Trapped Ions and the Cluster State Paradigm of Quantum Computing

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. • Unitary operations on any individual qubit: A+ B1  A + B1 ' ' • 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 • Projective measurement of each qubit: i.e. A0+ B1  0 (probability P0=|A|2) OR A0+ B1  1 (probability P1=|B|2) U Z Standard Paradigm for Quantum Computing You can do ANYTHING if you can do the following things with initialized qubits:

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

5. • 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 (9.0 106 sec) [1]. • Quantum factoring (without error correction) of a N-bit number requires ~ 544 N3 two qubit quantum gates [2]. • Sixth Commandment: for quantum computers to be useful, quantum gates need to take less than 1 microsecond. [1] Factorization of RSA-155, www.rsasecurity.com/rsalabs/challenges/factoring/rsa155.html [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). Do we need a 6th Commandment? • Shor’s Algorithm is the the “killer app”.

7. • Time for 2-ion logic gates is limited by need to resolve different oscillation modes in frequency [1]: • Trapping frequency is limited by the need to spatially resolve individual ions with the laser [2]: [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). What’s the Speed Limit for Trapped Ions? • 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. displacement of trap center Moving Trapped Ions Quickly

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

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 Lett. 86, 5188 (2001); (also unpublished notes by M. A. Nielsen).

12. • 1. Transport Circuit: m  Rz() H H Z XmH Rz()    Zm Z • 2. Discard Circuit m  Circuit Identities

13. Z Z Z  Zd Z Zc  Circuit Identities • 3. Indirect Entangling Gate:  c H  H d  H H 

14. 1.UA 2.UB 3.UC Notation: Unitary UA followed by measurement; then UB followed by measurement, then UC followed by measurement. 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.

15.  Rz() Rx() U 2. H Rz(’) 1. H Rz() H  m1 Rz() H m2 Rz(’)  H Z H Z  H Single Unitary

16. m  Rz() H H Z XmH Rz()  Remember the Circuit Identities: • 1. Transport Circuit:

17.  Rz() Rx() U m1 Rz() H m2 Rz(’) H Single Unitary 2. H Rz(’) 1. H Rz()  H Z  Z H Xm2H Rz(’)Xm1H Rz()  ’=(-1)m1 output becomes Rx()Rz()

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

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

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

21. Z Z Z  Zd Z Zc  Remember this one?  c H  H d  H H 

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