1. Introduction to Quantum Computing Lecture 3: Qubit Technologies Rod Van Meter rdv@tera.ics.keio.ac.jp June 27-29, 2005 WIDE University School of Internet with help from K. Itoh, E. Abe, and slides from T. Fujisawa (NTT)

2. Course Outline ● Lecture 1: Introduction ● Lecture 2: Quantum Algorithms ● Lecture 3: Devices and Technologies ● Lecture 4: Quantum Computer Architecture ● Lecture 5: Quantum Networking & Wrapup

3. Lecture Outline ● Brief review ● DiVincenzo's criteria & tech overview ● Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. ● Comparison

4. Superposition and ket Notation ● Qubit state is a vector – two complex numbers ● |0> means the vector for 0; |1> means the vector for 1; |00> means two bits, both 0; |010> is three bits, middle one is 1; etc. ● A qubit may be partially both! (just like the cat, but stay tuned for measurement...) complex numbers are wave fn amplitude; square is probability of 0 or 1

5. 1-qubit state and Bloch sphere(Phase) relative phase global phase has no effect 1-qubit basis states superposition Bloch sphere

6. Reversible Gates or NOT Control-NOT Control-Control-NOT (Toffoli gate) Control-SWAP (Fredkin gate) This set can be used for classical reversible computing, as well; brings thermodynamic benefits. Bennett, IBM JR&D Nov. 1973

7. Quantum Algorithms ● Deutsch-Jozsa(D-J) – Proc. R. Soc. London A, 439, 553 (1992) ● Grover's search algorithm – Phys. Rev. Lett., 79, 325 (1997) ● Shor's algorithm for factoring large numbers – SIAM J. Comp., 26, 1484 (1997) D. DeutschR. JozsaL. K. GroverP. W. Shor

8. Grover's Search items unordered items, we're looking for item “β” Classically, would have to check about N/2 items To use Grover's algorithm, create superposition of all N inputs N repetitions of the Grover iterator G will produce “β” Hard task!! β

9. Shor's Factoring Algorithm An “efficient” classical algorithm for this problem is not known. Using classical number field sieve, factoring an L-bit number is Using quantum Fourier transform (QFT), factoring an L-bit number is Superpolynomial speedup! (Careful, technically not exponential speedup, and not yet proven that no better classical algorithm exists.)

10. Summary: Characteristics of QC ● Superposition brings massive parallelism ● Phase and amplitude of wave function used ● Entanglement ● Unitary transforms are gates ● Measurement both necessary and problematic when unwanted So, can we surpass a classical computer with a quantum one? Depends on discovery of quantum algorithms and development of technologies

11. Lecture Outline ● Brief review ● DiVincenzo's criteria & tech overview ● Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. ● Comparison

12. DiVincenzo’s Criteria 1. Well defined extensible qubit array 2. Preparable in the “000…” state 3. Long decoherence time 4. Universal set of gate operations 5. Single quantum measurements Qubit initialization Read the result Must be done within decoherence time! Execution of an algorithm

13. Qubit Representations ● Electron: number, spin, energy level ● Nucleus: spin ● Photon: number, polarization, time, angular momentum, momentum (energy) ● Flux (current) ● Anything that can be quantized and follows Schrodinger's equation

14. Problems ● Coherence time – nanoseconds for quantum dot, superconducting systems ● Gate time – NMR-based systems slow (100s of Hz to low kHz) ● Gate quality – generally, 60-70% accurate ● Interconnecting qubits ● Scaling number of qubits – largest to date 7 qubits, most 1 or 2

15. A Few Physical Experiments ● IBM, Stanford, Berkeley, MIT (solution NMR) ● NEC (Josephson junction charge) ● Delft (JJ flux) ● NTT (JJ, quantum dot) ● Tokyo U. (quantum dot, optical lattice,...) ● Keio U. (silicon NMR, quantum dot) ● Caltech, Berkeley, Stanford (quantum dot) ● Australia (ion trap, linear optics) ● Many others (cavity QED, Kane NMR,...)

16. Physical Realization Cavity QED Magnetic resonance Ion trap Superconductor

17. Examples of Qubits Trapped ions Neutral atoms in optical lattices Solution NMR Crystal lattice Flux states in superconductors Charge states in superconductors Impurity spins in semiconductors Single-spin MRFM Atomic cavity QED Optically driven electronic states in quantum dots Electronically driven electronic states in quantum dots Optically driven spin states in quantum dots Electronically driven spin states in quantum dots Electrons floating on liquid helium Solid-state systems Others: Nonlinear optics, STM etc All-silicon quantum computer Shor (7)Entanglement (4)Rabi (1)

18. Lecture Outline ● Brief review ● DiVincenzo's criteria & tech overview ● Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. ● Comparison

19. Technologies Reviewed ● Liquid NMR ● Solid-state NMR ● Quantum dots ● Superconducting Josephson junctions ● Ion trap ● Optical lattice ● All-optical

20. Liquid Solution NMR Billions of molecules are used, each one a separate quantum computer. Most advanced experimental demonstrations to date, but poor scalability as molecule design gets difficult and SNR falls. Qubits are stored in nuclear spin of flourine atoms and controlled by different frequencies of magnetic pulses. Used to factor 15 experimentally. Vandersypen, 2000

21. All-Silicon Quantum Computer 10 5 29 Si atomic chains in 28 Si matrix work like molecules in solution NMR QC. A large field gradient separates Larmor frequencies of the nuclei within each chain. Many techniques used for solution NMR QC are available. No impurity dopants or electrical contacts are needed. Copies 1 2 3 4 5 6 8 7 9 T.D.Ladd et al., Phys. Rev. Lett. 89, 017901 (2002)

