Quantum State Protection and Transfer using Superconducting Qubits

1. Quantum StateProtection and Transferusing Superconducting Qubits Dissertation Defense of Kyle Michael Keane Committee: Department of Physics & Astronomy Alexander Korotkov June 29, 2012 Leonid Pryadko VivekAji

2. Journal Articles • A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A, 81, 040103(R), April 2010. • K. Keane and A. N. Korotkov, “Simple quantum error detection and correction for superconducting qubits,” arxiv:1205.1836, May 2012 (submitted to Phys. Rev. A). APS March Meeting Presentations • “Decoherence suppression of a solid by uncollapsing,” Portland, OR, March 15-19, 2010), Z33.00011. • “Currently realizable quantum error correction/detection algorithms for superconducting qubits” Dallas, TX, March 21-25, 2011), Z33.00011. • “Modeling of a flying microwave qubit” Boston, MA, Feb. 27-March 2, 2012, Y29.00010 Posters • “Theoretical analysis of phase qubits,” Quantum Computing Program Review (Minneapolis, MN, 2009) • “Suppression of -type decoherence of phase qubits using uncollapsing and quantum error detection/correction,” Coherence in Superconducting qubits(San Diego, CA, 2010)

3. Outline • Introduction • Decoherence by uncollapsing Korotkov and Keane, PRA 2010 • Repetitive N-qubit codes and energy relaxation Keane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012 • Two-qubit quantum error “correction” and detection Keane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012 • Qubit state transfer Keane and Korotkov, APS March Meeting, 2012 • Summary

4. Let’s begin with a basic introduction

5. Superconducting Phase Qubits state control Iμw meas. pulse Imeas flux bias Ib SQUID readout Isq Vsq δ δ quantum variable U qubit  I0 L C flux bias operation flux bias Z-rotations  |1 ΔU U |0 SQUID microwaves X-, Y-rotations  25 mK

6. State Measurement SQUID-based Measurement: Tunneling Not Detected = state has been projected onto |0 Tunneling Detected = state has been projected onto |1 and destroyed lower barrier for time t |0 relaxes |1 U readout w/ SQUID 

7. Weak Measurement Tunneling Detected = state has been projected onto |1 and destroyed Tunneling Not Detected = state projected onto|0 OR state was |1 and didn’t have enough time to tunnel lower barrier for shorttime t |0 relax There is a small change to the energy spacing during the lowering of the barrier |1 U readout w/ SQUID 

8. Uncollapsing State Prepared Partial Measurement Partial Measurement π-pulse π-pulse Projects state toward 0 Projects state toward 0 (was 1) Doesn’t Tunnel Doesn’t Tunnel If tunneling does not occur, the qubit state is recovered In experiment, only data for cases where tunneling does not occur is kept

9. Zero-Temperature Energy Relaxation The population of the excited state moves into the ground state |1 |0 This can be “unravelled” into discrete outcomes with probabilities

10. Project One Decoherence suppression by uncollapsing Korotkov and Keane, PRA 2010

11. Protection from Energy Relaxation Standard methods to protect against decoherence: • Quantum Error Correction (Shor/Steane/Calderbank circa 1995) • Requires larger Hilbert space and controllable entanglement) • Decoherence-Free Subspaces(Lidar 1998) • Requires larger Hilbert space and specfic subspaces • Dynamical Decoupling (Lloyd and Viola 1998) • Does Not Protect Against Markovian Processes (Pryadko 2008) Our proposed method • Simple modification of uncollapsing procedure • Our proposal was demonstrated in another system • Requires selection of only certain cases • Similar to probabilistic QEC and linear optics QC

12. Ideal Procedure Returned to initial value Similar protection for all density matrix elements 11 Initial value axis of π-rotation time storage period π-rotation π-rotation Prepared Partial meas. (p) Partial meas. (pu) Korotkov and Keane, PRA 2010

13. Results Yields a state arbitrarily close to initial Fidelity Some improvement even with naive uncollapsing strength Measurement Strength (p) Korotkov and Keane, PRA 2010

14. Process with Decoherence Returned to initial value Pure dephasing and energy relaxation during entire process 11 Initial value axis of π-rotation time storage period t π-rotation π-rotation Prepared Partial meas. (p) Partial meas. (pu) Korotkov and Keane, PRA 2010

15. Results Pure dephasing uniformly decreases fidelity Still works with relaxation during operations Fidelity and Probability Explains phase qubituncollapsing experiment (Katz, 2008) Perfect suppression requires small prob. of success Measurement Strength (p) Korotkov and Keane, PRA 2010

16. Experimental Demonstration Optical Circuit Results Weak Measurement polarization beam splitter, half wave plate, and dark port Nearly exact match to theory Relaxation similar components, (except no dark port) Jong-Chan Lee, et. al., Opt. Express 19, 16309-16316 (2011)

17. Protecting Entanglement Initially entangled state Entanglement is recovered WM π WM π Q1 Q1 Q2 Q2 WM π WM π Circumvents Entanglement Sudden Death Same optics group did this extension experiment Yong-Su Kim, et. al., Nature Physics, 8, 117-120 (2012)

18. Summary • Does not require a larger Hilbert space • Modification of existing experiments in superconducting phase qubits • Demonstrated using photonic polarization qubit • Extended to protect entanglement

19. Project Two Repetitive coding and energy relaxation • Keane and Korotov, arxiv 2012

20. Motivation Bit Flip T1 A bit flip looks like a more difficult error process than T1 AND PROTECTS Repetitive coding protects against bit flips ????????? THEREFORE…

