(Spring 2024)

## Links:## Problems/Assignments## Lecture notes |

Email:alice.quillen x*x rochester.edu

Course materials are usually posted on the website.

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.

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.

- 5 to 7 assignments. These will primarily involve analytical calculations.
- 1 project with a 5 page write-up. The project goal would be to explore and investigate a recent development in the field of quantum computing or information. The last two days of class will be devoted to project presentations.
- Contributing to in-class discussions on recent literature and other topics.
- No exams

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.

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.

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.

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.

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.