VIEWS: 55 PAGES: 54 POSTED ON: 7/9/2011
Adiabatic Quantum Computation with Noisy Qubits M.H.S. Amin D-Wave Systems Inc., Vancouver, Canada Collaborators: Experiment: Theory: Andrew Berkley (D-Wave) Paul Bunyk (D-Wave) Dmitri Averin (Stony Brook) Sergei Govorkov (D-Wave) Peter Love (D-Wave, Haverford) Siyuan Han (Kansas) Colin Truncik (D-Wave) Richard Harris (D-Wave) Andy Wan (D-Wave) Mark Johnson (D-Wave) Shannon Wang (D-Wave) Jan Johansson (D-Wave) Eric Ladizinsky (D-Wave) Sergey Uchaikin (D-Wave) Many Designers, Engineers, Technicians, etc. (D-Wave) Fabrication team (JPL) •2• Quantum requirements for gate model quantum computation: • Coherence • Superposition • Entanglement •3• For Adiabatic quantum computation: • Phase coherence is not required • Ground state superposition is required • Ground state entanglement is required Superposition and entanglement can be protected by the Hamiltonian •4• Single Qubit Example Measurement basis Hamiltonian: H 1 x 2 0 1 Superposition Energy eigenstates: 2 states Energy eigenvalues: E 2 Initializing in state “0”: (t 0) 0 2 Time evolution: eit / 2 eit / 2 t t (t ) cos 0 i sin 1 2 2 2 •5• Single Qubit Example Hamiltonian: H 1 x 2 Probability of finding the qubit in state “0”: 2 1 P0 (t ) 0 (t ) (1 cos t ) 2 Coherent Oscillations •6• Density Matrix 1 1 eit Closed system density matrix: 1 2it e 2 1 (in the energy basis) 2 2 Open system density matrix: weak coupling limit Dephasing (T2) Relaxation (T1) process process Peq ( 1 Peq )e t / T1 1 eit e t / T2 t Peq 0 2 1 it t / T2 2 t / T1 eq 2e e P ( 2 P )e eq 1 eq 0 P e E / T Equilibrium Peq E / T (Boltzmann) e e E / T Distribution •7• Coherent Tunneling Probability of finding the qubit in state “0”: Decoherence rate 1 / T2 P0 (t ) 0 (t ) 0 1 (1 e t cos t ), Coherent oscillations 2 0 1 = broadening Energy gap = well-defined gap •8• Incoherent Tunneling Probability of finding the qubit in state “0”: Decoherence rate 1 / T2 2 P0 (t ) 1 (1 e t ), Incoherent tunneling rate 2 0 1 Energy gap = = broadening No well-defined gap •9• Density Matrix Peq 0 Density matrix in energy basis: 0 eq P Density matrix in computation basis (“0” , “1”): Signature Superposition 1 1 P P eq eq of coherent eq (coherent mixture 2 P P eq mixture) 1 can persist in is diagonal only if equilibrium Peq Peq 1 2 i.e., T • 10 • Two-Qubit Example Hamiltonian: H 1 ( 1 x ) 1 J z z , 2 x 2 2 1 2 J Ferromagnetic coupling Lowest two 00 11 Entangled energy eigenstates: 2 states 2 Energy eigenvalues: E 2J • 11 • Two-Qubit Entanglement Equilibrium density matrix (J T , ): P P eq eq Concurrence Entanglement ( ) P eq P eq (entanglement measure): C W.K. Wootters, PRL 80, 2245 (1998) can persist in equilibrium C() 0, (i.e., unentangled) only if P P eq eq 1 2 i.e., T / J 2 • 12 • Summary: 1. Classical limit is large T (compared to energy spacings) and not long t (compared to decoherence time) 1. Without a Hamiltonian, the system will be classical after the decoherence time 1. With a well-defined Hamiltonian (stronger than noise) system may stay quantum mechanical at small T • 13 • Adiabatic Quantum Computation (AQC) E. Farhi et al., Science 292, 472 (2001) Energy Spectrum System Hamiltonian: H = (1 s) Hi + s Hf Linear interpolation: s = t/tf • Ground state of Hi is easily accessible. • Ground state of Hf encodes the solution to a hard computational problem. • 14 • Adiabatic Quantum Computation (AQC) E. Farhi et al., Science 292, 472 (2001) Energy Spectrum Effective System Hamiltonian: two-state system H = (1 s) Hi + s Hf Gap = Linear interpolation: s = t/tf • Ground state of Hi is easily accessible. • Ground state of Hf encodes the solution to a hard computational problem. • 15 • Adiabatic Theorem Landau Zener Error probabilit y : 2 / 2 E PLZ e ~ s ~ 1/ t f Success s To have small error probability: tf >> 1/2 • 16 • System Plus Environment Smeared out anticrossing Environment’s energy levels e.g., Harmonic oscillator Adiabatic theorem does not apply! • 17 • Environment at Zero Temperature At T=0 the excitation (Landau-Zener) probability is exactly the same as that for a closed system For spin environment: