1. Quantum computing with Rydberg atoms Klaus Mølmer Coherence school Pisa, September 2012

2. Quantum computing with Rydberg atoms Klaus Mølmer Coherence school Pisa, September 2012

3. Introduction to Quantum Computing Physical implementations Rydberg atom quantum computing Contents: Quantum Computing with Rydberg atoms

4. Intel® Pentium® Dual Core T4200-processor, 2,0 GHz, 3072 MB SDRAM. (250 GB harddisk) 650 Euros. Quantum computer Processor unknown 1 kHz is fine, 100 Hz is OK 1 kbit RAM would be great ! Buy at 10 8 Euros Reality:Dream:

5. Classical bit: 0 or 1 ”Quantum bit”: 0 and 1 How can a small, slow, ”quantum computer” outperform a faster PC or laptop?

6. Quantum computing Idea: ”quantum is weird”  ”quantum is useful !” A particle wave function occupies different locations  A computer register can deal with several numbers at the same time. x 1  f(x 1 ) and x 2  f(x 2 ) in two steps becomes: (x 1 and x 2 )  (f(x 1 ) and f(x 2 )) in just one step.

7. Measurements constitute a fundamental limitation. We cannot measure state, a|0>+b|1>. We can ask: ”Is the system in state |0> ?” and the answer is Yes with probability |a| 2, and the state becomes |0> No with probability |b| 2, and the state becomes |1> You get random answers, and you do not learn more by asking twice ! |a| 2 +|b| 2 =1

8. Parallel processing on a single quantum computer a|x 1 > +b|x 2 >  a|f(x 1 )> + b|f(x 2 )> All x, Σ x c x |x>  Σ x c x |f(x)>, all f(x) by a single iteration of a single quantum register. We only need to solve two problems: readout: how to get all f(x) and not just a random f(x) ? (algorithms) Shor and Grover ! construction: how to build and control a mikroscopic quantum system ? (physics)

9. Is a computational problem easy or hard? How do the ressources needed scale with the problem size? Addition: 1 1 1 1 1 2 4 6 3 8 9 1 3 5 + 3 5 8 9 0 1 2 4 6 = 6 0 5 2 9 0 3 8 1 needs L operations for L-digit numbers Multiplication: 2 4 6 3 8 9 1 3 5 x 3 5 8 9 0 1 2 4 6 = needs L 2 operations for L-digit numbers

10. Harder problems: Find an element out of N, who fulfils a certain condition: max: N trials, average: N/2 attempts Find factors of an integer N trial and error: does 2 factor M ? does 3 ?, … until √M (max): ~ √M attempts Let M ~ 10 L (L digit number), then √M ~ (√10) L ~ exp(L) (exponentially hard) Quantum computer  new scaling !!! Shor 1994, Grover 1997, other ”single-output” problems  √N  L 3

11. Shor’s algorithm Find factor of stort large N Pick random a Let f(x) = a x mod N f(x) is periodic, f(x+p)=f(x) Determine numbers a p/2 ± 1 One of these numbers factors N Check ! (easy) If you fail, try another a N=15 a=2 1, 2, 4, 8, 1, 2.. p=4 2 2 ± 1 = 3 or 5 3 factors15 5 factors 15 Yes ! 2 0 =1, 2 1 =2, 2 2 =4, 2 3 =8, 2 4 =16  1, 2 5 =32  2, …

12. Arizona Daily Star, 20. February 2005

13. x Grover: 1) Σc x |x>  Σ (-1) f(x) c x |x> (-1) if x matches the “marked“ x 0 2) Inversion of c x about their mean. Rydberg blockade and quantum information

14. x 1/√N ~3/√N Repeat √N times Grover: 1) Σc x |x>  Σ (-1) f(x) c x |x> (-1) if x matches the “marked“ x 0 2) Inversion of c x about their mean. Rydberg blockade and quantum information Quantum systems can explore the Hilbert space of superposition states, and they can ”converge” on solutions to hard problems. Must implement the calculations on a physics quantum system !

15. Re-cycle classical computing paradigm Computers represent information (data) in binary form (bits), example.: 5 =1*2 2 +0*2 1 +1*2 0 =’101’ All data manipulations are evaluations of functions based on operations that decompose as single-bit and bit-pair logical operations: NOT, AND, OR …. NOT |0> |1>

16. Logical operations on quantum bits One-bit operation, NOT: 0  1 must work ”without looking” Two-bit operation, C-NOT: (0, bit)  (0,bit) (1, bit)  (1, NOT bit)

17. Qubits and gates Orthogonal states: |0> and |1> Orthogonal spin ½ states: |↑> and |↓> Bloch sphere picture, Unitary operations  rotations. Transition: |0>  |1> = rotation around x-axis Phase:  a|0>+b e iφ |1> = rotation around z-axis Same operation can be built In many different ways.

