Please note that not all courses are offered every semester.
Fall 2025 - Click to expand
(Organized alphabetically by course title)
Modular grad course, second half of Fall 2025
This 6-week modular grad course will use a subset of the classic experiments on multi-photon interference as a window into quantum optics, quantum measurement, and quantum information. We will begin with a review of the relevant second-quantized theory for describing few-photon experiments, and discuss the regimes in which quantum & classical optics agree and those where they diverge. In particular, we will learn about photon correlations (the Hanbury-Brown–Twiss effect), what we mean when we say “a single photon has no phase” and “two photons never interfere with each other,” and how to reconcile those statements with the existence of single- and two-photon interferometers. We will study the Hong-Ou-Mandel interferometer and its many cousins and applications, ranging from tests of Bell’s Inequalities and demonstrations of the Quantum Eraser to applications as a Bell-state filter and for quantum teleportation. This will enable a discussion of the role of indistinguishability in quantum interference. We will talk about various sources of single- and two-photon states and nonclassical light more broadly, as well as techniques for characterizing them and their evolution. We will introduce both linear and nonlinear schemes for implementing quantum logic gates for photons. By the end of the course, we aim to touch on the protocols which underlie quantum-computing efforts such as those pursued by Xanadu and PsiQuantum.
PHY1520H F GENERAL Quantum Mechanics
Prof. John Sajeev
Review of Postulates of Quantum Mechanics
Hilbert Spaces, Density Matrix
Quantum Dynamics
Evolution Operator, Heisenberg vs. Schrodinger Picture
WKB (Semi-classical) Approximation
Coherent States
Electron in a Magnetic Field
Symmetry in Quantum Mechanics
Parity, Time Reversal
Translation, Electron in a Periodic Potential, Bloch’s Theorem
Disordered Lattices and Localized States
Green’s Function Method
Perturbation Theory
Time-independent: Rayleigh-Schrodinger and Brillouin-Wigner series expansions
Time-dependent perturbation theory: Fermi’s Golden Rule, Absorption and Emission of Light from Atoms
Variational Methods
Electron in a Deformable Elastic Medium
Scattering Theory
Lippman-Schwinger Integral Equation
Partial Waves
S-matrix and T-matrix
MAT1739 Introduction to PDE in physics and geometry - techniques and applications
Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.
Joint undergraduate/graduate course - APM421H1/MAT1723H
Prerequisite:
(MAT224H1/ MAT247H1, MAT337H1)/ MAT357H1
CSC2332H Introduction to Quantum Algorithms
This course will provide a rigorous introduction to quantum computing for computer science students. Topics covered will include: the quantum circuit model; quantum query complexity; basic quantum algorithms including Grover's algorithm and amplitude amplification, the quantum Fourier transform and applications, phase estimation, Deutsch Josza, the quantum linear systems algorithm, quantum error correction, and quantum simulation. Modern approaches to quantum algorithm design will also be covered including linear combinations of unitaries, block encodings, and quantum singular value transformations.
Prerequisites: Good knowledge of linear algebra and elementary real and complex analysis
PHY1487H F INTRODUCTORY Quantum Theory of Solids I
Prof. Stephen R. Julian
Introduction to the physics of solids and their electronic and thermal properties. Topics include crystal structure and symmetry, X-ray and neutron diffraction, electronic band structure, lattice vibrations and phonons. Selected advanced topics such as electron interactions and anharmonic effects will be also covered. A good understanding of undergraduate quantum mechanics and statistical mechanics is expected.
Topics:
- Bonding, crystal structure and diffraction.
- covalent, ionic, metallic, van der Waals bonding
- periodic structure, the reciprocal lattice, diffraction condition
- Lattice vibrations
- elementary excitations of a periodic lattice
- Electron wave functions in periodic crystals
- simplifying the many-body hamiltonian
- energy bands, approximate methods, nearly free electrons, tight binding
- relating band structure to properties: metals, semiconductors and insulators
- Electron dynamics
- effective mass tensor, Berry curvature
- Electron-electron interactions (Time permitting)
- Hartree-Fock approximation, local density approximation
MAT1723 Mathematical Foundations of Quantum Mechanics and Quantum Information Theory
Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.
