1. Trapped Ions and the Cluster State Paradigm of Quantum Computing Universität Ulm, 21 November 2005 Daniel F. V. JAMES Department of Physics, University of Toronto, 60, St. George St., Toronto, Ontario M5S 1A7, CANADA Email: dfvj@physics.utoronto.ca

2. Standard Paradigm for Quantum Computing You can do ANYTHING if you can do the following things with initialized qubits: Unitary operations on any individual qubit: A  + B  1   A  + B  1  '' U Two qubit gates such as the “Controlled Z gate” a  + b  1   c  1  + d  1  1   a  + b  1   c  1  - d  1  1  Z Projective measurement of each qubit: i.e.A  0  + B  1    0  (probability P 0 =|A| 2 ) ORA  0  + B  1    1  (probability P 1 =|B| 2 )

3. DiVincenzo’s Five Commandments* 1. Scalable physical system with well-characterized qubits. 2. Ability to initialize the state of the qubits in some fiducial state. 3. Long (relative) decoherence times, much longer than gate-operation time. 4. Universal set of quantum gates (e.g. arbitrary one qubit operations + CNOT with any two qubits). 5. Qubit-specific measurement capability. *D. P. DiVinzenco, Fortschr. Phys. 48 (2000) 771-783

4. Roadmap Traffic-Light Diagram (Apr 2004) - updated watch these spaces

5. Do we need a 6th Commandment? Shor’s Algorithm is the the “killer app”. [1] Factorization of RSA-155, www.rsasecurity.com/rsalabs/challenges/factoring/rsa155.html State of the Factoring Art with Conventional Computers: RSA-155 (512 bits) factored on a distributed network with a number field sieve in 3.7 months (9.0 10 6 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 useful, quantum gates need to take less than 1 microsecond.

7. What’s the Speed Limit for Trapped Ions? Time for 2-ion logic gates is limited by need to resolve different oscillation modes in frequency [1]: [1] D. F. V. James, Appl. Phys. B 66, 181 (1998). [2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek, Phys. Rev. Lett. 77, 3240 (1996). Trapping frequency is limited by the need to spatially resolve individual ions with the laser [2]: Bottom line: you’re limited to ~10 MHz

8. It gets worse... Gates in scalable (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.

9. Moving Trapped Ions Quickly displacement of trap center

10. 10/19 displacement operator width of the ground state Solution: Fidelity of the Ground State after motion: L T<<1/ 

11. Are cluster states the answer?* * R. Raussendorff and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); (also unpublished notes by M. A. Nielsen). 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.

12. Circuit Identities 1. Transport Circuit:  H ZH Rz()Rz()  X m H R z (  )  m 2. Discard Circuit m Z   ZmZm 

13. Circuit Identities 3. Indirect Entangling Gate:     H H ZZZ H H c d   Z ZcZc ZdZd

14. 3x4 Cluster State Each circle represents a qubit. Prepare each qubit in state |0 . Perform Hadamard Gate (AKA  pulse) on each qubit. Perform Controlled-Z between neighbors. 1.U A 2.U B 3.U C Notation: Unitary U A followed by measurement; then U B followed by measurement, then U C followed by measurement.

15.    Single Unitary  Rx()Rx() Rz()Rz() U  H Rz()Rz() m1m1 1. H R z (  ) 2. H R z (  ’) H Rz(’)Rz(’) m2m2 ZH H Z H

16. Remember the Circuit Identities: 1. Transport Circuit:  H ZH Rz()Rz()  X m H R z (  )  m

17.    Single Unitary H Rz()Rz() m1m1  Rx()Rx() Rz()Rz() U  1. H R z (  ) 2. H R z (  ’) X m 2 H R z (  ’)X m 1 H R z (  )  ZH H Z H Rz(’)Rz(’) m2m2  ’=(-1) m 1    output becomes R x (  )R z (  ) 

18. Simple 2 Qubit Circuit   Z U  U  This needs a 4 x7 Cluster State: 1.I Step 1: measure indicated qubits and correct for discard

19. Remember the Circuit Identities Again 2. Discard Circuit m Z   ZmZm  (so we’ll need to correct for the phase shifts on some of the qubits)

20. You get this Cluster State: Step 2 &3: perform single qubit unitary as before 2.HR z (  ) 2.H3.H 3.HR z (  ’) Step 4: Measurement on linking qubits to perform two qubit gate operation 4.H

21. Remember this one?     H H ZZZ H H c d   Z ZcZc ZdZd

22. 2.HR z (  ) 2.H3.H 3.HR z (  ’) 4.H 5.H 6.H Step 5&6: propagate the quantum information Step 7&8: perform second unitary 7.HR z (  ) 8.HR z (  ’) 7.H8.H

23. Implications Quantum Computing is reduced to initially creating a big-ass entangled state, then local unitatries and measurement. This is a natural for optical quantum computing. What about trapped ions? - Number of Controlled Z gates reduced to 4 total! - Trap configuration can be optimized for cluster state creation - Will need a lot more ions - Basis requirements (read out and fast feed-forward) already demonstrated in teleportation experiment. - Can measurement be fast enough?