18. Basic building blocks Hadamard gate: |0>  |0>+|1> |1>  |0> - |1> H 2 = NOT (cannot do that classically): |0>  |1> |1>  |0> Phase gate: |0>  |0>, |1>  e iφ |1>. π/8 gate (T-gate): |1>  e i π/4 |1>. T and H do not commute: they span all rotations! C-Phase/C-NOT: same operations, but only carried out of ”control-qubit” is in state |0>.

19. Universality Classical computing: Gate alphabet, that allows computation of any function (fan-out + NAND). Surprising facts: Hadamard, π/8 gate and C-NOT is quantum universal. i.e. this set allows construction of any unitary operation on full register. (operating on any qubit and qubit pair). Hadamard, NOT and C-NOT gives no advantage for QC. We do not know which quantum property is needed for QC.

20. AARHUS UNIVERSITET Errors happen. Can we detect and correct them ? Classical approach: use copies and ”vote”: 0  000, 1  111, error: 001  000 by majority. In QM, copying is forbidden, and checking by measurements collapse superposition states a|0>+b|1>  0> or |1> Solution: a small algorithm |0>  |000>, |1  |111> 00: OK 01: NOT_3 10: NOT_2 11: NOT_1: Quantum Error correction (Steane and Shor)

21. AARHUS UNIVERSITET Errors happen. Can we detect and correct them ? Classical approach: use copies and ”vote”: 0  000, 1  111, error: 001  000 by majority. In QM, copying is forbidden, and checking by measurements collapse superposition states a|0>+b|1>  0> or |1> Quantum Error correction (Steane and Shor) 5-bit code can correct one flip or phase error  multi-bit codes, topological states, gapped states

22. AARHUS UNIVERSITET How many errors can we correct? N logical bits N  N 10 K physical bits G operations NG = 10 L  Critical parameter surface Error ε = 10 - 3 NG=10 10 L K The five 9’s: 1-10 -5 =0.99999 succes

23. AARHUS UNIVERSITET Quantum circuit and other models of QC Quantum circuit: Bits and sequence of gates Cluster state computing: Initial entangled grid of qubits. Perform measurements only, Program= measurement strategy: adapt observables. Dissipative quantum computing: Gates  decay channels, Stationary (dark) state solves problem.

24. AARHUS UNIVERSITET QC is a great theoretical idea … … but it only works if we build one.

25. Criteria and strategies for quantum computers ? 1, 2 many qubits Initial state00.. 0 Lifetime >> gatetime One- and two-bit gates Effective detection Coupling to flying bits Transmission

26. Carles Babbage (1792-1871) on atoms: ”…. Every atom, impressed with good and with ill, retains at once the motions which philosophers and sages have imparted to it, mixed and combined in ten thousand ways with all that is worthless and base.” Charles Babbage, Ninth Bridgewater Thesis, (1837).

27. Experiments by Drewsen, Aarhus.

28. 7-bit quantum computer, 15=5*3 (i 2002) many identical (natural) computers, majority vote C 11 H 5 F 5 O 2 Fe

29. Solid state computers - gallery. Quantum Dots (InAs/GaAs)

30. Atom chips

31. Hybrid technologies ”Self-hybridization”: Major decoherence due to coupling of electron spin to nuclear spin bath.  Identify and use the nuclear spins (individual or collective) as the qubits

32. Atomic systems Atoms, ions, el. and nucl. spins Long coherence times Couple weakly Identical systems Scaling is difficult Handling can be difficult Solid, man-made, systems Quantum dots, superconductors, cantilevers Short coherence times Couple strongly Different (inhomogeneous env./prod.) Scaling via microfabrication Fixed in material Hybrid systems for quantum processing

33. Atomic systems Atoms, ions, el. and nucl. spins Long coherence times Couple weakly Identical systems Scaling is difficult Trapping is difficult Solid, man-made, systems Quantum dots, superconductors, cantilevers Short coherence times Couple strongly Different (inhomogeneous env./prod.) Scaling via microfabrication Fixed in material Memory  HYBRID  Processing Hybrid quantum systems Coupling is a problem: Quite incompatible systems Different natural frequencies Bad spatial matching  Weak coupling strengths WORKSCALE

34. Full hybrid technologies Yale, Oxford, Zurich, Santa Barbara, Vienna, Saclay, Chalmers, …

35. And what if we fail completely… Niels Bohr (about Quantum Mechanics): ” … if we should one day wake up, and realize that it had all been only a dream, then I am absolutely convinced that we would still have learned something !”