Joint undergraduate/graduate course - APM421H1/MAT1723H
Prerequisite:
(MAT224H1/ MAT247H1, MAT337H1)/ MAT357H1
ECE1385H Selected Topics in VLSI Systems: Quantum Computing Hardware
A review of the principles and practical implementation of quantum processors based on solid state superconducting and semiconductor spin qubits. The focus is on hardware with no overlap with existing Quantum Information or proposed Quantum Algorithms undergraduate EngSci or CompSci courses. A top-down approach is taken starting from the quantum processor architecture and building block specification, to qubit and control and readout circuit modelling, design, fabrication and testing. Topics include the basics of quantum mechanics and quantum computing, superconducting and semiconductor spin qubit physics, fabrication and characterization techniques for qubits, and classical control and readout of qubits. Students will gain hands-on experience with the engineering of a quantum computer, deriving specifications for its quantum and classical hardware building blocks, and designing, modelling, simulating, and testing qubits, control and readout circuits for quantum processors.
PHY2403H F SPECIALIZED Quantum Field Theory I
Prof. Yonatan (Yoni) Kahn
An introduction to Quantum Field Theory and Quantum Electrodynamics. Topics include: Failure of single particle relativistic quantum mechanics, multi-particle quantum mechanics and quantum field theory, canonical and path-integral quantization, symmetries and conservation laws, interacting fields and Feynman diagrams, spin 1/2 fields and the Dirac Lagrangian, gauge invariance and QED.
PHY2203H F SPECIALIZED Quantum Optics I
PHY2203H explores atom-photon interactions with a semi-classical treatment: how does a quantum system respond to a classical drive field? We begin by discussing how an atom driven by an optical field reduces to a dipole interaction Hamiltonian. The atom-photon problem can then be mapped onto a spin one-half electron in a magnetic field, since both are driven two-level quantum systems. We develop the Bloch equations, Rabi oscillations, and magnetic resonance. Returning to the optical regime a treatment using density matrices is necessary to include the effects of damping. Dynamics of the density operator are described by the optical Bloch equations, with which one can understand a wide range of current experiments in atomic, molecular, and optical physics and solid-state physics. These quantum dynamics are contrasted to classical (Lorentz-model) dynamics, such as quantum saturation. In the context of a diagonalized atom-photon Hamiltonian, we discuss inversion, dressed states and light shifts. Applications of this foundational material include electromagnetically induced transparency, slow light, dark states, and laser cooling.
Prerequisite
PHY456 and PHY350, or equivalent
Winter 2026 - Click to expand
Winter 2026
(Organized alphabetically by course title)
PHY1491H S (FAS PHY491H) Current Interpretations of Quantum Mechanics
Realist, phenomenalist, and pragmatist perspectives on scientific theories. Review of conventional textbook quantum mechanics. Local causality and signal locality. Elements of formal measurement theory and wave function collapse; decoherence and the classical/quantum boundary. Copenhagen interpretation of quantum mechanics. Operationalist quantum mechanics. Hidden variable theories: possibilities and problems. Contextuality and nonlocality; Bell’s theorem. Bohm deBroglie theory and generalizations. Modal interpretations and consistent histories quantum mechanics of Gell-Mann, Hartle, Omnes. Relative state interpretations (Everett’s “many worlds,” more recent work by Wallace and Carroll). Quantum Darwinism. QBism. Relational Quantum Mechanics. A sketch of collapse theories.
This cross listed course is offered at together with PHY491S. For course schedule please see: PHY491H1/1491H
Prerequisite
PHY456 or equivalent.
MSE 1073H – Semiconductors for Quantum Information: Theory and Realisation
The course introduces modern aspects of non-classical electron transport including an introduction to single electron transport, Landauer theory, quantum interference in nanostructures; heterostructures that consider such systems including those with topological properties and impose magnetic, superconducting order, and other phenomena into such systems, are discussed. The course also outlines the role of modern semiconductor fabrication technology in realising the required nanostructures suitable for building such devices. Students are required to prepare a paper from a selected set of topics and current literature and present their work to the class. As part of the course, students will visit the Centre for Advanced Nanotechnology, where they will observe modern deposition and processing systems and gain practical insights into semiconductor fabrication techniques. This graduate-level technical course is applicable toward degree requirements for M.Eng., MASc, and Ph.D. students in MSE and ECE.
