Pulse Methods for Preserving Quantum Coherences T. S. Mahesh Indian Institute of Science Education and Research, Pune Criteria for Physical Realization of QIP 1. Scalable physical system with mapping of qubits 2. A method to initialize the system 3. Big decoherence time to gate time 4. 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 Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Closed and Open Quantum System Environment Hypothetical Environment 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 Effect of environment Quantum System – Environment interaction Evolution U(t) |0|E U(t) |0|E0 System Environment |1|E U(t) |1|E1 System Environment ||E = (c0|0 + c1|1)|E U(t) c0|0|E0 + c1|1|E1 System Environment Entangled Decoherence r = ||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* E0|E1 c1c0* E1|E0 c0c1* c1c1* Coherence Population Coherence decays irreversibly |E1(t)|E0(t)| = eG(t) Decoherence Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Signal Decay 13-C signal of chloroform in liquid Signal x Time Frequency Signal Decay Decoherence Amplitude decay T1 process Phase decay T2 process Relaxation Incoherence Depolarization Signal Decay Decoherence Amplitude decay T1 process Phase decay T2 process Relaxation Incoherence Depolarization Incoherence Individual (30 Hz, 31 Hz) Net signal – faster decay Time Hahn-echo or Spin-echo (1950) Echo Signal y /2-x t t + d d y Symmetric distribution of pulses removes incoherence Signal Decay Decoherence Amplitude decay T1 process Phase decay T2 process Relaxation Incoherence Depolarization Bloch’s Phenomenological Equations (1940s) 1 Mx 0 0 0 M x T2 d My 0 0 0 M y 0 dt 0 0 Mz 0 M z 0 0 1 T2 0 0 M x 0 M y eq 1 M M z z T1 M zeq T1 T2 Time to reach equilibrium, (energy of spin-system is not conserved) Lifetime of coherences, (energy of spin-system is conserved) M Bloch’s Phenomenological Equations (1940s) 1 Mx 0 0 0 M x T2 d My 0 0 0 M y 0 dt 0 0 Mz 0 M z 0 0 1 T2 0 Solutions in rotating frame: M zeq t M x (t ) M x (0) exp T2 0 t M y (t ) M y (0) exp T2 0 M z ( t ) M z M z ( 0) M z eq 0 M x 0 M y eq 1 M M z z T1 M eq exp Tt 1 Mz eq Signal Decay Decoherence Amplitude decay T1 process Phase decay T2 process Relaxation Incoherence Depolarization Effect of environment r r’ = E(r) = ∑ Ek r Ek† k (operator-sum representation) Amplitude damping (T1 process, dissipative) g(t) is net damping : eg., g(t) = 1 et/T1 E0 = p1/2 1 0 E1 = p1/2 0 0 0 (1g1/2 1/2 E2 = (1 p)1/2 (1g 0 0 1 E3 = (1 p)1/2 0 g1/2 E(r) = ∑ Ek r Ek† k Asymptotic state (t , g 1 : p r = 0 0 1p In NMR, p= 1 1 + eE/kT ~ 0.5 + 104 g1/2 0 0 0 Amplitude damping (T1 process, dissipative) Measurement of T1: Inversion Recovery Equilibrium t Inversion M(t) = 1 2exp( t/T1) Signal Decay Decoherence Amplitude decay T1 process Phase decay T2 process Relaxation Incoherence Depolarization Phase damping (T2 process, non-dissipative) g(t) is net damping : eg., g(t) = 1 et/T2 E0 = 1 0 0 (1g1/2 E(r) = ∑ Ek r Ek† E1 = 0 0 r(t) = k Stationary state (t , g 1 : r = 0 g1/2 a b b* 1-a a 0 0 1-a Phase damping (T2 process, non-dissipative) Transverse magnetization: Mx(t) Re{r01(t)} Bloch’s equation : dMx(t) Mx(t) = dt T2 Spin-Spin Relaxation Solution : Mx(t) = Mx(0) exp( t/T2) Signal envelop: s(t) = exp( t/T2) FWHH = /T2 Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Carr-Purcell (CP) sequence (1954) Signal y /2y t y t t y t t t Shorter t is better (limited by duty-cycle of hardware) H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954) t Meiboom-Gill (CPMG) sequence (1958) Signal x /2y t x t t x t t t Robust against errors in pulse !!! S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958) t CPMG Dynamical effects are minimized t t 1 t Dynamical decoupling t t 2 t 3 t Sampling points t 4 time j = T(2j-1) / (2N) Linear in j Signal No pulse Hahn Echo CP CPMG Time S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958) Dynamical Decoupling (DD) CPMG (1958): Uniformly distributed pulses Uhrig 2007 (UDD): Optimal distribution of pulses for a system with dephasing bath j = T sin2 ( j /(2N+1) ) T = total time of the sequence N = total number of pulses Götz S. Uhrig PRL 98, 100504 (2007) Carr & Purcell, Phys. Rev (1954) . Meiboom & Gill, Rev. Sci. Instru. (1958). Carr Purcell Sequence 0 1 2 3 j = T(2j-1) / (2N) Uhrig Sequence 4 Linear in j 5 6 7 time T Was