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Pulse Methods for Preserving Quantum Coherences T. S. Mahesh

Pulse Methods for Preserving Quantum Coherences T. S. Mahesh Indian Institute of Science Education and Research, Pune. Criteria for Physical Realization of QIP. Scalable physical system with mapping of qubits A method to initialize the system Big decoherence time to gate time

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Pulse Methods for Preserving Quantum Coherences T. S. Mahesh

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  1. Pulse Methods for Preserving Quantum Coherences T. S. Mahesh Indian Institute of Science Education and Research, Pune

  2. Criteria for Physical Realization of QIP • Scalable physical system with mapping of qubits • A method to initialize the system • Big decoherence time to gate time • Sufficient control of the system via time-dependent Hamiltonians • (availability of universal set of gates). • 5. Efficient measurement of qubits DiVincenzo, Phys. Rev. A 1998

  3. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  4. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  5. Closed and Open Quantum System Environment Environment Hypothetical

  6. Coherent Superposition An isolated 2-level quantum system | = c0|0 + c1|1, with |c0|2 + |c1|2 = 1 Density Matrix rs = || = c0c0*|0 0| + c1c1*|1 1|+ c0c1*|0 1| + c1c0*|1 0| c0c0*c0c1* c1c0* c1c1* Coherence = Population

  7. Effect of environment Quantum System – Environment interaction Evolution U(t) System  Environment System  Environment ||E  = (c0|0 + c1|1)|E  c0|0|E0 + c1|1|E1 Entangled U(t) |0|E  |0|E0 |1|E  |1|E1 U(t) U(t) System  Environment

  8. Decoherence • = ||E |E| • = c0c0*|0 0||E0 E0| + c1c1*|1 1||E1 E1| + • c0c1*|0 1||E0 E1| + c1c0*|1 0||E1 E0| • rs = TraceE[r] = c0c0*|0 0| + c1c1*|1 1|+ • E1|E0 c0c1*|0 1| + E0|E1 c1c0*|1 0| • c0c0*E1|E0 c0c1* • E0|E1 c1c0* c1c1* Coherence = Population Coherence decays irreversibly |E1(t)|E0(t)| = eG(t) Decoherence

  9. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  10. Signal Decay 13-C signal of chloroform in liquid Signal  x Time Frequency

  11. Signal Decay Decoherence Incoherence Amplitude decay Phase decay Depolarization T1 process T2 process Relaxation

  12. Signal Decay Decoherence Incoherence Amplitude decay Phase decay Depolarization T1 process T2 process Relaxation

  13. Incoherence Individual (30 Hz, 31 Hz) Net signal – faster decay Time

  14. Hahn-echo or Spin-echo (1950) Echo Signal y /2-x   + d  d y Symmetric distribution of  pulses removes incoherence

  15. Signal Decay Decoherence Incoherence Amplitude decay Phase decay Depolarization T1 process T2 process Relaxation

  16. Time to reach equilibrium, (energy of spin-system is not conserved) T1 Lifetime of coherences, (energy of spin-system is conserved) T2 Bloch’s Phenomenological Equations (1940s)

  17. Bloch’s Phenomenological Equations (1940s) Solutions in rotating frame: 0 0

  18. Signal Decay Decoherence Incoherence Amplitude decay Phase decay Depolarization T1 process T2 process Relaxation

  19. Effect of environment ’ = E() = ∑ Ek Ek† (operator-sum representation) k

  20. Amplitude damping (T1 process, dissipative) g(t) is net damping : eg., g(t) = 1  et/T1 1 0 0 (1g)1/2 p 0 0 1  p E0 = p1/2 = 0g1/2 0 0 0 0 g1/20 E3 = (1  p)1/2 E1 = p1/2 (1g)1/20 0 1 E2 = (1  p)1/2 E()= ∑ Ek Ek† k Asymptotic state (t , g  1) : In NMR, p = ~ 0.5 + 104 1 1 + eE/kT

  21. Amplitude damping (T1 process, dissipative) Measurement of T1: Inversion Recovery Equilibrium  Inversion M() = 1 2exp( /T1)

  22. Signal Decay Incoherence Decoherence Amplitude decay Phase decay Depolarization T1 process T2 process Relaxation

  23. Phase damping (T2 process, non-dissipative) g(t) is net damping : eg., g(t) = 1  et/T2 1 0 0 (1g)1/2 0 0 0 g1/2 a 0 0 1-a a b b* 1-a E0 = E1 = = (t) = E()= ∑ Ek Ek† k Stationary state (t , g  1) :