PHY2404H S SPECIALIZED Quantum Field Theory II
Prof. David Curtin
This QFT2 course continues the study of quantum field theory started with PHY2403 (QFT1) into more advanced topics. Rough list of topics: - path integral formulation of quantum field theory - loops and renormalization - the renormalization group - non-abelian gauge theories - the Higgs mechanism - anomalies - effective field theory, effective potentials.
Prerequisite
- QFT1 (PHY2403) or equivalent, roughly corresponding to Peskin & Schroeder Ch 1-5. - QFT1 prerequisites: Lagrangian and Hamiltonian formulations of classical mechanics; Maxwell equations, energy and momentum of the electromagnetic fields, Lorentz invariance (special relativity and four vectors); Nonrelativistic quantum mechanics, in particular, angular momentum theory will be relied upon.
PHY2204H S SPECIALIZED Quantum Optics II
This course will consider the physics of the quantum electromagnetic field and how it interacts with matter. The matter will sometimes be treated classically, as in many devices used for the generation of nonclassical states of light, and sometimes quantum mechanically, as in the interaction of atoms, molecules, and quantum dots with the quantum electromagnetic field. We will focus both on the basic physics and on applications, such as integrated quantum photonics, Heisenberg-limited interferometry, and quantum information processing.
Prerequisite
PHY2203 (Quantum Optics I)
PHY2303H S SPECIALIZED Quantum Theory of Solids II
Quick review on band theory and introduction of correlated systems
Multi-electron Atoms and Molecules; spin-orbit coupling, exchange interactions, Hund’s rule (Ch. 2)
Crystal Field Theory; point group, Jahn-Teller effect, Kramers theorem, double group (Ch. 3)
Metal-Insulator Transition, Breakdown of Band Theory, and Hubbard Model (Ch. 4)
Mott Insulators (Ch. 5)
Heisenberg Magnets and Spin Liquids (Ch. 6)
Itinerant Electron Magnetism (Ch. 7)
Superconductivity (BCS theory)
Prerequisite
Required background: Quantum mechanics, Statistical mechanics, and Quantum theory of Solid I
I teach the theory of statistical mechanics hand in hand with its applications to molecular and materials simulations, covering both algorithms and hands-on computer implementation exercises. Importantly - and that should be the main appeal to quantum students, tools that we cover in the course have direct quantum extensions with applications to open quantum systems, including Monte Carlo and molecular dynamics simulations, Langevin and Fokker Planck equations and the Master equation formalism.
Past Courses Fall 2024 | Winter 2025
(Organized alphabetically by course title)
CSC 2332 Introduction to Quantum Algorithms
Provides an introduction to modern algorithmic techniques in quantum computing as well as an introduction to the quantum formalism appropriate for computer scientists or mathematicians. Topics covered include: universality of quantum computation, quantum query complexity, block encoding and linear combinations of unitaries, quantum simulation, quantum phase estimation, quantum walks and amplitude amplification and finally the quantum singular value transformation.
PHY2321H F SPECIALIZED Many Body Physics I
Denis Dalidovich
Topics may include: Free fermions (bands and band topology), Linear response theory for many-body systems, Coherent state path integrals for bosons, Superfluidity and superfluid-to-Mott insulator transition, Coherent state path integrals for fermions, Density and spin response of free fermions, Interacting fermions: Collective modes
MAT1723 Mathematical Foundations of Quantum Mechanics and Quantum Information Theory
Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.
ECE1531 Quantum Information Theory
This is a first course on quantum information and communications theory. Topics covered include: (1) basics of quantum mechanics and quantum information, (2) resource model of quantum information processing, (3) entanglement and entanglement distillation protocols, (4) quantum cryptography and security proofs.
