1. Quantum Computing: Solving Complex Problems David DiVincenzo, IBM Fermilab Colloquium, 4/2007

2. One of the motivating ideas of quantum computation was that there could be a new kind of machine that would solve hard problems in quantum mechanics. There has been significant progress towards the experimental realization of these machines (which I will review), but there are still many questions about how such a machine could solve computational problems of interest in quantum physics. New categorizations of the complexity of computational problems have now been invented to describe quantum simulation. The bad news is that some of these problems are believed to be intractible even on a quantum computer, falling into a quantum analog of the NP class. The good news is that there are many other new classifications of tractability that may apply to several situations of physical interest.

3. Rules for quantum computing Consider this form of two-bit boolean logic gate: D. Deutsch, Proc. R. Soc. A 400, 97 (1985) D. Deutsch, Proc. R. Soc. A 425, 73 (1989) x y y  x add the bits mod 2 = “controlled-NOT” (CNOT) 1 01 1

4. Rules for quantum computing Quantum rules of operation : D. Deutsch, Proc. R. Soc. A 400, 97 (1985) D. Deutsch, Proc. R. Soc. A 425, 73 (1989) x y y  x = “controlled-NOT” (CNOT) |0|0 output not factorizable! Creation of entanglement |0|0 U one-qubit rotations – create superpositions (| 0  + | 1  ) 1 2

5. Fast Quantum Computation P. Shor, AT&T, 1994

6. Physical systems actively considered for quantum computer implementation 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

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

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

9. Simple electric circuit… small L C 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.

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

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

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

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

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

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

17. Feb. 2007: D-wave systems announces 16 qubit processor 4x4 array of “flux qubits” (RF SQUIDs, tunable by flux biasing) DC SQUID couplers, also flux-tunable Great progress? Hold on a minute… They propose to use an alternative approach to quantum computing:

18. Hartmann, Phys. Rev. B 59, 3617 - 3623 (1999) Slow variation of couplings… Spins on a lattice, or other quantum two-level systems Quantum Simulated Annealing Optimization problems encodable in problem of finding lowest energy state of Ising spin glass. Uniform transverse field Variable out-of-plane field Ferromagnetic exchange Antiferromagnetic exchange Adiabatic evolution of the quantum system in its ground state

19. Farhi et al., quant- ph/0001106. Possible gap scalings: Very unlikely to be constant with n. Problems with small gaps: Landau-Zener tunneling Thermal excitations Constraints on Adiabatic Computation

20. Richard Feynman: First paper on Quantum computers

21. Quantum simulation will be useful, but there will be limitations. 2007 Perspective:

22. An overview of complexity NC P BPP NP NC: logarithmic time P: polynomial time BPP: polynomial time, allowing for some error NP: “nondeterministic polynomial”, answer can be confirmed in polynomial time

23. Classical Spin Glass: NP-complete That is, Barahona ’82: (H) is the ground-state energy. Is there a similar result for quantum spin glasses? Hartmann, Phys. Rev. B 59, 3617 - 3623 (1999) Variable out-of-plane field Ferromagnetic exchange Antiferromagnetic exchange

24. Quantum complexity NC P BPP NP MA/AM: “Merlin-Arthur” – NP with interaction BQP: polynomial time on a quantum computer QMA: answer can be confirmed in polynomial time on a quantum computer: quantum analog of NP MA AM PostBPP QMA BQP

25. Decision Problems in Computational Complexity Merlin. The all-powerful prover which cannot necessarily be trusted. The goal of Merlin is to prove to Arthur that the answer to his decision problem is YES. If this can be achieved the proof is complete. If the answer is NO, Merlin can still try to convince Arthur that the answer is yes. The proof is sound if Arthur cannot be convinced. Arthur. A mere mortal who can run polynomial time algorithms only. Wants to solve the decision problem by possible interaction(s) with Merlin.

26. The bad news: finding the ground state energy of the “quantum spin glass”: is QMA-complete. (Kitaev, Kempe, Regev, Oliveira, Terhal) Implication: there are quantum simulation problems that we expect to be hard, even on a quantum computer. (arbitrary quantum couplings)

27. Conclusions -- Hardware for quantum computing is progressing steadily -- Small working machines will be with us in the coming years -- Simulation of hard quantum problems with a quantum computer is clearly possible -- Some problems will be tractable, some intractable -- Progress will be empirical rather than rigorous