brian_Quantum information basics

1. Quantum information basics

2. What is a qubit? • A qubit is a quantum mechanical system with two energy levels. • This system can be manipulated with a control and entangled with other systems. • Generally are in the form of ensembles • Example: electron spin

3. What do we need to use a qubit? 3. Entanglement with other systems 4. Measurement 1. Isolated two-level system 2. State control

4. Qubit problems • Very susceptible to noise • Often “leaky” systems create decoherence times • T1 is the relaxation time of the state to a ground or rest state • T2 is the dephasing time of the state; the amount of time the phase stays intact • There are only so many particles with only two energy levels: most qubits deal with many level systems that may cause interference Question: does a system in the state experience T2?

5. Quantum info conventions dt Quantum algorithms involve precessions and operations

6. Measuring a qubit Let’s say you can only measure a spin ½ system in the Z direction… Easy. Now what if it is on the X-Y plane? T/F: T How do we distinguish these states?

7. Gates for completeness Distinguishing between states of equal θ: Use Hadamard: Note: many measurement schemes work differently than a spin system Other gates needed for completeness include the π/4 (also called S or phase) and π/8 gates (also called T) New problem: Convert a 0 to a 1 (NOT gate) Answer: This is a simple not gate.

8. Gates for completeness (single qubit)

9. Physical meaning of gates: NOT • Generally there are two axes of qubit control (X and Z) • Otherwise known as a “π pulse”, the X gate is just a π rotation about the X axis • Lets explore more complicated pulses

10. Physical meaning of gates: Hadamard • Demonstrates the need for completeness • Sometimes you only have specific gates (maybe X and Z) • Sequence:X, Y, S, S • Note that different times distinguish the X and S gates • How do you execute Y gate with only Z and X coils?

11. Kronecker product: A⊗B Series gates are multiplied, parallel gates undergo Kronecker Product or Tensor Product

12. Entanglement Bits represented as Gates are 4x4 matrices Define C-U gate for any gate U (in operator form, i.e. ) .

13. Back to superconducting qubits Brian

14. Josephson junction Hamiltonian Josephson relations: where is phase difference between the two sides is often just called • Hamiltonian: (1 dropped by shifting 0 to Ej) • Energy of JJ goes like an inductor to the first order • The rest can is the nonlinear part QHO Hamiltonian: From these: And Experimentally:

15. Josephson junction oscillator Hamiltonian • Hamiltonian: (1 dropped by shifting 0 to Ej) • Energy of JJ goes like an inductor to the first order • The rest can is the nonlinear part QHO Hamiltonian:

16. Josephson junction oscillators • Hamiltonian: • One important difference: energylevels are not equally spaced

17. Oscillators compared QHO JJ oscillator https://youtu.be/jUPAeOoZpEU

18. Sources • https://qiskit.org/learn/intro-qc-qh/ • Kjaergaard, M., Schwartz, ... Oliver, W. D. (2020)Superconducting Qubits: Current State of Play Annual Reviews of Condensed Matter Physics 11, 369-395 • Krantz, P., Kjaergaard, M., Yan, F., ... & Oliver, W. D. (2019)A quantum engineer’s guide to superconducting qubitsApplied Physics Reviews, 6(2), 021318 • Clarke, J., & Wilhelm, F. K. (2008)Superconducting quantum bits. Nature, 453(7198), 1031–1042