have acquired. If you are auditing the class, let me know and I will send you a link to these.

Course description: The quantum mechanical nature and capabilities of a Qubit based quantum computer will be introduced and explored. Topics covered include: Two state quantum systems, qubits, as components of a quantum computer. Quantum measurements. Tensor products and entanglement. Quantum gates and quantum circuits. Quantum information and von-Neumann entropy. Density operators, partial traces, quantum channels and decoherence. Realizing logical operations and universality on a quantum computer. Black box problems, such as the Bernstein-Vazirani and Simon's problems. The quantum Fourier transform. Quantum algorithms such as Shor's factoring algorithm. Types of quantum computing complexity. Quantum error correction. Quantum search algorithms. Variational methods. Simulations on a Quantum computer. Prospects for realizing quantum computing.

Overview: In this class we will explore the rapidly developing fields associated with quantum information and quantum computing. Quantum computing includes rapidly-emerging technology that leverages the unique behavior of quantum physics to extend the capabilities of computers and develops algorithms on quantum computational platforms to solve complex problems. Quantum information theory is an interdisciplinary field merging quantum physics, computer science, information theory, philosophy and cryptography. PHY 265 is a new class (2024 is third time!) in our department devoted to this topic.

Prerequisites: Modern physics including some quantum mechanics. Linear algebra at the level of the Math 161-165 or the MATH 171-174 sequences.

Level: upper level for PHYS/PAS majors.
This course will not fullfill the computational literacy requirement for PHYS/PAS majors.
This course would be considered an elective for PHYS/PAS majors.
This course is not cross-listed with any other class.

Office hours: None officially. However the instructor tends to be available from 9-5 on weekdays. Since the class meets late, it will be fun to chat after class, though people are going to want dinner. We could think about setting up a time to get together to work on problems. The instructor will be pleased when students offer to chat and is happy to work through problems together. If you want to be sure that the instructor is available please propose some convenient times (for you) via email.

TI/TA: None. We are on our own!

Draft Course requirements:
Where to turn things in: Blackboard

Rules: You can talk about your assignments with your fellow students and other people. Solutions to problems, projects and reports should not be directly copied from other students or any other source. Sources used and collaborators should be identified on assignments.

Credit hour policy: This course follows the College credit hour policy for four-credit courses. This course meets twice weekly for 2.5 academic hours per week. The course includes weekly meetings outside of this regular class time. The course includes weekly independent out-of-class assignments.


Introduction to Quantum Mechanics:
Quantum systems. Basis vectors and quantum states. The qubit. The Bloch sphere. Pauli matrices. Single qubit quantum gates. Exponentials of matrices. Unitary transformations. Relative vs Global phase. Measurements. Measurement postulates. Projective operators. The Quantum Zeno effect. Product Spaces and multiple qubits. Tensor products. The Bell or EPR pair state. Entanglement. Quantum operations on 2 qubits. Controlled quantum gates. Partial measurement - measurement of 1-qubit of a 2-qubit system. The no-cloning theorem. 2 qubit quantum circuits. Dense coding. Interpretations of Quantum mechanics. The EPR paradox and Shrodinger's cat. Copenhagen and many worlds interpretations. EPR polarization measurements with hidden variables. Bell's inequalities. Teleportation with a Bell pair. Entanglement as a resource.

Introduction to Quantum Information:
The density operator. The density operator for pure and mixed states. Transformations of the density matrix. The density matrix for a product space. Partial traces in a bipartite system. The positivity of the density matrix. Schmidt decomposition of a bipartite pure state. Entanglement and the Schmidt number. Evolution of density matrices: quantum channels/superoperators. Generalized measurements (POVM). Operator sum decompositions. Decoherence. Channel-state duality. Entropy and information. Entropy in statistical physics. Shannon entropy. Von Neuman entropy. Quantifying the extent of entanglement. Distance measures for quantum information. The GHZ 3-qubit state and the monogamy of entanglement. The Holevo bound. Noiseless compression. The Lindblad master equation.

Introduction to Quantum Computing:
Logical operations on a quantum computer. Universal gate sets for logical operations on a classical computer. 3-qubit gates and the Toffoli gate. Quantum versions of classic logical operations. Realizing Unitary transformations and universality on a quantum computer. What is meant by universality? How to approximate any single qubit unitary transformation with a finite set of simple gates. Universal gate sets for multiple qubits. Solovay-Kitaev theorem. Reversibility and reclaiming used qubits.

Introduction to Quantum Algorithms:
Black box problems. Deutsch's problem. The N-bit Wasch-Hadamard operation and the Deutsch-Jozca problem. What is a quantum oracle? Quantum parallelism. The Bernstein-Vazirani algorithm. The Quantum Fourier Transform. The Discrete Fourier Transform and the Fast Fourier Transform. A product representation for the Quantum Fourier transform. An efficient circuit for the Quantum Fourier Transform. Quantum phase estimation. Shor's factoring algorithm. Simon's problem. Period finding on a quantum computer using the Quantum Fourier Transform. Reduction of factoring to period finding. The efficiency of the Shor factoring algorithm. Computational Complexity. Advantages of quantum computing. What is a quantum compiler? Quantum search and counting algorithms. Solving linear systems on a quantum computer. Solving quadratic pseudoBoolean optimization problems on a quantum computer. Adiabatic algorithms. Quantum Variational methods.

Optional Additional topics:
Error correction. Shor's 9-bit code. Stabilizer codes. Random benchmarking to measure errors on a quantum computer. Realizing quantum computing with real physical systems. Simulating the behaviour of quantum computers. Simulating quantum systems with quantum computers. Weak measurements. Quantum random walks. Hidden subgroup problem.