Mini-course MC4 "Introduction to linear-optical quantum computation", Feb. 2-4, 2026. Instructor: Ernesto Galvão (Universidade Federal Fluminense)
Here's a preliminary version of the course slides.
Lecture 1 on 2/2/2026: Introduction, stabilizer formalism, measurement-based quantum computation
Summary:
Intro to quantum computation platforms, differences between linear optics and other physical systems.
Clifford circuits. Definitions of Pauli and Clifford groups of unitaries. Simulability of Clifford circuits. Magic state injection as a way to upgrade a Clifford circuit to universal quantum computation.
Measurement-based quantum computation (MBQC). One-bit teleportation circuit. Gate teleportation. Concatenating MBQC gates.
Resources for MBQC: graph and cluster states, growing graph states via Pauli measurements.
MBQC application: blind quantum computation.
References: stabilizer formalism and measurement-based quantum computation
Intro to Clifford circuits - D. Gottesman, The Heisenberg representation of quantum computers. https://arxiv.org/abs/quant-ph/9807006
Stabilizer formalism: book “Quantum computation and quantum information” by M. Nielsen and I. Chuang (Cambridge U. Press), section 10.5.
Intro to MBQC – section 2.1 of https://arxiv.org/abs/1309.5675
Graph states - https://arxiv.org/abs/quant-ph/0602096
Growing graph states: fusion based computation https://arxiv.org/abs/2101.09310
Lecture 2 on 3/2/2026: linear optics, multimode interferometers, boson sampling
Summary:
States and transformations of linear optics on Fock states. Dual-rail qubits. Worked out example: the Hong-Ou-Mandel effect.
The Mach-Zehnder interferometer, the quantum interrogation task, also known as interaction-free measurement, or the Elitzur-Vaidman bomb-testing protocol.
Amplitudes of Fock-state experiments with linear optics are permanents of matrices built out of the matrix describing the linear-optical transformation. Complexity of computation of permanents. Partial photonic indistinguishability.
Boson sampling as a proposal for demonstration of quantum computational advantage. Overview of integrated photonic interferometers, reconfigurability, single-photon sources, Gaussian boson sampling.
References:
Intro to LO, boson sampling, computational complexity. D. J. Brod, Revista Brasileira de Ensino de Física, vol. 43, suppl. 1, e20200403 (2021).
https://www.scielo.br/j/rbef/a/JDtGnH8jDn8yCKwXBfFXMpR/?format=pdf&lang=en
Boson sampling, theory and experiments overview. D. J. Brod et al., Photonic implementation of boson sampling: a review.
Elitzur-Vaidman bomb-testing set-up using a Mach-Zehnder interferometer. A. C. Elitzur, L. Vaidman "Quantum mechanical interaction-free measurements“, Foundations of Physics. 23 (7): 987–997 (1993).
Sylvester interferometers: A. Crespi, https://arxiv.org/abs/1502.06372
Quantum flytrap - a fun online linear-optical simulator:
https://lab.quantumflytrap.com/lab/mach-zehnder?mode=waves
https://www.quandela.com/our-tutorials/
https://strawberryfields.ai/photonics/demos/run_boson_sampling.html
Lecture 3 on 4/2/2026: Fusion gates, scalable architectures, classical simulation techniques
Summary:
We reviewed why Bell measurements are important for fusion gates, and reviewed some results on how to implement a Bell state measurement for dual-rail qubits. We worked out part of the action of a 50%-efficiency Bell state measurement, and left the remainder of the calculation as an exercise. How a more complex set-up using 4 auxiliary single photons implements a boosted Bell state discriminator, with 75% efficiency.
Review of two different approaches to quantum simulation. The first, the Schrodinger approach, stores and updates the wave-function after each round of gates. We then looked at the Feynman approach, which computes different computational paths, assigning an amplitude for each, and summing over amplitudes corresponding to different paths to obtain the final amplitude.
We then discussed how to apply the Feynman sum over paths to multimode interferometers, first with two locally connected layers of beam-splitters, then two non-locally connected layers of beam splitters, then more than two layers of BSs. We went over the idea of using tensor contraction to make the calculation faster, at the cost of increasing the memory cost as we increase the number of layers.
If time allows, I will talk a bit about continuous-variable photonic systems, and how one can try to implement photonic quantum computation with them, in particular two approaches: one using Gottesman-Kitaev-Preskill (GKP) encoding as pursued by Canadian company Xanadu; and one using CV cluster states.
References:
Original proposal of photonic fusion gates. D. Browne, T. Rudolph, Physical Review Letters 95, 010501 (2005).
Improved fusion gates with efficiency ¾: Ewert and van Loock, https://arxiv.org/abs/1403.4841
Scalable architectures based on fusion gates: S. Bartolucci et al. Fusion-based quantum computation. Nat. Commun. 14, 912 (2023).
Feynman paths for simulation of quantum computers, interpolation between Schrodinger and Feynman schemes: Aaronson and Chen https://arxiv.org/abs/1612.05903 (see section 4).
Short review article on photonic quantum computation: Romero and Milburn, https://arxiv.org/abs/2404.03367v2
Preprint on Feynman path simulation for linear optics: W. F. Balthazar, Q. M. B. Palmer, A. E. Jones, J. F. F. Bulmer, E. F. Galvão. Preprint arXiv:2510.26408 [quant-ph].
Xanadu's 2025 demonstration of networked CV photonic quantum computing: https://www.nature.com/articles/s41586-024-08406-9
Perspective article on photonic QC using CV systems: Takeda, Furusawa, APL Photonics 4, 060902 (2019).