1. Ancilla-Assisted Quantum Information Processing Indian Institute of Science Education and Research, Pune T. S. Mahesh

2. Acknowledgements Abhishek Shukla Swathi Hegde Hemant Katiyar Koteswara Rao Manvendra Sharma Ravi Shankar Prof. Anil Kumar Dr. Vikram Athalye Prof. Usha Devi Prof. A. K. Rajagopal PhD students MS students Collaborators

3. system ancilla Ancillary staff: Provide necessary support to the primary activities or operation of an organization, system, etc. Dictionary meaning: ancilla system

4. 1.Spin-Systems and NMR 2.Measurements a.Extracting expectation values b.Extracting probabilities c.Noninvasive measurements d.Ancilla Assisted State-Tomography e.Ancilla Assisted Process-Tomography 3.Quantum Simulations a.Particle in a potential b.Introducing quantum noise 4.Phase Encoding (Quantum Sensors) a.Diffusion in liquids b.Mapping-out electromagnetic fields 5.Summary Outline

5. Nuclear Spin and Magnetic Resonance Spin ½ (qubit) Chloroform B0B0 EM energy (Radio waves) 00 11 1H1H1H1H

6. Nuclear Spin and Magnetic Resonance B0B0 EM energy (Radio waves) 00 11

7. NMR Signal  x   Tr[   x ] Net transverse magnetization xx Procedure: Prepare  xx t Nuclear Spin and Magnetic Resonance

8. Ancilla assisted measurement: 1H1H1H1H 13 C Prepare  Prepare |+  Prepare |+  A1A1 A2A2 xx System qubit Ancilla qubit  x  =  A 1 A 2  A m    AmAm O. Moussa et al, PRL,104, 160501 (2010) Prepare  Prepare |+  Prepare |+  A xx System qubit Ancilla qubit  x  =  A   Unitary observable

9. Example: Evaluating Leggett-Garg inequality t = 0 tt2t2t x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ time  x (0)  x (  t)  = C 12  x (  t)  x (2  t)  = C 23  x (0)  x (2  t)  = C 13 00 00 00 Hamiltonian : H = ½  z Macrorealistic: K 3 = C 12 + C 23  C 13  1 For spin ½ : K 3 = 2cos(  t)  cos(2  t) (-3  K 3  -1.5) Athalye, S. S. Roy, TSM, PRL-2011  t 1H1H1H1H 13 C A. J. Leggett and A. Garg, PRL-1985 Johannes Kofler, PhD Thesis, 2004

10. Example: Evaluating Leggett-Garg inequality 1H1H1H1H 13 C Athalye, S. S. Roy, TSM, PRL-2011 t = 0 tt2t2t x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ time  x (0)  x (  t)  = C 12  x (  t)  x (2  t)  = C 23  x (0)  x (2  t)  = C 13 00 00 00 Hamiltonian : H = ½  z Macrorealistic: K 3 = C 12 + C 23  C 13  1 For spin ½ : K 3 = 2cos(  t)  cos(2  t) (-3  K 3  -1.5) A. J. Leggett and A. Garg, PRL-1985 Johannes Kofler, PhD Thesis, 2004

11. Extracting probabilities (in computational basis) crusher incoherence convert measure Arbitrary 1q density matrix Diagonal density matrix Single quantum density matrix xx Prepare  t U U U (dephasing channel)

12. Extracting joint probabilities t t+  t System qubit q(t) q(t+  t) p( q(t),q(t+  t) ) ? U(  t) xx System qubit Ancilla qubit Prepare  Prepare |0  xx U(t) Suppose Q be an observable, with eigenvalues q = 0 or 1 

13. Extracting joint probabilities: Noninvasive method (Negative Result) Suppose Q be an observable, with eigenvalues q = 0 or 1 t t+  t System qubit q(t) q(t+  t) p( q(t),q(t+  t) ) ? U(  t) xx System qubit Ancilla qubit Prepare  Prepare |0  xx U(t) U(  t) xx System qubit Ancilla qubit Prepare  Prepare |0  xx U(t) Discord q = 1 --------------------- p(0,0) & p(0,1) Discord q = 0 --------------------- P(1,0) & p(1,1)

14. p(q 1,q 2 )p(q 1,q 3 ) time Q1Q1 Q2Q2 Q3Q3 t2t2 t3t3 t1t1 Hemant, Abhishek, Koteswar, TSM, PRA-2013 Extracting joint probabilities C H system ancilla

15. Entropic Leggett-Garg Inequality Information Deficit: time Q1Q1 Q2Q2 Q3Q3 t2t2 t3t3... t1t1 System state: 1 /2 Dynamical observable : S z (t) = U t S z U t † Time Evolution: U t = exp(i  S x t) Hemant, Abhishek, Koteswar, TSM, PRA-2013 C H system ancilla A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, PRA-2013

16. Reason for LGI violation: Classical Probability Theory: P’(q 1,q 2 ) = P(q 1,q 2,q 3 )  q3q3 P’(q 1,q 3 ) = P(q 1,q 2,q 3 )  q2q2 P’(q 2,q 3 ) = P(q 1,q 2,q 3 )  q1q1 P(q 1,q 2 )  P(q 1,q 3 )  P(q 2,q 3 )  MarginalsGrand Quantum systems do not obey this rule !! A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, PRA-2013

17. Extracting GRAND probabilities: Suppose Q be an observable, with eigenvalues q = 0 or 1 0 tt System qubit Q(0) q(  t) p(q(0),q(  t), ,q(n  t)) ? (n-1)  t q((n-1)  t) ntnt Q(n  t) xx System qubit n ancilla qubits xx U(  t) Prepare  Prepare |0  U(  t)