CHM 1478H Quantum Mechanics for Physical Chemists
A course in Quantum Mechanics in Hilbert Space with a focus on operator formalism. Covering Schrodinger, Heisenberg and Interaction Pictures, Coordinate, Momentum and Phase Space Representations, Symmetry and Conservation Laws, Angular Momentum Coupling, Density Matrices, Introduction to Entanglement, Non-locality, Bell's Theorem, etc.
PHY2203H F SPECIALIZED Quantum Optics I
PHY2203H explores atom-photon interactions with a semi-classical treatment: how does a quantum system respond to a classical drive field? We begin by discussing how an atom driven by an optical field reduces to a dipole interaction Hamiltonian. The atom-photon problem can then be mapped onto a spin one-half electron in a magnetic field, since both are driven two-level quantum systems. We develop the Bloch equations, Rabi oscillations, and magnetic resonance. Returning to the optical regime a treatment using density matrices is necessary to include the effects of damping. Dynamics of the density operator are described by the optical Bloch equations, with which one can understand a wide range of current experiments in atomic, molecular, and optical physics and solid-state physics. These quantum dynamics are contrasted to classical (Lorentz-model) dynamics, such as quantum saturation. In the context of a diagonalized atom-photon Hamiltonian, we discuss inversion, dressed states and light shifts. Applications of this foundational material include electromagnetically induced transparency, slow light, dark states, and laser cooling.
PHY2109 0.25 FCE Special Topics in Physics - Nuclei
This half-term course is an introduction to the physics of atomic nuclei. The purpose of the course is to explore ways in which the techniques of quantum optics and atomic physics can be applied to the strongly-interacting fermion soup within a nucleus. The selection of topics covered in this seminar course will be idiosyncratic. Background preparation from PHY 2203/04 will be useful, but is not essential.
PHY2108 0.25FCE SECTION Special Topics in Physics: Ultracold Atoms I
The first of the two special-topics courses offered this year on ultracold atoms will have an emphasis on bosons. We begin with a discussion of Bose-Einstein condensation of non-interacting bosons, held in a harmonic trap. We discuss basic experimental techniques, including trapping, evaporative cooling, imaging, and Bragg scattering. With weak interactions, a mean-field treatment leads to the Gross-Pitaevskii equation and Bogoliubov excitations. Time permitting, we will discuss bosons in optical lattices, and under quasi-two-dimensional confinement. Overall, this special topic will allow for an elegant synthesis of optics, atomic physics, statistical mechanics, and field theory.
The course will assume fluency in electromagnetism, quantum mechanics, statistical mechanics, and rudimentary field theory (such as raising and lowering operators). The course will not be based on a textbook, but a good reference is Bose-Einstein condensation in Dilute Gases by Pethick and Smith.
(Organized alphabetically by course title)
PHY1491H S Current interpretations of quantum mechanics
Realist, phenomenalist, and pragmatist perspectives on scientific theories. Review of conventional textbook quantum mechanics. Local causality and signal locality. Elements of formal measurement theory and wave function collapse; decoherence and the classical/quantum boundary. Copenhagen interpretation of quantum mechanics. Operationalist quantum mechanics. Hidden variable theories: possibilities and problems. Contextuality and nonlocality; Bell’s theorem. Bohm deBroglie theory and generalizations. Modal interpretations and consistent histories quantum mechanics of Gell-Mann, Hartle, Omnes. Relative state interpretations (Everett’s “many worlds,” more recent work by Wallace and Carroll). Quantum Darwinism. QBism. Relational Quantum Mechanics. A sketch of collapse theories.
ECE1365S High Frequency Integrated Circuits Design
A design intensive overview of high-speed, RF, mm-wave monolithic, and silicon photonics integrated circuits for wireless, automotive radar sensors, optical fiber systems and quantum processors, with an emphasis on specific high-frequency circuit analysis and design methodologies, device-circuit topology interaction and optimization. Small-signal, noise, large-signal, high-frequency common-mode and differential-mode matching and stability, digital control of tuned circuits, methodologies for maximizing circuit bandwidth, high speed CML gate design, as well as layout and isolation techniques will be discussed. Students will participate in assignments on mm-wave circuits, optical fiber circuits and classical control circuits for qubits using 22nm FDSOI technology and Cadence Analog Artist.