22. Overview dB z /dz = 1.4 T/ m m B 0 = 7 T Active region 100  m x 0.2  m

23. Keio Choice of System and Material Two incompatible conditions to realize quantum computers: 1. Isolation of qubits from the environment 2. Control of qubits through interactions with the environment System: NMR 1. Weak ensemble measurement 2. Established rf pulse techniques for manipulation Material: silicon 1. Longest possible decoherence time 2. Established crystal growth, processing and isotope engineering technologies

24. Fabrication: 29 Si Atomic Chain 28 Si 29 Si J.-L.Lin et al., J. Appl. Phys 84, 255 (1998) Regular step arrays on slightly miscut 28 Si(111)7x7 surface (~1º from normal) Steps are straight, with about 1 kink in 2000 sites. STM image

25. Kane Solid-State NMR Qubits are stored in the spin of the nucleus of phosphorus atoms embedded in a zero-spin silicon substrate. Standard VLSI gates on top control electric field, allowing electrons to read nuclear state and transfer that state to another P atom. Kane, Nature, 393(133), 1998

26. Semiconductor nanostructure High electron mobility transistor (HEMT) http://www.fqd.fujitsu.com/ Quantum dot structure N < 30 # of electrons

27. Quantum dot in the Coulomb blockade regime Coulomb blockade region, (the number of electrons is an integer, N) orbital degree of freedom & spin degree of freedom double quantum dot bonding & anti-bonding orbitals (charge qubit) (spin qubit) Single-electron transistor (SET)

28. Quantum dot artificial atom S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996). 1s2p3d |  (r)| 2 500 nm

29. Environment surrounding a QD coherency of the system dissipation (T 1 ), decoherence (T 2 ) & manipulation

30. Double quantum dot device A double quantum dot fabricated in AlGaAs/GaAs 2DEG (schematically shown by circles) Approximate number of electrons: 10 ~ 30 Charging energy of each dot: ~ 1 meV Typical energy spacing in each dot: ~100 ueV Electrostatic coupling energy:~200 ueV Lattice temp. ~ 20 mK (2 ueV) Electron temp. ~100 mK (9 ueV) cf. CPB charge qubit (Y. Nakamura ’99)

31. Measurement system 1 mm dilution refrigerator T lat ~ 20 mK T elec ~ 100 mK B = 0.5 T Double quantum dot AlGaAs/GaAs 2DEG EB litho., fine gates, ECR etching “thin coax” filtering

32. Josephson Junction Charge (NEC) Two-qubit device Pashkin et al., Nature, 421(823), 2003 One-qubit device can control the number of Cooper pairs of electrons in the box, create superposition of states. Superconducting device, operates at low temperatures (30 mK). Nakamura et al., Nature, 398(786), 1999

33. JJ Phase (NIST, USA) J. Martinis, NIST Qubit representation is phase of current oscillation. Device is physically large enough to see!

34. JJ Flux (Delft) The qubit representation is a quantum of current (flux) moving either clockwise or counter-clockwise around the loop.

35. Ion Trap NIST, Oxford, Australia, MIT, others Ions (charged atoms) are suspended in space in an oscillating electric field. Each atom is controlled by a laser.

36. Ion Trap NIST, Oxford, Australia, MIT, others Number of atoms that can be controlled is limited, and each requires its own laser. Probably <100 ions max.

37. Optical Lattice (Atoms) Deutsch, UNM Neutral atoms are held in place by standing waves from several lasers. Atoms can be brought together to execute gates by changing the waves slightly. Also used to make high-precision atomic clocks.

38. All-Optical (Photons) All-optical CNOT gate composed from beam splitters and wave plates. O'Brien et al., Nature 426(264), 2003

39. Lecture Outline ● Brief review ● DiVincenzo's criteria & tech overview ● Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. ● Comparison

40. Comparison ● NMR (Keio, Kane): excellent coherence times, slow gates – nuclear spin well isolated from environment – Kane complicated by matching VLSI pitch to necessary P atom spacing, and alignment ● Superconducting: fast gates, but fast decoherence ● Quantum dots: ditto – electrons in solid state easily influenced by environment

41. Comparison (2) ● Ion trap: medium-fast gates, good coherence time (one of the best candidates if scalability can be addressed) ● Optical lattice (atoms): medium-fast gates, good coherence time; gates and addressability of individual atoms need work ● All-Optical (photons): well-understood technology for individual photons, but hard to get photons to interact, hard to store

42. By the Numbers Apples-to-apples comparison is difficult, and coherence times are rising experimentally.

43. Quantum Error Correction and the Threshold Theorem Our entire discussion so far has been on “perfect” quantum gates, but of course they are not perfect. Various “threshold theorems” have suggested that we need 10^4 to 10^6 gates in less than the decoherence time in order to apply quantum error correction (QEC). QEC is a big enough topic to warrant several lectures on its own.

44. Wrap-Up ● Qubits can be physically stored on electrons (spin, count), nuclear spin, photons (polarization, position, time), or phenomena such as current (flux); anything that is quantized and subject to Schrodinger's wave equation. ● More than fifty technologies have been proposed; all of these described are relatively advanced experimentally.

45. Wrap-Up (2) ● Many technologies depend on VLSI ● Most are one or two qubits ● Have not yet started our own Moore's Law doubling schedule ● Several years yet to true, controllable, multi- qubit demonstrations ● 10-20 years to a useful system?

46. Upcoming Lectures ● Quantum computer architecture – how do you scale up? how do you build a computer out of this? what matters? ● Quantum networking – Quantum key distribution – Teleportation ● Wrap-Up and Review