21. Repetitive Quantum Codes and Energy Relaxation | T1(i) tomography Syndrome Result All “N-1” are 0: good Any in 1: either discard (detection) or try to correct (correction) |0N-1 X X FAILS c-X gate (cNOT) Encoding by N c-X gates | |0 | | cNOT| |0 |0 |0N-1 X |0 cNOT|

22. Two-Qubit Encoding syndrome Fidelity Two qubits Equal decoherence strength T1(i) Decoherence Strength (p) • Keane and Korotov, arxiv 2012

23. N-Qubit Error Detection | T1(i) tomography All “N-1” are 0: keep Any in 1: discard |0N-1 X X Fidelity detect Since the procedure works ignore single optimal, but 2 qubits are sufficient p Decoherence Strength (p) • Keane and Korotov, arxiv 2012

24. N-Qubit Error Correction | T1(i) tomography All “N-1” are 0: keep Any in 1: cannot correct! |0N-1 X X QEC is impossible In our paper we show that no unitary operation can improve the fidelity for p<0.5 Fidelity correct ignore single p Decoherence Strength (p) • Keane and Korotov, arxiv 2012

25. Summary • Can be used for QED, but not for QEC of energy relaxation • 3 qubits are optimal, but 2 qubits are sufficient

26. Project Three Two-qubit quantum error detection/correction • Keane and Korotov, arxiv 2012

27. Two-Qubit Error “Correction”/Detection Notations: | E1 tomography = c-Z |0 Y/2 -Y/2 E2 0: good 1: either discard (only detection) or correct (if know which error) E1 = X-rotation of main qubit by arbitrary angle 2: good X-correction needed E1 = Y-rotation of main qubit: good E2 = Y-rotation of ancilla qubit: Y-correction needed good E2 = Z-rotation of ancillaqubit: Z-correction needed no correction needed (insensitive) • Keane and Korotov, arxiv 2012

28. Two-Qubit Error “Correction”/Detection Notations: | E1 tomography = c-Z |0 Y/2 -Y/2 E2 0: good 1: either discard (only detection) or correct (if know which error) Various Decoherence Strengths All Four Errors Fidelity Fidelity det det corr corr ign ign Rotation Strength (2θ/π) Rotation Strength (2θ/π) • Keane and Korotov, arxiv 2012

29. QED for Energy Relaxation store in resonators Notations: 0: good 1: discard | T1 = c-Z tomography |0 Y/2 -Y/2 Y/2 -Y/2 T1 Almost “repetitive” QED of real decoherence Fidelity The fidelity is improved by selection of measurement result 0 detect ignore Relaxation Strength Keane and Korotov, arxiv 2012

30. Summary • QEC is possible for intentional errors • QED is possible for energy relaxation • Experiments can be done with superconducting phase qubits

31. Project Four Quantum State transfer Keane and Korotov, APS 2012

32. System Initially here Sent here High-Q Storage Tunable Couplers Example from UCSB Resonator or Phase Qubit Transmission Line 1 Tunable Inductance and are tuned by varying Superconducting Waveguide 0 Tunable Parameter Korotkov, PRB 2011

33. Ideal Procedure Qubit transferred to here Qubit initially is here Transmission Coefficients Time (t) Duration ns resonators, Typical parameters (UCSB) Desired Efficiency ON/OFF Korotkov, PRB 2011

34. Main idea “into line” A B “into resonator” PERFECT TRANSFER B A Transmission line Receiving resonator transmission coefficient resonator round trip time Korotkov, PRB 2011

35. Procedural Robustness Transmission Coefficients We vary the parameters in the above equations: , , and Time (t) Keane and Korotkov, APS 2012

36. Shaping of Control Time (t) Transmission Coefficients Robustness shaping error efficiency loss No Problem! of 33.3 ns is 2.5 ns) Keane and Korotkov, APS 2012

37. Switching Time Vary and together Vary only Time (t) Transmission Coefficients Robustness timing error efficiency loss ns No Problem! of 230 ns is 11.5 ns) Keane and Korotkov, APS 2012

38. Maximum Transmission Coefficient Vary only Vary and together Time (t) Transmission Coefficients Robustness amplitude error efficiency loss No Problem! (experiments have good control of tunable coupler) Keane and Korotkov, APS 2012

39. Frequency Mismatch resonator frequency

40. Frequency Mismatch Robustness frequency error efficiency loss Requires Attention (resonator frequencies should be kept nearly equal throughout procedure) Keane and Korotkov, APS 2012

41. Summary • Robust to procedural errors (timing, shaping, maximum transmission coefficient) • Requires active maintenance of nearly equal resonator frequencies The second conclusion is very important for experiments — For the current solid-state tunable couplers there is an effective frequency shift during modulation of the transmission coefficient

42. recapitulation Closing remarks

43. Summary • Decoherence suppression by uncollapsing • Probabilistically suppresses Markovian energy relaxation • After our proposal, it was demonstrated by another group • Extended in another experiment to entangled qubits • N-qubit repetitive codes and relaxation • Can be used for QED, but not for QEC (2 qubits are sufficient) • Two-qubit “QEC”/QED experiments • Can be performed with current technology • Quantum state transfer • Robust against procedural errors • Requires resonator frequencies to be kept nearly equal THANK YOU!

44. Just in case appendices

45. Representations of Errors-Example: Energy Relaxation Master Equation RETURN Need to derive this from commutator!!!!! Need to derive this from somewhere!!!!! From the normalization requirement Solving these equations and combining into an operation Choosing a specific operator sum decomposition If you initially have a pure state, the classical mixture created by this process becomes explicit LINK This can be done for any operation however only some give physically meaningful interpretations