believed to be optimal for N flips in duration T Uhrig, PRL (2007) time 0 1 2 3 j = T sin2 ( j /(2N+1) ) 4 5 6 7 T Proved to be optimal for N flips in duration T Dynamical Decoupling (DD) Hahn-echo (1950) CPMG (1958) PDD (XY-4) (Viola et al, 1999) CDDn = Cn = YCn−1XCn−1YCn−1XCn−1 C0 = t (Lidar et al, 2005) UDD (Uhrig, 2007) Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary DD performance ION-TRAP qubits M. J. Biercuk et al, Nature 458, 996 (2009) DD performance Electron Spin Resonance (g-irradiated malonic acid single crystal) J. Du et al, Nature 461, 1265 (2009) Time (s) Time (s) DD performance Solid State NMR 13C of Adamantane Dieter et al, PRA 82, 042306 (2010) Dynamical Decoupling in Solids 13C of Adamantane D. Suter et al, PRL 106, 240501 (2011) Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Sources of decoherence – dipole-dipole interaction Randomly fluctuating local fields Spin in a coherent state Sources of decoherence – dipole-dipole interaction Randomly fluctuating local fields Spin looses coherence Source of Phase-damping – chemical shift anisotropy B0 Redfield Theory: semi-classical System - > Quantum, Lattice - > Classical System System+ Random field (coarse grain) dr i[ H , r ] dt Completely reversible No decoherence d r iH , r R r r eq dt Auto-correlation Local field X(t) time Auto-correlation function G(t) = X(t) X*(t+t) = dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, t) Fluctuations have finite memory: G(t) = G(0) exp(|t|/ tc) tc Correlation Time Spectral density J() = G(t) exp(-it) dt = G(0) 2tc 1+ 2tc2 Spectral density J() = G(0) 2tc 1+ 2tc2 J() d r J ( )X X , r dt tc = 1 (after secular approximation) Spectral density J() = G(0) 2tc 1+ 2tc2 J() c0c0* eGt c0c1* eGt c1c0* c1c1* Dipolar Relaxation in Liquids 1 J(2) + J() T1 1 3 J(2) + 15 J() + 3 J(0) T2 8 4 8 tc = 1 G= 2 0 J() d 2 Effect of decoupling pulses L. Cywinski et al, PRB 77, 174509 (2008). M. J. Biercuk et al, Nature (London) 458, 996 (2009) Time-dependent Hamiltonian exp(-i H(t) dt ) 0 Magnus expansion Filter Functions |x(t)|= e(t) = 2 0 Cywiński, PRB 77, 174509 (2008) M. J. Biercuk et al, Nature (London) 458, 996 (2009) J() F() d 2 Fourier Transform of Pulse-train F() F(t) t Filter Functions = 2 0 J() F() d 2 Modified Spectral density: J’() = J() F() Residual area contributes to decoherence Cywiński, PRB 77, 174509 (2008) M. J. Biercuk et al, Nature (London) 458, 996 (2009) J(t) Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Two-qubit DD Two-qubit DD Electron-nuclear entanglement (Phosphorous donors in Silicon) Wang et al, PRL 106, 040501 (2011) No DD PDD Two-qubit DD – in NMR Levitt et al, PRL, 2004 Hamiltonian: H = h1Iz1 + h2Iz2+ hJ I1 S. S. Roy & T. S. Mahesh, JMR, 2010 Hz I2 HE Eigenbasis of HE Eigenbasis of Hz 90x , 1 , , 90y , 2J |11 |00 |01+|10 |11 2 |10 |01 |00 |01−|10 2 Fidelity = 0.995 Two-qubit DD – in NMR 27s 2 ms 2 ms j = Nt sin2 ( j /(2N+1) ) t = 4.027 ms 5-chlorothiophene-2-carbonitrile UDD-7 on 2-qubits Singlet Fidelity S. S. Roy, T. S. Mahesh, and G. S. Agarwal, Phys. Rev. A 83, 062326 (2011) UDD-7 on 2-qubits Product state 0110 01+10 Entanglement 0011 00+11 S. S. Roy, T. S. Mahesh, and G. S. Agarwal, Phys. Rev. A 83, 062326 (2011) Dynamical Decoupling in Solids CPMG UDD RUDD Uhrig, 2011 Abhishek et al Dynamical Decoupling in Solids DD on single-quantum coherences 1H of Hexamethylbenzene Abhishek et al Dynamical Decoupling in Solids RUDD No DD 1H of Hexamethyl Benzene Abhishek et al Dynamical Decoupling in Solids 2q 4q 6q 8q Abhishek et al Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Noise Spectroscopy |x(t)|= e(t) F(t) Alvarez and D. Suter, arXiv: 1106.3463 [quant-ph] (t) = 2 0 J(t) F(t) d 2 Contents 1. Coherence and decoherence 2. Sources of signal decay 3. Dynamical decoupling (DD) 4. Performance of DD in practice 5. Understanding DD 6. DD on two-qubits and many qubits 7. Noise spectroscopy 8. Summary Summary 1. Dynamical decoupling can greatly enhance the coherence times, some times by orders of magnitude 2. Various types of pulsed DD sequences are available. Best DD depends on the spectral density of the bath, the state to be preserved, robustness to pulse errors, etc. 3. Filter-functions are useful tools to understand the performance of DD. 4. DD on large number of interacting qubits also shows improved performance.