## PHY 310: Quantum Computation and Information

3 credits | Prerequisites: PHY 222 for Major; PHY 230 for Minor

### Course rationale

Quantum Mechanics is the most successful description of the atomic world. The ideas from Quantum Mechanics can be applied to computer science, and there is a nascent branch of computing called Quantum Computation where the ideas of Quantum Mechanics are applied to computing. This course will describe the basics of Quantum Computation and Information.

### Course content

Essential Quantum Mechanics: vector space, basis and linear independence, linear operator and matrices, Pauli matrices, eigenvalues and eigenvectors, adjoint, Hermitian and unitary operators, state space, evolution of states, measurement, POVM measurements, phase, Bloch sphere, density operator, Schmidt decomposition, purification, EPR and Bell inequality; Quantum Circuits: quantum algorithms, single qubit operations, controlled operations, universal unitary gates; Quantum Fourier Transform: definition, phase estimation, order finding, factoring, period finding, discrete logarithms; Quantum Search Algorithm: oracle, the procedure, geometric visualization, performance, quantum search as quantum simulation, optimality of the search algorithm; Physical Realization of Quantum Computers: conditions for quantum computation, performance if unitary transformations, harmonic oscillator quantum computers, physical apparatus, Hamiltonian, quantum computation, optical photon quantum computer-physical apparatus, quantum computation, cavity quantum electrodynamics-physical apparatus, Hamiltonian, ion traps, nuclear magnetic resonance; Quantum Noise: classical noise and Markov process, environment and quantum operations, operator-sum representation, trace and partial trace, geometric picture of single qubit operation, quantum channels, phase damping; Quantum Error Correction: three qubit bit and phase flip code, the Shor code, discretization of errors, independent error model, degenerate codes, quantum Hamming bound, classical linear codes, Calderbank-Shor-Steane codes, stabilizer formalism, unitary gates and the stabilizer formalism, measurement in the stabilizer formalism, the Gottesman-Knill formalism, quantum circuits for encoding, decoding, and correction, fault-tolerant quantum computation; Entropy and Information: Shannon entropy, binary and relative entropy, conditional entropy and mutual information, Von Neumann entropy, basic properties, subadditivity, concavity of the entropy, strong subadditivity; Quantum Information Theory: the Holevo bound, example applications, data compression, Shannon’s noiseless channel coding theorem, classical information over noisy quantum channels, quantum information over noisy quantum channels, entanglement as a physical resource, quantum cryptography.

### Course objectives

- Understand the basic concepts of Quantum Mechanics.
- Use the language of Linear Algebra to represent quantum states and operations.
- Familiarize with the basic quantum gates.
- Use Quantum Mechanics to design basic quantum computation.
- Use Quantum Mechanics in information theory.

### References

- Quantum Computation and Quantum Information (10th anniversary edition) by Michael A. Nielsen, and Isaac L. Chuang
- Quantum Computing: An Applied Approach (2nd edition) by Jack D. Hidary
- Introduction to Classical and Quantum Computing by Thomas G Wong
- Quantum Computing for Everyone by Chris Bernhardt
- Dancing with Qubits: How quantum computing works and how it can change the world by Robert S.Sutor