A.T.S. Wan, M.H.S. Amin, S.X. Wang, cond-mat/0703085 For harmonic oscillator model: M. Wubs et al., PRL 97, 200404 (2006) • 18 • Environment at Finite Temperature Probabilit y of success: t f /t c P0 (t f ) 1 (1 e 2 ), Energy level 1 tf Broadening = W dt Γ(t ) tc 0 1 1 ds 0 s Γ(s) If W > , transition will be via tf incoherent tunneling process Incoherent tunneling rate • 19 • Directional Tunneling Rates “0” “1” “0” “1” 01 10 01 and 10 can be extracted from the initial slopes: • 20 • Macroscopic Resonant Tunneling (MRT) “0” “1” “0” “1” 01 10 1st resonant peak 2nd resonant peak 10 01 wp • 21 • Calculating Incoherent Tunneling Rate Two-State Model: E e 0 1 System Hamiltonian: H S (e z x ) 2 Interaction Hamiltonian: H int Q z Heat bath operator • 22 • Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint 1. For a Gaussian environment up to second order in : Noise spectral density: 2. If S(w) is peaked at low frequencies, one can expand eiwt • 23 • Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint Gaussian line-shape 1 Width ~ * T2 Shift t 2 / 2T2*2 Decoherence ~ e 1 sin 2 (wt / 2) cos dw 2 S (w ) ~ cos2 dwS (w ), arctan( / e ) T2*2 (wt / 2) 2 • 24 • Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint Gaussian line-shape Width depends on symmetric part of S(w) Shift depends on anti-symmetric part of S(w) For a classical noise, S(w) is symmetric, hence Symmetric 01 • 25 • Quantum Noise Let e 0 01(e) 01(e) Absorption Tunneling Tunneling Emission Bose-Einstein distribution: 01(e) 01(e) The peak is shifted toward positive e • 26 • Equilibrium Environment S(w) = Ss(w) Sa(w) depends on symmetric part Ss(w) depends on anti-symmetric part Sa(w) Can be tested w S (w ) S (w ) experimentally S (w ) coth 2T 2T w a Fluctuation-Dissipation Theorem: s a W2 W 2 2Te p Teff Effective 2e p Temperature • 27 • Experiment Magnetic flux Tunable rf-SQUID qubit: Josephson junctions: F2 F1 Double-well potential: Low F2 F0 / 2 High F2 0 barrier: barrier: ~ wp wp ~0 wp = plasma frequency • 28 • Pulse Sequence High barrier Initializing Tunneling Low barrier Measurement • 29 • Transition Rate Measurements R. Harris et al., preprint available Tunneling to the 2nd level Tunneling to the 1st level W 10 01 (mF0) Asymmetric tunneling Quantum Noise • 30 • Fit to Theory R. Harris et al., preprint available Tunneling to the 2nd level Tunneling to the 1st level W (mF0) Good agreement with theory • 31 • Width and Shift Measurements ep W W (1 / T2* ) weakly depends on T e p decreases with T as ~ 1/T • 32 • Effective Temperature Equilibrium Saturation distribution Temperature Mixing chamber thermometer • 33 • Tunneling Amplitude One can extract from data using Why T-dependent? Why increase with T? is renormalized by high frequency environmental modes • 34 • Adiabatic Renormalization of Consider the Hamiltonian H 1 0 x Q z 2 1 At 0 0, the two states 0 g ,0 1 g ,1 2 are degenerate Environment states Degeneracy is lifted by H, with a splitting: H H 0 g ,0 g ,1 HF Renormalization can only decrease • 35 • Adiabatic Renormalization of In terms of spectral density Ohmic environment: increases with T ! The larger the T, the less the environment cares about the system • 36 • Sub-Conclusions: • The noise is coming from a quantum source, e.g. two-state fluctuators; J. Martinis et al. PRL. 95, 210503 (2005) • Low frequency part of the noise spectrum is peaked at zero frequency (e.g. 1/f noise). • High frequency noise is likely to be ohmic. 1/f In agreement with: Astafiev et al. PRL 93, 267007 (2004) ohmic • 37 • Back to AQC M.H.S. Amin and D.V. Averin, arXiv:0708.0384 t f /t c Probability of success: P0 (t f ) (1 e 1 2 ) 1 t f dt Γ(t ) 2 de 1 Characteristic time scale: tc 0 t e Γ(e ) f For a non-Markovian environment: (e ) 01 (e ) 10 (e ) 2 8 W (e ( e e p ) 2 / 2W e ( e e p ) 2 / 2W ) 2E Linear interpolation (global adiabatic evolution): e const. tf E tc 2 • 38 • Computation Time Scale M.H.S. Amin and D.V. Averin, arXiv:0708.0384 2E Open system: tf ~ 2 Normalized Closed system: 2E tf ~ 2 (Landau-Zener probability) 0 Not normalized Broadening (low frequency noise) does not affect the computation time Incoherent tunneling