Quantum Communication & Computation Using Spin Chains

Transcription

1 Quantum Communication & Computation Using Spin Chains Sougato Bose Institute for Quantum Information, Caltech & UCL, London Quantum Computation Part: S. C. Benjamin & S. Bose, quant-ph/02057 (to appear in PRL) Quantum Communication Part: S. Bose, quant-ph/02204

2 D Bulk Magnets are atural Spin Chains (Examples): Cu (spin ½ sites) Isotropic Heisenberg Antiferromagnet: P. R. Hammar et. al., PRB 59, 008 (999).

3 Quantum computation using a D magnet Quantum computation by applying a time varying and inhomogeneous magnetic field to a spin chain.

4 Heisenberg Chain to Ising Chain Conversion: Heisenberg A, B = Zeeman Energies, A-B >> J Ising Where, If Then Implies

5 Case A: When Universal Local Gates Are Possible: Positions of Qubits & Barrier Spins A barrier spin A qubit i ε are variable energies, set to B in the passive state when single qubit gates are performed. The Ising interaction on each qubit is then completely cancelled at all times. ote: Both barrier spins could be in the same state (which is easier to initialize, with periodic cancellation of Ising effects.

6 Case A: When Universal Local Gates Are Possible: (t) For a Gate between X & Y, is changed (fast) to ε( t ) = B ε ( t ) = A+ J Then 3 becomes resonant with 2 & 4 (2,3,4 become a small Heisenberg chain).

7 Case A (Contd.) At time 2 & 4 disentangle from 3. An entangling gate between X and Y! Use techniques of: M. J. Bremner et. al., quant-ph/ J. L. Dodd et. al., PRA 65, (2002).

8 Case B: When Only Zeeman Energy Tuning is Possible Locally: Method for one qubit gates:

9 Case B: When Only Zeeman Energy Tuning is Possible Locally: Method for two qubit gates

10 Global control quantum computation schemes of Lloyd & Benjamin S. Lloyd, Science 26, 569 (993); S. C. Benjamin, PRL 88, (2002). One Qubit Gates Two Qubit Gates

11 Case B: When o Local Ability is Present: Control Switch of Six Settings Control Through the Strength of a Single Field

12 Quantum Communication through a Spin Chain Alice Bob Alice Bob Avoids interfacing solid state systems with optics for the purpose of short-distance communication: Quantum Computer Quantum Computer

13 Definition of Spin-Chains: (A) D array of spins (B) Always On (untunable) interactions Makes it much easier to fabricate such systems with qubit arrays (especially in solid state) than to perform arbitrary quantum computations.

14 First consider arbitrary graphs with ferromanetic Heisenberg interactions 2 r s Initialized in the ground state with HB = B σ, j j z H = J ij < i, j> σ 0 = , 0 i. σ j

15 2 r s At t = 0, Ψ ( 0) = θ (cos sin ) θ e iφ ,2,..,s- s s+,,

16 Time evolution of the spin-graph: θ iφ θ 2iBt Ψ ( t) = cos 0 + e sin e f j, s( t) j 2 2 where, 0 = 00 0 and f ( t) = j e j, s j=..., j = iht s j th spin is the transition amplitude of an excitation from the s th to the j th spin due to H. ote that only the ground & one-excitation states of the graph are invloved (because H does not create excitations, only propagates excitations).

17 2 r s ρ ( t) = P( t) ψ ( t) ψ ( t) + ( P( t)) 0 0 ψ r out out r r out ( t) 2 θ 2 2 θ P( t) = cos + f ( t) sin, r,s 2 2 = θ iφ θ ibt (cos e sin e f ( t) ) P( t) r r,s r 2 where, B should be chosen so that f ( t) f ( t) r,s r,s

18 The graph of Heisenberg interacting spins behaves as an amplitude damping quantum channel: M 0 0 = 0 f ( t) r,s, M = 0 f t ( ) r,s Fidelity averaged over the Bloch Sphere: F = ψ in ρout ( t) ψ in = + fr,s( t) + fr,s( t) 4π Entanglement (Concurrence) for input of one half of a Ψ ( + ) E f ( t) C = r,s Exceptionally simple formulae in terms of a single transition amplitude 2

19 We will consider two cases: A linear chain with communicating parties at opposite ends (most natural and readily implementable case): Alice H = J 2 j= Bob A closed loop with the communicating parties at diametrically opposite ends (to compare): σ j. σ j+ Alice H J J = σ j. σ j + σ. σ 2 2 j= Bob /2

20 Eigenstates in the one excitation sector: (A) Linear (open chain) case: j= ~ π m = a cos ( )( ) m m 2 j 2 for m =,..., with a = and a (B) Closed chain case: (A Quantum Cosine Trans.) m> = j 2 (A Quantum Fourier Trans.) ( ) ~ e i 2π = m j j = m j for m =,...,

21 Transition amplitudes in terms of readily computable transforms: (A) Linear (open chain) case: where r,s f = DCT (, v ( r, t)) π ( m ) i2 Jt cos v ( r, t) = a cos ( r -) e m m 2 2 (B) Closed chain case: r,s f = DFT (, v ( t)) r-s where s m m v ( t) = e m a = and a π m> = ( m ) 2π ( m ) i2 Jt cos 2

22 Fidelity, Entanglement Log of Scaled time

23 Alternative formulas in terms of Bessel functions: Open chain: 2 k k ' f, = 2 ( ) J ( k ) ( 2Jt) ( ) J ( k ) ( 2Jt) k = 0 k = J ( 2Jt) at the maximum near 2Jt= / 2, 2 J ( 2Jt) / 2 k = 0 Closed chain: ( / 2) k f = 2 ( ) J ( 2Jt) ( / 2)( 2k + ) at the maximum near 2Jt=/2. Closed chain 2 Open chain 2. Can find high fidelity transfer at 2Jt= 3 = 0 E ( 2Jt = 005) = 03. C 4 = 0 E ( 2Jt = 007) = C Distillable

24 Q.Comm part: Possible Future Work.Sending higher D systems (Ex: 4 state systems by using up to 2 spin excitations --- with Korepin). 2. Study Graphs which improve comm. fidelity (suggested by Preskill) 3. Direct qubit comm. (without distillation) over arbitrary distances by using spin- chain (--- with Thapliyal). 4. Can measurements on the chain improve transfer? (suggested by Verstraete). Q.Compu part: 2D extensions, extensions to clusters replacing single qubits etc. --- for greater fault tolerance and robustness.