Solid State Quantum Computation. 1. Spins David DiVincenzo, IBM Can. Summer School, 5/2004

1. Solid State Quantum Computation. 1. SpinsDavid DiVincenzo, IBMCan. Summer School, 5/2004

2. Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Electrons on liquid He Small Josephson junctions “charge” qubits “flux” qubits Spin spectroscopies, impurities in semiconductors Coupled quantum dots Qubits: spin,charge,excitons Exchange coupled, cavity coupled Physical systems actively consideredfor quantum computer implementation

3. Kane (1998)  Concept device: spin-resonance transistorR. Vrijen et al, Phys. Rev. A 62, 012306 (2000)

4. Quantum-dot array proposal

5. 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.

6. Ideal quantum measurement for quantum computing: For the selected qubit: if its state is |0, the classical outcome is always “0” if its state is |1, the classical outcome is always “1” (100% quantum efficiency) If quantum efficiency is not perfect but still large (50%), desired measurement is achieved by “copying” (using cNOT gates) qubit into several others and measuring all. If q.e. is very low, quantum computing can still be accomplished using ensemble technique (cf. bulk NMR) Fast measurements (10-4 of decoherence time) permit easier error correction, but are not necessary 5. Measurement requirement

7. Loss & DiVincenzo quant-ph/9701055

8. Variant on spin-charge conversion mechanism. Single spin measurement now achieved by Delft group (Elzerman, Vandersypen, Hansen, Kouwenhoven, Dec. 2003) QPC detector 1 electron quantum dot

9. Spin Read-out via Spin Polarized Leads Quantum dot attached to spin-polarized leads* P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 85, 1962 (2000) Thus: Is = 0 spin up ( Ic << Is ) Is > 0 spin down => single-spin memory device, US patent PCT/GB00/03416 * - magnetic semiconductors [R. Fiederling et al., Nature 402, 787 (1999); Y. Ohno et al., Nature 402, 790 (1999)] - Quantum Hall Edge states [M. Ciorga et al., PRB 61, R16315 (2000)],

10. Quantum algorithms are specified as sequences of unitary transformations U1,U2, U3, each acting on a small number of qubits Each U is generated by a time-dependent Hamiltonian: 4. Universal Set of Quantum Gates • Different Hamiltonians are needed to generate the desired quantum gates: 1-bit gate • many different “repertoires” possible • integrated strength of H should be very precise, 1 part in 10-4, • from current understanding of error correction • (but, see topological quantum computing (Kitaev, 1997))

11. Quantum-dot array proposal Gate operations with quantum dots (1): --two-qubit gate: Use the side gates to move electron positions horizontally, changing the wavefunction overlap Pauli exclusion principle produces spin-spin interaction: Model calculations (Burkard, Loss, DiVincenzo, PRB, 1999) For small dots (40nm) give J 0.1meV, giving a time for the “square root of swap” of t 40 psec NB: interaction is very short ranged, off state is accurately H=0.

12. Making the CNOT from exchange: Exchange generates the “SWAP” operation: More useful is the “square root of swap”, = Using SWAP:  CNOT

13. Quantum-dot array proposal Gate operations with quantum dots (2): --one-qubit gate: Desired Hamiltonian is: One approach: use back gate to move electron vertically. Wavefunction overlap with magnetic or high g-factor layers produces desired Hamiltonian. If Beff= 1T, t 160 psec If Beff= 1mT, t 160 nsec

14. Can we get CNOT with just Heisenberg exchange? Conventional answer– NO: --because Heisenberg interaction has too much symmetry --it cannot change S (total angular momentum quantum number) Sz (z component of total angular momentum) Correct answer (Berkeley, MIT, Los Alamos) – YES: --the trick: encode qubits in states of specific angular momentum quantum numbers

15. Specific scheme to get quantum gates with just Heisenberg exchange: Most economical coding scheme: 1 qubit = 3 spins: (i.e., singlet times spin-up) (triplet on first two spins) Because quantum numbers are fixed (S=1/2, Sz=+1/2), all gates on These logical qubits can be performed using SWAP:

16. Economical coded-gate implementations—results of simulations By varying interactions times shown, all 1-qubit gates on coded qubits can be obtained with no more than 4 exchange operations (if only nearest-neighbor interactions) or 3 exchange interactions (if interactions between spin 1 and spin 3 are possible)

17. k k k k k k k k k k k k k k k k k k k k k k k k k k k k CNOT on two coded qubits -minimal solution 19 interactions, doable in 13 time steps -essentially unique -gate accuracy 10-5 with precision shown -nearest-neighbor seems best

18. Simple features of scheme for coded computation --Initialization: turn on uniform B field and strong antiferromagnetic Heisenberg exchange between spins 1 and 2. Then is the ground state of the system. --Measurement: coded qubit is measured by determining whether spins 1 and 2 are in a relative singlet or triplet. Somewhat easier than single-spin measurements.