rate ~ 1/ W Cancel each other Width of transition region ~ W • 39 • Gap Renormalization Effect High frequency modes: T w wc Low frequency modes: w W If W T , the two regions overlap Gap renormalization may not happen for large n, or may happen differently from the simple one-qubit example • 40 • AQC vs. Classical Annealing Optimization problem: H (1 s) Hi s Hf n n H f hi z j J jk zj zk Diagonal j 1 j , k 1 1 n H i j xj Non-diagonal 2 j 1 sH f Hf j 0 Boltzmann factor: e e T /s T Effective temperature Teff = T/s changes from to T Classical annealing • 41 • AQC vs. Classical Annealing Optimization problem: H (1 s) Hi s Hf n n H f hi z j J jk zj zk Diagonal j 1 j , k 1 1 n H i j xj Non-diagonal 2 j 1 j T Quantum regime j T Thermal regime j ~ T Mixed regime! • 42 • Noise Requirements for AQC 1. Away from classical limit: j >T For Gaussian distribution of the energy levels the 2. Stable final ground state: hj , Jjk > T number of the levels near the ground state will be 3. Well defined tunnel splitting: j > Wj Single polynomial (in n) qubit 4. Accurate final Hamiltonian: hj , Jjk > Wj noise 5. Polynomial number of energy levels within energy T from the ground state during the evolution • 43 • Our Multi-Qubit System Tunable rf-SQUID qubit + coupler: R. Harris et al., PRL 98, 177001 (2007) Qubits Tunable coupler Fa 2 F1 a Fc F1 b Fb 2 Controls j Controls Jjk Controls j Controls hj • 44 • Annealing Process Double-well potential: F2 F 0 / 2 F2 0 Annealing j ~ wp wp j ~ 0 wp = plasma frequency Initial Hamiltonian Final Hamiltonian • 45 • Annealing Regime wp ~ 400 mK 2nd resonant peak 1st resonant peak wp Effective Temperature: Teff ~ 20 mK W ~ 80mK Unlikely to be j ~ wp classical annealingsplittings >W Well-defined energy j ~ wp >> T Quantum regime hj , Jjk > W,T Well-defined final Hamiltonian • 46 • Conclusions 1. Our environment model correctly describes resonant tunneling in superconducting devices 2. Theoretical + experimental MRT investigations: dominant low frequency noise has quantum origin 3. Phase coherence (energy basis) is not required (global AQC not for local adiabatic evolution) 4. Ground state superposition and entanglement (computation basis) are required for AQC; protected by the Hamiltonian 5. Our multi-qubit system is in the right regime for AQC/Quantum Annealing • 47 • Additional Slides • 48 • Classical and Quantum Annealing Annealing Disorder Order Slow transition Classical annealing: Disorder = Thermal mixing (introduced by entropy) Quantum annealing: Disorder = Superposition (introduced by a Hamiltonian) • 49 • AQC vs. Quantum Annealing (QA) Optimization problem: H (1 s) Hi s Hf n n H f hi z j J jk zj zk Problem Hamiltonian (Diagonal) j 1 j , k 1 1 n H i j xj Disordering Hamiltonian (Non-Diagonal) 2 j 1 Quantum phase transition Large Small superposition superposition z 1 ( 0) Quantum (t f ) z solution N z{ 0,1} n Critical point • 50 • AQC vs. Quantum Annealing (QA) 1. For optimization problems: AQC = QA 2. Superposition and entanglement are important for AQC/QA Single 3. Quantum critical point happens qubit when j(s) ~ hj(s) , Jjk(s) noise Condition: j(s) , hj(s) , Jjk(s) > Wj • 51 • Global vs. Local Adiabatic Evolution Adiabatic Grover search: E J. Roland and N.J. Cerf, PRA 65, 042308 (2001) Gap size: N Closed system: N 2 n , n number of qubits Quantum Global adiabatic evolution: advantage only possible viat f ~ 2 O(N ) 1 E The same as s const. tf local evolution classical Local adiabatic evolution: s gap(s ) 2 1 tf ~ O N ( ) The same as Grover search • 52 • Global vs. Local Adiabatic Evolution M.H.S. Amin and D.V. Averin, arXiv:0708.0384 1 Open system: 2 de tc e Γ(e ) t f No Quantum Global adiabatic evolution: advantage! s only prefactor t f ~ 2 O(N ) 1 E The same as const. t f speedup classical Local adiabatic evolution: The same as t f ~ 2 O( N ) W s gap(s ) 2 classical • 53 • Global vs. Local Adiabatic Evolution M.H.S. Amin and D.V. Averin, arXiv:0708.0384 Quantum advantage is only possible The same as if W gate model decoh. on the W ~ 1/d quantum time tf < d other : computation hand tf ~ 1/ Global coherence is required for local AQC • 54 •