Five criteria for physical implementation of a quantum computer

1. Well defined extendible qubit array -stable memory Preparable in the “000…” state Long decoherence time (>104 operation time) Universal set of gate operations Single-quantum measurements Five criteria for physical implementation of a quantum computer D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” Fort. der Physik 48, 771 (2000), quant-ph/0002077.

2. Well defined extendible qubit array -stable memory Preparable in the “000…” state Long decoherence time (>104 operation time) Universal set of gate operations Single-quantum measurements Interconvert stationary and flying qubits Transmit flying qubits from place to place Five criteria for physical implementation of a quantum computer& quantum communications

3. Quantum-dot array proposal

4. Science 296, 886 (2002) Josephson junction qubit -- Saclay Oscillations show rotation of qubit at constant rate, with noise. Where’s the qubit?

5. Delft qubit: PRL (2004) small -Coherence time up to 4lsec -Improved long term stability -Scalable?

6. Nature, 2004 “Yale” Josephson junction qubit Coherence time again c. 0.5 ls (in Ramsey fringe experiment) But fringe visibility > 90% !

7. IBM Josephson junction qubit 1 “qubit = circulation of electric current in one direction or another (????)

8. IBM Josephson junction qubit “qubit = circulation of electric current in one direction or another (xxxx) Understanding systematically the quantum description of such an electric circuit…

9. small Good Larmor oscillationsIBM qubit -- Up to 90% visibility -- 40nsec decay -- reasonable long term stability What are they?

10. small L C Simple electric circuit… harmonic oscillator with resonant frequency Quantum mechanically, like a kind of atom (with harmonic potential): x is any circuit variable (capacitor charge/current/voltage, Inductor flux/current/voltage) That is to say, it is a “macroscopic” variable that is being quantized.

11. small Textbook (classical) SQUID characteristic: the “washboard” w Energy F 1. Loop: inductance L, energy w2/L 2. Josephson junction: critical current Ic, energy Ic cos w 3. External bias energy (flux quantization effect): wF/L w Josephson phase

12. small Textbook (classical) SQUID characteristic: the “washboard” w w Energy Energy F 1. Loop: inductance L, energy w2/L 2. Josephson junction: critical current Ic, energy Ic cos w 3. External bias energy (flux quantization effect): wF/L w Josephson phase Junction capacitance C, plays role of particle mass

13. small Quantum SQUID characteristic: the “washboard” w Energy Quantum energy levels w Josephson phase Junction capacitance C, plays role of particle mass

14. small But we will need to learn to deal with… --Josephson junctions --current sources --resistances and impedances --mutual inductances --non-linear circuit elements? G. Burkard, R. H. Koch, and D. P. DiVincenzo, “Multi-level quantum description of decoherence in superconducting flux qubits,” Phys. Rev. B 69, 064503 (2004); cond-mat/0308025.

15. small Josephson junction circuits Practical Josephson junction is a combination of three electrical elements: Ideal Josephson junction (x in circuit): current controlled by difference in superconducting phase phi across the tunnel junction: Completely new electrical circuit element, right?

16. small not really… What’s an inductor (linear or nonlinear)? (instantaneous) Ideal Josephson junction: is the magnetic flux produced by the inductor is the superconducting phase difference across the barrier (Josephson’s second law) (Faraday) flux quantum

17. small not really… What’s an inductor (linear or nonlinear)? Ideal Josephson junction: is the magnetic flux produced by the inductor is the superconducting phase difference across the barrier (Josephson’s second law) (Faraday) Phenomenologically, Josephson junctions are non-linear inductors.

18. small So, we now do the systematic quantum theory

19. small Strategy: correspondence principle --Write circuit equations of motion: these are equations of classical mechanics --Technical challenge: it is a classical mechanics with constraints; must find the “unconstrained” set of circuit variables --find a Hamiltonian/Lagrangian from which these classical equations of motion arise --then, quantize! NB: no BCS theory, no microscopics – this is “phenomenological”, But based on sound general principles.

