DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 1/22 Whither Quantum Computing? 2007 CQCT ANNUAL WORKSHOP Sydney, Australia 8 February 2007 funding by ARO, NSERC gratefully acknowledged Department of Physics and Center for Quantum Information and Quantum Control (CQIQC) University of Toronto Daniel F. V. JAMES

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 2/22 DiVincenzo’s Five Commandments* 1. Scalable, well-characterized qubits. 2. Ability to initialize qubits in some pure state. 3. Long decoherence time compared to gate time. 4. Universal set of quantum gates. 5. Measurement capability. *D. P. DiVinzenco, Fortschr. Phys. 48 (2000) Are these obsolescent? Strong vs. Weak measurement in #5? KLM redefines #4. Cluster states redefines #2, #4

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 3/22 -updated Clock states, DFS SET detectors (  > 80%) QLD, APL gates NMR Algorithmic cooling Entanglement at UCSB

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 4/22 Trapped Ions Cannot trap ions electrostatically (Earnshaw’s Theorem) Do it dynamically (Paul trap) Effectively a harmonic well in all 3D oscillating saddle potential

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 5/22 Phonon Modes Ions coupled by Coulomb force  ions’ oscillations have normal modes. Lowest mode: center-of-mass (CM): Next mode is “stretch” mode: Number of modes in each direction = number of ions. D.F.V. James, Appl Phys B 66, (1998)

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 6/22 Trapped Ion QC* Laser pulses can: Perform Rabi flips between and. Excite phonons of the longitudinal vibration modes. *J. I. Cirac and P. Zoller, Phys Rev Lett 74, 4091 (1995) Harmonic potential

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 7/22 Dephasing due to Magnetic Fields Random drift of the ‘static’ magnetic field (the “Britney Spears effect”) B (collective operator) average over random B field

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 8/22 Two Qubits Linear response to a zero mean, stationary B field: White noise limit :

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 9/22 chose B 0 so this term is zero B 0 = Gauss Fixes 1: NIST 1. “Magic” Magnetic Fields* *C. Langer, et al. [die Winelandern], "Long-lived qubit memory using atomic ions," Phys. Rev. Lett. 95, (2005) T 2 > 10 seconds bias field random field Breit-Rabi Formula: = effective hyperfine B field I = nuclear spin = magnetic quantum numbers of levels

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 10/22 Fixes 2: IQOQI (Innsbruck) 2. Decoherence Free Spaces* *H. Häffner, et al. [die Blattern], "Robust Entanglement", Appl. Phys. B 81, 151 (2005). Consider dephasing of 2 ions: there are two eigenstates with zero eigenvalue: they don’t dephase, regardless of the value of B: -use this pair of two-qubit states as a single logical qubit. -use sub-levels of 2 S 1/2 to avoid spontaneous decay. T 2 = 34 ± 3 seconds

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 11/22 Non-resonant gates Large detuning + temporal averaging of dynamics*: remove Stark-shift term by destructive interference with another field at -  : Mølmer-Sørensen gate. *DFVJ and J. Jerke, "Effective Hamiltonian Theory and Its Applications in Quantum Information,” Can. J. Phys., in the press (2007)

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 12/22 *L. Wu and DFVJ, in preparation Phase shift, realized by A.C. Stark effect Quantum A.C. Stark gate Logical CZ Gate*

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 13/22 So: Near-term prospects $64,000,000 question: Is it scalable? Logical gates between logical qubits Small scale factoring: 5 qubits, 3 CNOTs. Comparisons of different error mitigation techniques in the same environment. Proof-of-principal for parallel operations. Cluster state preparation/p.o.p. experiment.

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 14/22 Do we need a 6th Commandment? Shor’s Algorithm is the the “killer app”. [1] State of the Factoring Art with Conventional Computers: RSA-640 (640 bits) factored on a distributed network with a number field sieve in 5 months ( sec) [1]. [2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek, Phys. Rev. Lett. 77, 3240 (1996), eq.(7). Quantum factoring (without error correction) of a N-bit number requires ~ 544 N 3 two qubit quantum gates [2]; Sixth Commandment: for quantum computers to be really useful, quantum gates need to take ~1 nanosecond. Can we do it in, say, 5 minutes?

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 15/22

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 16/22 What’s the Speed Limit for Trapped Ions? Bottom line: T gate > ~10  sec. [2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek, Phys. Rev. Lett. 77, 3240 (1996). Resolve ions spatially [2]: Resolve modes in frequency [1]: [1] D. F. V. James, Appl. Phys. B 66, 181 (1998). 

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 17/22 *J.J. Garcia-Ripoll, P. Zoller and J. I. Cirac, Phys. Rev. Lett. 91, (2003) High Speed Gates* Ultrafast pulses interacting with two trapped ions. CM and stretch modes interfere to create a CZ gate. Cost: considerable increase in complexity N pulses ~ (T gate f trap ) -3/2 ; if T gate ~ 1 nsec, f trap ~ 10 MHz, N pulses ~ 10 3 : can this be handled if need be? No need for spatial resolution; insensitive to heating. individual pulses ~ spontaneous decay rate of upper level (i.e. ~10 nsec). BUT: Results of Monroe et al: 5 nsec Rabi oscillations; psec non-deterministic atom-photon entanglement

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 18/22 It gets worse... Gates in multi-trap architectures have five-steps: 1. Extract two ions from “storage” trap. 2. Move ions to “logic” trap. 3. Sympathetic cooling. 4. Perform logic gate. 5. Return ions to “storage” trap. Speed issues in moving ions around: 1. Move fast + sympathetic cooling OR move slow. 2. Steep potentials  small traps  heating.

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 19/22 Moving Trapped Ions Quickly displacement of trap center Ground State Fidelity:

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 20/22 Are cluster states the answer? * * R. Raussendorff and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). Definitions: Number of qubits in a circuit = breadth, m Number of gates in a circuit = depth, n Claim: For any quantum circuit there exists a pure state  (m,n) such that:  (m,n) involves O(m.n) qubits  (m,n) can be prepared with poly(m.n) resources Local measurement in an appropriate basis + feed forward simulates the quantum circuit.

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 21/22 Cluster States: Create 2D cluster state with only 4 CZ gates (6 for a 3D cluster) done in parallel. Do not worry about heating after Cluster State created. readout time and memory give new speed limit. Good: Bad: Need a bunch more qubits. Cannot make a 3D cluster needed (?) for fault-tolerance. Ugly:

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 22/22 Conclusion Progress in the decade since publication of Cirac-Zoller proposal has been excellent, but there is still a still a long, long way to go... My bet with Andrew White (made in 1998): will there be a quantum computer which can break 64 bit encryption in 30 minutes by 2023? (Anyone want a piece of the action?)

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 23/22 “The English always win one battle......the last” E. Venizelos, 1919 Parthian shot (cricket)