  24. Phase damping (T2 process, non-dissipative) Spin-Spin Relaxation dMx(t)Mx(t) dt T2 = Signal envelop: s(t) = exp( t/T2) Transverse magnetization: Mx(t) Re{01(t)} FWHH = /T2 Bloch’s equation : Solution : Mx(t) = Mx(0) exp( t/T2)

  25. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  26. Carr-Purcell (CP) sequence (1954) Signal y /2y y y        Shorter  is better (limited by duty-cycle of hardware) H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954)

  27. Meiboom-Gill (CPMG) sequence (1958) Signal x /2y x x        Robust against errors in  pulse !!! S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

  28. CPMG Dynamical effects are minimized Dynamical decoupling     Sampling points         1 2 3 4 time j = T(2j-1) / (2N) Linear in j Time Signal CP CPMG No pulse Hahn Echo S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

  29. Dynamical Decoupling (DD) Götz S. Uhrig PRL 98, 100504 (2007) CPMG (1958): Uniformly distributed  pulses Uhrig 2007 (UDD): Optimal distribution of p pulses for a system with dephasing bath j = T sin2 ( j /(2N+1) ) T = total time of the sequence N = total number of pulses

  30. Carr & Purcell, Phys. Rev (1954) . Meiboom & Gill, Rev. Sci. Instru. (1958). Carr Purcell Sequence        time 0 T Was believed to be optimal for N  flips in duration T 1 2 3 4 5 6 7 j = T(2j-1) / (2N) Linear in j Uhrig Sequence Uhrig, PRL (2007)        time 2 1 3 4 5 6 7 0 T Proved to be optimal for N  flips in duration T j = T sin2 ( j /(2N+1) )

  31. Dynamical Decoupling (DD) Hahn-echo (1950) CPMG (1958) PDD (XY-4) (Viola et al, 1999) UDD (Uhrig, 2007) CDDn = Cn = YCn−1XCn−1YCn−1XCn−1 C0 =  (Lidar et al, 2005)

  32. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  33. DD performance ION-TRAP qubits M. J. Biercuk et al, Nature 458, 996 (2009)

  34. DD performance Electron Spin Resonance (-irradiated malonic acid single crystal) J. Du et al, Nature 461, 1265 (2009) Time (s) Time (s)

  35. DD performance Solid State NMR 13C of Adamantane Dieter et al, PRA 82, 042306 (2010)

  36. Dynamical Decoupling in Solids 13C of Adamantane D. Suter et al, PRL 106, 240501 (2011)

  37. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  38. Sources of decoherence – dipole-dipole interaction Randomly fluctuating local fields Spin in a coherent state

  39. Sources of decoherence – dipole-dipole interaction Randomly fluctuating local fields Spin looses coherence

  40. Source of Phase-damping – chemical shift anisotropy B0

  41. Redfield Theory: semi-classical System - > Quantum, Lattice - > Classical System Completely reversible No decoherence System+ Random field (coarse grain)

  42. Auto-correlation Local field X(t) time Auto-correlation function  G() =  X(t) X*(t+)  = dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, ) Fluctuations have finite memory: G() = G(0) exp(||/ c) c  Correlation Time 2c 1+ 2c2  Spectral density J() = G() exp(-i) d = G(0)

  43. Spectral density 2c 1+ 2c2 J() = G(0) J()  c = 1 (after secular approximation)

  44. Spectral density 2c 1+ 2c2 J() = G(0) J()  c = 1 c0c0*eGtc0c1* eGtc1c0* c1c1*  2 J()  2   = d 3 8 15 4 3 8 1 T2 1 T1  J(2) + J() + J(0)  J(2) + J() 0 Dipolar Relaxation in Liquids

  45. Effect of decoupling pulses L. Cywinski et al, PRB 77, 174509 (2008). M. J. Biercuk et al, Nature (London) 458, 996 (2009)   0 Time-dependent Hamiltonian exp(-iH(t) dt) Magnus expansion

  46. Filter Functions Cywiński, PRB 77, 174509 (2008) M. J. Biercuk et al, Nature (London) 458, 996 (2009) |x()|= e() Fourier Transform of Pulse-train F()  F()  2 J()  2   = F() d 0

  47. Filter Functions J() Modified Spectral density: J’() = J() F()  2 J()  2   = F() d 0 Residual area contributes to decoherence Cywiński, PRB 77, 174509 (2008) M. J. Biercuk et al, Nature (London) 458, 996 (2009)

  48. Contents • Coherence and decoherence • Sources of signal decay • Dynamical decoupling (DD) • Performance of DD in practice • Understanding DD • DD on two-qubits and many qubits • Noise spectroscopy • Summary

  49. Two-qubit DD

  50. Two-qubit DD Electron-nuclear entanglement (Phosphorous donors in Silicon) No DD PDD Wang et al, PRL 106, 040501 (2011)

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