Quantum circuits are the basic model for describing quantum computations. Many body quantum systems are subject to high level of noise, which is the main obstacle for building universal quantum computers. In order to perform useful computations on noisy devices it is necessary to carry out error correction procedures. The development of the theory of fault-tolerance has led to the study of stabilizer circuits. Such circuits are error-proof and they allow maximal entanglement, but they are not universal and, in particular, are efficiently simulated on classical computers. To gain advantage in the computational power of quantum computers, one can add the resource of «magic» to stabilizer circuits. The course is devoted to mathematical description and investigation of computational properties of fault-tolerant quantum circuits.

COURSE PROGRAMME

1. Basic circuit elements: states, gates, measurements, control.
2. Models of probabilistic and quantum computations, the concept of simulation.
3. Pauli group and stabilizer formalism.
4. Clifford group, stabilizer circuits, Gottesman-Knill theorem.
5. Entanglement of stabilizer states, graph states.
6. Stabilizer circuits simulation by quadratic forms expansion.
7. Projective and unitary designs for quantum algorithms.
8. T-gates, Solovay-Kitaev theorem, Clifford hierarchy.
9. Magic states, magic distillation protocols.
10. Strong and weak simulation of stabilizer circuits with magic.
11. Qudit systems, discrete Wigner function.
12. Quasiprobability representations, negativity as a resource.
13. Hidden variable models, contextuality.
14. Simulation of bosonic and fermionic linear optics.

Lecturer
Yashin Vsevolod Igorevich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center