19. Spin Orbitfirst-order effect (Dzyaloshinski-Moriya) Bonesteel, Stepanenko, DiVincenzo, Phys. Rev. Lett. 87, 207901 (2001) see also G. Burkard and D. Loss, Phys. Rev. Lett. 88, 047903 (2002) y y z z R x Heuristic: tunneling motion produces effective B field Kavokin, cond-mat/0011340

20. Cancellation of spin-orbit effects Burkard & Loss, Phys. Rev. Lett. 88, 047903 (2002) Exchange Spin-orbit (Moriya-Dzyaloshinski; Kavokin ’01; Bonesteel et al., q-phys/0106161) Quantum gate i.e. for ideal case with identical pulse shapes ( ), spin-orbitcancels exactly in XOR Gate

21. Big issue with solid state qubits: do we have effective and robust controls available (does the electrical engineering work)? Theoretical focus on fundamental decoherence problems: hyperfine effects (nuclear spins) and spin-orbit For these two, there are many possible problems and (almost) as many theoretically proposed solutions. Physics Issues for Spin Qubits

22. Solid State Quantum Computation. 2. SuperconductorsDavid DiVincenzo, IBMCan. Summer School, 6/2004

23. “Typical” flux qubit (Delft) – Chioescu, Nakamura, and Mooij, Science (2004). Not just a Josephson junction, but a complex electric circuit, operating quantum-mechamically.

24. Science 296, 886 (2002) Recent progress – Josephson junction qubit

25. small L C Simple electric circuit… harmonic oscillator with resonant frequency Quantum mechanically: 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.

26. small But we will need to learn to deal with… --Josephson junctions --current sources --resistances and impedances --mutual inductances --non-linear circuit elements?

27. 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?

28. 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

29. 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.

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

31. 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.

32. small First, the graph formalism (6 steps) • 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

33. small graph formalism, continued • Enumerate the loops 1,2,…F, assign each a direction • Enumerate the branches 1,2,…B, assign each a direction • Construct fundamental loop matrixFij, i=1..F, j=1..B Branch in loop, same direction graph Branch in loop, opposite direction Branch not in loop

34. small Branch in loop, same direction graph Branch in loop, opposite direction Branch not in loop graph formalism, continued NB: numbering of branches and loops can always be organized so that F(L) has the form M (each chord defines one loop) F = -MT Define also “cutset matrix”:

35. 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:

36. small graph formalism, continued 5. Identify submatrices of F: No C chords.

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

38. small graph formalism, continued 6. Write down inductance matrix L and capacitance matrix C. NB: inductances are both in the tree (K) and among the chords (L), L should be partitioned into these submatrices:

39. 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):

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

41. small the equation of motion (continued): The lossless parts of this equation arise from a simple Hamiltonian:

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

43. small Connecting Cadeira Leggett to circuit theory:

44. 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:

45. small Application to “Delft qubit” Gives “particle in 5-dimensional potential” coupled to environment --reducible to usual two-level description; exposes approximations

46. small Application to “Delft qubit” (cont.) Previous work (Wilhelm, van der Wal, …) quantum theory as particle in 2D potential Present approach: 5 JJs, so 5D quantum mechanics? Relation between these?

47. small 5D theory to 2D theory: • One of measuring SQUID degrees of freedom is heavily damped by external impedance, not really quantum mechanical • 4D theory is really right! But, because effective inductances of the SQUID and qubit are fairly small, • the “particle” is fairly tighly confined to a 2D sub-plane of the 4D space. • ALSO – this plane is not quite flat, it is slightly warped because of the nonlinearity of the qubit potential. The warping is very important, because it is the only thing that couples the 2D particle to the external impedance – tells us the “decoherence operator” of the qubit.

48. small Overview: • A “user friendly” procedure: automates the assessment of • different circuit designs • Gives some new views of existing circuits and their analysis • A “meta-theory” – aids the development of approximate theories • at many levels • BUT – it is the “orthodox” theory of decoherence – exotic effects • like nuclear-spin dephasing not captured by this analysis.

49. New application: May, 2004, cond-mat

50. Chioescu, Nakamura, and Mooij, Science (2004). --intended to have top-bottom symmetry. --observed departure from symmetry is very large. Why?