1. BASIS Foundation Summer School 2018 "Many body theory meets quantum information" Simulation of many-body physics with existing quantum computers Walter Pogosov Dukhov Research Institute of Automatics (VNIIA), Rosatom State Corporation In collaboration with A. Zhukov (VNIIA), E. Kiktenko (Steklov Inst. RAS & RQC), A. Elistratov (VNIIA), S. Remizov (VNIIA), and Yu. E. Lozovik (VNIIA & Inst. Spectroscopy RAS)

2. Outline 1. Quantum logic gates 2. Simulation of spin models 3. Simulation of fermionic models 4. Introduction to the error correction 5. Analog simulation of nonadiabatic cavity quantum electrodynamics

3. 1. Quantum logic gates Walter Pogosov Dukhov Research Institute of Automatics (VNIIA), Rosatom State Corporation

4. Introduction / Overview Dramatic progress in the construction of quantum computers and simulators based on different physical realizations (superconducting Josephson circuits, trapped ions, neutral atoms, etc.) Georgescu et al RMP (2014)

5. Quantum computers and simulators are believed to be extremely useful, for example, in simulation of many-body systems: novel materials, quantum chemistry, drugs. Arguments: first-principle simulation of quantum systems is difficult due to the exponential explosion of Hilbert space size (2 N states for spin systems). (Yu. Manin 1980, R. Feynman 1982) Georgescu et al RMP (2014)

6. Strengths and weaknesses of various physical realizations

7. IBM Quantum Experience 5-qubit chip (composer)16-qubit chip (QISKIT) 20 qubit chip (IBM Q Network) Composer https://quantumexperience.ng.bluemix.net/qx/editor

8. Bloch sphere representation Probabilities: A convenient representation

9. Pauli-X gate

10. Hadamard gate Hadamard gate produces quantum superpositions Rotation on pi around x-axis followed by rotations on pi/2 around y-axis

11. Measurements In general case Using multiple measurements, we can extract theta, but not phi

12. Measurements In general case By measuring in X basis, we can extract also phi. Apply H before the standard measurement The state of the qubit after H: The probability to measure 0:

13. Pauli-Y gate Pauli-Z gate

14. Rotations around x, y, z IBM Q single qubit gates

15. Toy model Hamiltonian of a single spin Evolution operator The action of the evolution operator is equivalent to the rotation An example of the circuit:

16. Two-qubit gate CNOT Control qubit and target qubit CNOT does not change the state of the target qubit provided control qubit is in the state 0 CNOT inverts the state of the target qubit provided control qubit is in the state 1

17. CNOTs on real quantum chips

18. A warm-up exercise Prove that this circuit gives an identity gate, which does not change an input state

19. Bell states Two-qubit states with the maximum entanglement (non separable states)

20. Preparation of Bell states Basic circuit:

21. Preparation of Bell states Basic circuit:

22. Preparation of Bell states Basic circuit: -CNOT produces entanglement between qubits in the computational basis. -Entanglement is a key recourse for quantum computation and quantum technologies, in general. -In quantum modeling CNOT is crucial for the simulation of many-body systems.

23. Measurement in Bell basis

24. Exercise: measurement in the Bell basis for Schrödinger’s cat?

25. Measurement in Bell basis

26. A set of quantum logic gates - CNOT produces entanglement between qubits. -It is crucial for algorithmic simulation of many-body systems. -With arbitrary single-qubit unitarities and CNOT you can construct a universal quantum computer (an arbitrary unitary operator can be realized). Fidelities in SC realization -Single-qubit gates: Fidelities are already high – 0.999… -Two-qubit gates: error rates in multiqubit chips are relatively large, > 1 %. Progress is very slow if any. Scaling? Architecture problem - Bottleneck! -Readouts: Errors are also relatively large, > 1 %. Progress is slow. -Error rates are very important characteristics of a quantum computing hardware apart of longitudinal and transversal relaxation times of qubits

29. Exercise: SWAP (interchange of quantum states of two qubits) IBM Q User Guide

30. Exercise: inversion of CNOT IBM Q User Guide

31. QISKIT Open-source framework for quantum computing A Python library for the IBM Quantum Experience https://github.com/Qiskit

32. We acknowledge use of the IBM Quantum Experience for this work. The viewpoints expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience team.

33. Pavlov’s dog?