1. Quantum simulators and hybrid algorithms
Aleksei Dmitriev, LAQS, MIPT
Введение. Зачем нужен семинар. Формат свободной дискуссии. Введение в тему.

2. Contents A (very) brief introduction to quantum computing
Algorithms and simulators
Hybrid (classical-quantum) algorithm: Variational Quantum Eigensolver (VQE)
Quantum Chemistry
Quantum Subspace Expansion (QSE) algorithm
Paper
Хотелось сделать введение но тема очень обширна.

3. Quantum computer Quantum Turing Machine
Quantum circuit model - equivalent
Set of machine’s states Q – Hilbert space of head (qubit?)
Alphabet symbols Γ – H.s. of states of individual cells (qubits)
Blank symbol b ∈ Γ - zero state vector
Transition function δ: Q ⊗ Γ → Q ⊗ Γ x {L,R,N} realized, e.g., by interaction of a head qubit with a cell qubit
Компьютер. Машиа Тьюринга – все умеет считать. Квантовый аналог? Их есть у меня?
David Deutsch

4. Quantum computer
Solves arbitrary tasks (classical or quantum) and requires long coherence and evolution of many qubit system
Shor’s algorithm: factorization O((log N)2(log log N)(log log log N)) = 104 gates for 100- digit N
Quantum phase estimation: for given H define its eigenvector |ψa〉and eigenvalue λ O(p-1) «evolutional» ( ) operations for precision p
Quantum simulators: to mimic an evolution of arbitrary system Ha
Peter Shor
Чем занимается компьютер. Чем плохи квантовые алгоритмы. Есть еще симуляторы. Это не совсем алгоритмы. 4

5. Quantum simulator
Simulator without interactions (only adjusting knobs) Allows to simulate an evolution like where is algebra generated by commutation
Any (nonlinear) interaction – arbitrary evolution!
Simulation of unitary evolution of N qubits is not in fact efficient
logic operations
System with local interactions and many others total number of operations is grows linearly with N and t
Seth Lloyd 5

6. Hybrid algorithms
Idea: is basis in 2x2 complex Hilbert space polynomial number of terms O(P(N)): quantum Ising, Heisenberg models, any k-sparse Hamiltonian
Separate optimization all of reduced states – QMA Hard
we dramatically reduce the coherence time requirement while maintaining an exponential advantage over the classical case, by adding a polynomial number of repetitions with respect to QPE costs local qubit measurements
Quantum expectation estimation (QEE) 6

7. VQE algorithm
Ground state vector: is minimized classically (but using QEE as a subroutine)
Good anzatz is needed for the parametrization (only polynomial number of parameters!) Jordan-Wigner Transformation – from fermionic operators to qubits 7

8. VQE algorithm 8

9. Quantum chemistry 9 Exact solution – nearly impossible LCAO
Slater-type orbitals (STO):
Calculation of kinetic energy, nuclear attraction, Coulomb repulsion – integrals – hard computation
STO-nG basis – to simplify computation, STO is presented as a combination of Gaussians 9

10. Quantum chemistry: H2 molecule
g depend parametrically on R 10

11. Concept of experiment 11

12. Measurement scheme

13. B-swap gate 12
S. Poletto et al. PRL 2012

14. Particle swarm optimization
Avoids to be trapped in local minima
Robust to noisy functions
Very flexible. Swarm interaction is adjustable 13

15. Results of convergence
VQE has intrinsic property to correct coherent errors 14

16. Quantum subspace expansion
VQE: gives only ground state. How to compute excited? fermionic representation qubit representation minimizes Rayleigh quotient 15

17. Main result: a spectrum of molecule16

18. Result of QSE protocol 17

19. QSE with nonunitary evolution18