Quantum Tomography with an application to a CNOT gate

1. Quantum Tomography with an application to a CNOT gate

2. Quantum State Tomography • Finite Dimensional • Infinite Dimensional (Homodyne) • Quantum Process Tomography (SQPT) • Application to a CNOT gate • Related topics Outline

3. QST “is the process of reconstructing the quantum state (density matrix) for a source of quantum systems by measurements on the system coming from the source.” • The source is assumed to prepare states consistently Quantum State Tomography

4. Simply put: Do this a lot Quantum State Tomography

5. Typically easier to work with • Know a priori how many coefficients to expect • The value of n is known Finite Dimensional Space

6. . • Easily approached via linear inversion • Ei is a particular measurement outcome projector • S and T are linear operators Finite Dimensional Space

7. . • Use measured probabilities and invert to obtain density matrix • Sometimes leads to nonphysical density matrix! Finite dimensional space

8. “the likelihood of a set of a parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values” The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state Maximum Likelihood Estimation

9. Example from class: 1 qubit Repeatedly measure sigma x QST for one qubit

10. FOUND r1! Finite Dimensional Space

11. The value of n is unknown! • Make multiple homodyne measurements • Obtain Wigner function • Find density matrix Infinite Dimensional Space

12. Homodyne measurements • Analogous to constructing 3d image from multiple 2d slices • Goal is to determine the marginal distribution of all quadratures

13. In QPT, “known quantum states are used to probe a quantum process to find out how the process can be described” Quantum Process Tomography

14. In essence: Quantum Process Tomography

15. In practice: Quantum Process Tomography

16. J.L. O’Brien: “The idea of QPT is to determine a completely positive map ε, which represents the process acting on an arbitrary input state ρ” Am are a basis for operators acting on ρ QPT

17. Choose set of operators: Use input states: QPT

18. Form linear combination Do QST to determine each Write them as a linear combination of basis states QPT

19. Solve for lambda Now write And solve for beta (complex) QPT

20. Combine to get Which follows that for each k: QPT

21. Define kappa as the generalized inverse of beta And show that satisfies QPT

22. Operators Basis QPT for a single qubit

23. Use input states Now QST on output QPT for a single qubit

24. Use QST to determine QPT for a single qubit

25. Results correspond to Now beta and lambda can be determined, but due to the particular basis choice and the Pauli matrices: QPT for a single qubit

26. Finally arriving to: QPT for a single qubit

27. J.L. O’Brien et al used photons and a measurement-induced Kerr-like non-linearity to create a CNOT gate Application to CNOT

28. CNOT

29. Φa are input states Ψb are measurement analyzer setting cab is the number of coincidence detections QPT in practice

30. Average gate fidelity: 0.90 Average purity: 0.83 Entangling Capability: 0.73 Results

31. Ancilla-Assisted Process Tomography (AAPT) • d2 separable inputs can be replaced by a suitable single input state from a d2-dimensional Hilbert space • Entanglement-Assisted Process Tomography (EAPT) • Need another copy of system • Tangle Related Topics

32. “Quantum Process Tomography of a Controlled-NOT Gate” • http://quantum.info/andrew/publications/2004/qpt.pdf • Quantum Computation and Quantum Information • Michael A. Nielsen & Isaac L. Chuang • Wikipedia Sources