Part 3 Circuit complexity of code states | Jeongwan Haah (Microsoft Research)
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 04/08/2023

Topological aspects of quantum codes
Lecture notes 3 Circuit complexity of code states
https://www.ias.edu/sites/default/files/Lecture%203%20Circuit%20complexity%20of%20code%20states.pdf

Perhaps the most important quantum code of all is the surface code for expected practical reasons and for its connection to topological phases of matter. In various aspects the two-dimensional surface code is extreme, which I plan to discuss in these lectures. Specifically, I will explain that (i) local codes in one-dimension cannot achieve large code distance, (ii) two-dimensional local codes can only achieve code distance that is linear in the linear system size L, (iii) the number of logical qubits in d-dimensional local codes can only be O(L^{d-2}), and (iv) two-dimensional translation invariant codes are always surface codes. In the last lecture, I will change the theme and discuss triorthogonal codes that are useful for magic state distillation.

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The 2023 Program: Quantum Computation
Organizers: David Gosset, University of Waterloo; Aram Harrow, MIT; Stacey Jeffery, CWI and QuSoft; Ryan O'Donnell, Carnegie Mellon University; and Thomas Vidick, Caltech.

Very recently we have seen experiments at the boundary of the "quantum computing advantage", where quantum computers can massively outperform classical ones at certain tasks.  These advances highlight the need for further mathematical understanding of the computational power of near-term quantum devices.  The goal of the 2023 GSS is to dive deeply into the mathematics relevant for building near-term quantum computers, analyzing their power, and putting them to use.  Minicourses will include: overviews of quantum learning, information theory, and linear-algebraic algorithms; recent advances in quantum error-correcting codes; and, the complexity theory of random circuits and Hamiltonians.

Structure:
The Graduate Summer School at PCMI consists of a series of several interwoven minicourses on different aspects of the main research theme of that summer.  These courses are taught by leading experts in the field, chosen not only for their stature in the field but their pedagogical abilities. Each minicourse comprises three to five lectures. Each course is accompanied by a daily problem session, structured to help students develop facility with the material.

2023 Schedule
Week 1 
  Andras: Quantum Fourier transform beyond Shor's algorithm
  Omar: Quantum information theory
  Srinivasan: Overview of quantum learning theory
Week 2 
  Ewin: Quantum and quantum-inspired linear algebra
  Nicolas: Quantum LDPC codes
  Yassine: Quantum query complexity
Week 3 
  Bill: Computational complexity of near-term quantum experiments
  Jeongwan: Topological aspects of quantum codes
  Sandy: Quantum Hamiltonian complexity
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The GSS takes place within the broader structure of PCMI, so there are many researchers at all levels in the field in attendance, as well as participants in the other PCMI programs.

ias.edu/PCMI