PHY1485H S (FAS PHY485H) Laser Physics
This course covers a broad range of advanced topics in classical optics, with the laser as a unifying theme. Topics include atom-photon interactions (absorption, radiation, and stimulated emission), how a laser works (gain, pumping, rate equation models, threshold, and gain clamping), optical resonators (their spectrum, finesse, stability, and transverse modes), propagation of Gaussian beams and paraxial rays, and the statistics of optical fields (spatial and temporal coherence). Time permitting, pulse propagation and pulsed lasers will be discussed.
CHM1449HS Machine Learning and Physics Based View on Chemical Compound Space
This is an advanced, continuously updated research-oriented course for students with interests in computational and theoretical chemistry/physics/materials. Prerequisites include undergraduate knowledge in terms of: statistical mechanics, computer programming, quantum mechanics, applied math (linear algebra, differential equations), and atomistic simulation.
This course offers an introduction to the concepts underlying nonlinear optical phenomena. Topics include: Basic formalism and classification of nonlinear optical processes through the framework of nonlinear susceptibilities: Non-phase-matched processes (e.g. rectification, Kerr effect, soliton generation, Pockels effect, two-photon absorption, degenerate four-wave mixing); phase-matched processes (parametric conversion, harmonic and difference frequency generation); Raman and Brillouin scattering. Microscopic (quantum) origin of nonlinear susceptibilities, and subtleties associated with their calculation for solids associated with band structure topology. The use of nonlinear optics to generate nonclassical states of light for quantum information processing, particularly in integrated photonics, and connections between the classical and quantum regimes.
Students taking this course should be thoroughly familiar with the material covered in PHY1510 (Electromagnetism) and PHY1520 (Quantum mechanics). It is also recommended that they have taken PHY1485 (Laser Physics) or its equivalent.
PHY2204 Quantum Optics II
This course will examine the physics of the quantum electromagnetic field, and its interaction with other quantum mechanical objects. The broad purpose of the course is to equip students with the tools and background needed to connect with current research in quantum optics. Outline of topics: Quantization of the electromagnetic field; Quantum Coherence Theory; Representation of Quantum States; Squeezed Light; Master Equations; Light Matter Interactions.
PHY2303H S SPECIALIZED Quantum Theory of Solids - II
We will discuss various concepts relevant to a modern understanding of quantum materials, including the physics of Mott insulators, magnetism, superconductivity, and Kondo effect.
MSE1022 Special Topics in Materials Science I: Quantum Transport
The course is concerned with quantum transport and focuses on semiconductor nanostructures. Applications of this concepts are relevant to next generation electronics and quantum computing. The course will provide an introduction to important relevant concepts in solid state physics as well as to the fabrication of such nanostructures. The course will cover structures for electron transmission, tunnelling, and interference. Students will be responsible for preparing a critical review on the current relevant literature, presented as a term paper and a class presentation.
PHY2109H S 0.25FCE Special Topics in Physics: Ultracold Atoms II
The second of the two special-topics courses offered this year on ultracold atoms will have an emphasis on strongly interacting fermions. We start with the scattering problem, to relate interaction strength to the scattering length. The scattering phase can be tuned using Feshbach resonance. We discuss how to generalize thermodynamics to isolated systems, and identify the contact parameter as the conjugate of the scattering length. We then discuss the Cooper problem and fermionic superfluidity. Time permitting, we discuss the Hubbard model and fermions in one dimension.
I teach the theory of statistical mechanics hand in hand with its applications to molecular and materials simulations, covering both algorithms and hands-on computer implementation exercises. Importantly - and that should be the main appeal to quantum students, tools that we cover in the course have direct quantum extensions with applications to open quantum systems, including Monte Carlo and molecular dynamics simulations, Langevin and Fokker Planck equations and the Master equation formalism.
About Quantum Graduate Courses: One objective of CQIQC is to foster interactions between students in QIS at UofT, and expose them to the breadth of topics in QIS beyond their research field. This is achieved through the Centre’s programs and via involvement in graduate curriculum, which is aimed at advocating the adoption of courses and coordinating courses in various departments to ensure a good coverage of topics.