18. Illegitimate Joint Probability P(q 1,q 2,q 3 ) is illegitimate !! Violation of Entropic LGI Hemant, Abhishek, Koteswar, TSM, PRA-2013

19. Quantum State Tomography Tomography:

20. Quantum State Tomography Complete characterization of complex density matrix - Requires a series of measurements all starting from same initial condition =+ Obtained by measuring  z Obtained by measuring  x and  y 9 different experiments carried out 3-unknowns 15-unknowns Measure:  x (1)  |0  0|,  x (1)  |1  1|, |0  0|   x (2), |1  1|   x (2), After rotations: II, XI, YI, IX, IY, XX, XY, YX, YY Complex signal of Two-qubits

21. Quantum State Tomography: Scaling n-qubit system: n 2 n Number of experiments~ Observables per experiment 2 2n Number unknowns in the density matrix = n 2n2n n-qubits number of experiments 2 2 3 4 7 11 19 2 n x 2 n density matrix

22. System qubits  System qubits  ancilla qubits |00…0  ancilla qubits |00…0  System qubits |00…0  System qubits |00…0  ancilla qubits U comp System qubits ancilla qubits U tomo xx Ancilla Assisted Quantum State Tomography: (n+a)-qubit system: n 2 (n+a) Number of experiments~ Observables per experiment 2 2n Number unknowns in the density matrix = n 2 n - a Nieuwenhuizen & coworkers, PRL-2004

23. Ancilla Assisted Quantum State Tomography: Scaling (a) (n) n 2 n - a Abhishek, Koteswar, TSM, PRA-2013

24. Ancilla Assisted Quantum State Tomography: Fidelity: 0.95 3-system qubits, 2-ancilla qubits Abhishek, Koteswar, TSM, PRA-2013

25. Ancilla Assisted Quantum State Tomography: Noisy Measurements Abhishek, Koteswar, TSM, PRA-2013

26. Quantum Process Tomography: - Characterizes the process (unitary or nonunitary) Standard method: 1 1 1 1   matrix    tomo b1b1 b2b2 b3b3 b4b4  (  ) =  mn E m  E n †  mn

27. Ancilla Assisted Process Tomography: - Characterizes the process (unitary or nonunitary) Using a single ancilla qubit 11 11  matrix  (on system) tomo  (  ) =  mn E m  E n †  mn Altepeter et al, PRL-2003

28. Single-Shot Process Tomography: - Characterizes the process (unitary or nonunitary) Using two ancilla qubits 11 11  matrix process (on system) xx  (  ) =  mn E m  E n †  mn

29. Schrodinger equation: iħ (d/dt) |  (t)  = H |  (0)  |  (t)  = exp(-iHt) |  (0)  H = T + V Kinetic P 2 /2m Potential Do not commute exp(-i H dt)  exp(-i V/2 dt). exp(-i T dt). exp(-i V/2 dt) Trotter approximation: Quantum Simulation: Particle in a potential (1D)

30. (with spin-1/2 nuclei) |111  |110  |101  |100  |011  |010  |001  |000  x exp(-i H dt)  exp(-i V/2 dt). exp(-i T dt). exp(-i V/2 dt) Circuit for Diagonal Unitary Trotter form: Quantum Simulation: Particle in a potential (1D) exp(-i H dt)  exp(-i V/2 dt).U iqft. exp(-i T’ dt). U qft. exp(-i V/2 dt) position

31. Ancilla Assited Quantum Simulation: Initial state Final state (after Simulation) Ravi Shankar, Swathi Hegde, TSM, PLA-2013

32. Ancilla Assited Quantum Simulation: Ravi Shankar, Swathi Hegde, TSM, PLA-2013 ExperimentsTheory

33. chloroform 1 H (system) 13 C (ancilla: environment) System Ancilla Time System Ancilla Time kicks Cory & coworkers PRA, 2003 Simulating quantum noise:

34. chloroform 1 H (system) 13 C (environment) Simulating quantum noise: Has applications in optimizing dynamical decoupling sequences Swathi & TSM (on-going work)

35. Measuring diffusion B0B0 |0  +|1  |0  +e i  |1  Price, Concepts in NMR-1997

36. Measuring diffusion B0B0 |0  +|1  |0  +e i  |1  Price, Concepts in NMR-1997

37. 31 P Trimethylphosphite (300 K, DMSO, fixed conc.) Measuring diffusion Abhishek, Manvendra, TSM, CPL-2013

38. B0B0 |0…0  +|1…1  |0…0  +e in  |1…1  Measuring diffusion: NOON states 31 P Trimethylphosphite (300 K, DMSO, fixed conc.) Preparing NOON states Converting to single-quantum states Abhishek, Manvendra, TSM, CPL-2013 10-qubits

39. 31 P Trimethylphosphite (300 K, DMSO, fixed conc.) Measuring diffusion: NOON states Abhishek, Manvendra, TSM, CPL-2013

40. Mapping RF Intensity with NOON states: Abhishek, Manvendra, TSM, CPL-2013 31 P

41. Summary: Ancilla qubits play an important role in practical quantum processors  Provide efficient ways to measure expectation values and joint probabilities  Assist in Quantum State Tomography and Quantum Process Tomography  Assist in direct read-out of probabilities in quantum simulation  Can induce controlled quantum noise on the system qubits  Can participate in preparing large NOON states – have applications in quantum sensors