20. small Graph formalism • Identify a “tree” of the graph – maximal subgraph containing • all nodes and no loops Branches not in tree are called “chords”; each chord completes a loop tree graph

21. small graph formalism, continued NB: this introduces submatix of F labeled by branch type e.g.,

22. small Circuit equations in the graph formalism: Kirchhoff’s current laws: V: branch voltages I: branch currents F: external fluxes threading loops Kirchhoff’s voltage laws:

23. small With all this, the equation of motion: The tricky part: what are the independent degrees of freedom? If there are no capacitor-only loops (i.e., every loop has an inductance), then the independent variables are just the Josephson phases, and the “capacitor phases” (time integral of the voltage): “just like” the biassed Josephson junction, except…

24. small the equation of motion (continued): All are complicated but straightforward functions of the topology (F matrices) and the inductance matrix

25. small Analysis – quantum circuit theory tool Burkard, Koch, DiVincenzo, PRB (2004). Conclusion from this analysis: 50-ohm Johnson noise not limiting coherence time.

26. the equation of motion (continued): small The lossless parts of this equation arise from a simple Hamiltonian: H; U=exp(iHt)

27. small the equation of motion (continued): The lossy parts of this equation arise from a bath Hamiltonian, Via a Caldeira-Leggett treatment:

28. small Connecting Cadeira Leggett to circuit theory:

29. small Overview of what we’ve accomplished: We have a systematic derivation of a general system-bath Hamiltonian. From this we can proceed to obtain: • system master equation • spin-boson approximation (two level) • Born-Markov approximation -> Bloch Redfield theory • golden rule (decay rates) • leakage rates For example:

30. IBM Josephson junction qubit Results for quantum potential of the gradiometer qubit…

31. IBM Josephson junction qubit:potential landscape --Double minimum evident (red streak) --Third direction very “stiff”

32. IBM Josephson junction qubit:effective 1-D potential x --treat two transverse directions (blue) as “fast” coordinates using Born-Oppenheimer

33. Extras

34. IBM Josephson junction qubit:features of 1-D potential well asymmetry barrier height x

35. IBM Josephson junction qubit:features of 1-D potential Well energy levels, ignoring tunnel splitting

36. IBM Josephson junction qubit:features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states

37. IBM Josephson junction qubit:features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states

38. IBM Josephson junction qubit:features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states

39. IBM Josephson junction qubit:features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states

40. IBM Josephson junction qubit:features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states

41. IBM Josephson junction qubit:scheme of operation: --fix e to be zero --initialize qubit in state --pulse small loop flux, reducing barrier height h well asymmetry barrier height x

42. IBM Josephson junction qubit:scheme of operation: --fix e to be zero --initialize qubit in state --pulse small loop flux, reducing barrier height h energy splitting

43. IBM Josephson junction qubit:scheme of operation: --fix e to be zero --initialize qubit in state --pulse small loop flux, reducing barrier height h energy splitting

44. IBM Josephson junction qubit:scheme of operation: --fix e to be zero --initialize qubit in state --pulse small loop flux, reducing barrier height h --state acquires phase shift --in the original basis, this corresponds to rotating between L and R: energy splitting “100% visibility”

45. IBM Josephson junction qubit:scheme of operation: --fix e to be small --initialize qubit in state --pulse small loop flux, reducing barrier height h energy splitting N.B. – eigenstates are and

46. The idea of a “portal”: --portal = place in parameter space where dynamics goes from frozen to fast. It is crucial that residual asymmetry e be small while passing the portal: energy splitting portal where tunnel splitting D exp. increases in time, D = D0exp(t/ t). and

47. IBM Josephson junction qubit:analyzing the “portal” --e cannot be fixed to be exactly zero --full non-adiabatic time evolution of Schrodinger equation with fixed e and tunnel splitting D exponentially increasing in time, D = D0 exp(t/ t), can be solved exactly … the spinor wavefunction is Which means that the visibility is high so long as

48. Tunnel splitting exponentially sensitive to control flux Flux noise will seriously impair visiblity Solution  Problem:

49. IBM Josephson junction qubit Couple qubit to harmonic oscillator (fundamental mode of superconducting transmission line). Changes the energy spectrum to:

50. IBM Josephson junction qubit Couple qubit to harmonic oscillator (fundamental mode of superconducting transmission line). Changes the energy spectrum to: