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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: [email protected] 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+ B1 A' + B '1 U • Two qubit gates such as the “Controlled Z gate” a + b1 c1 + d11 a + b1 c1 - d11 • Projective measurement of each qubit: i.e. A0+ B1 0 (probability P0=|A|2) OR A0+ B1 1 (probability P1=|B|2) Z 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 DiVincenzo’s Criteria 1. S ch cab ar le ac , w te e riz ll2. ed In itia qu liz bit 3. at s ion Lo tim ng es de co 4. he Un re nc ive e rs a lg 5. at M es ea su re m en t Roadmap Traffic-Light Diagram (Apr 2004) - updated • watch these spaces NMR Experimental reality Trapped Ion Theoretical Possibility Neutral Atom No known viable approach Optical Solid State Superconducting Cavity QED Do we need a 6th Commandment? • Shor’s Algorithm is the the “killer app”. • 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]. 9.0 106 544 512 3 123 sec • 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). 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]: Tgate ftrap 2 1 2.6 • Trapping frequency is limited by the need to spatially resolve individual ions with the laser [2]: 1/ 3 2 q ions F 2 4 M 0 ions 2ftrap • Bottom line: you’re limited to ~10 MHz Appl. Phys. B 66, 181 (1998). [1] D. F. V. James, [2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek, Phys. Rev. Lett. 77, 3240 (1996). It gets worse... • Gates in scalable (multi-trap) architectures have fivesteps: 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. Moving Trapped Ions Quickly 1 2 1 2 2 ˆ ˆ ˆ H p m x s t 2m 2 displacement of trap center • Solution: t e i t Dˆ t 0 t i t m s t e it dt 2 0 displacement operator Fidelity of the Ground State after motion: 2 t 1 F t exp 2 sÝt e it dt 4 0 width of the ground state 2m s t L T<<1/ 10/19 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). Circuit Identities • 1. Transport Circuit: Rz() H XmH Rz() Z • 2. Discard Circuit Z m H m Zm Circuit Identities • 3. Indirect Entangling Gate: H H Z Z H c H d Z Zd Z Zc 3x4 Cluster State 1.UA 2.UB 3.UC • Each circle represents a qubit. • Prepare each qubit in state |0. • Perform Hadamard Gate (AKA pulse) on each qubit. • Perform Controlled-Z between neighbors. Notation: Unitary UA followed by measurement; then UB followed by measurement, then UC followed by measurement. Single Unitary Rz() Rx() U 1. H Rz() H H H 2. H Rz(’) Rz() m1 H Rz(’) Z Z H m2 Remember the Circuit Identities: • 1. Transport Circuit: Rz() H Z H m XmH Rz() Single Unitary Rz() Rx() U 1. H Rz() 2. H Rz(’) Rz() H ’=(-1)m1 m1 H Rz(’) Z H Z H m2 Xm2H Rz(’)Xm1H Rz() output becomes Rx()Rz() Simple 2 Qubit Circuit U Z U 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 Remember the Circuit Identities Again • 2. Discard Circuit Z Zm m shifts on some of (so we’ll need to correct for the phase the qubits) You get this Cluster State: 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 Remember this one? H H Z Z H c H d Z Zd Z Zc 2.HRz() 3.HRz(’) 5.H 6.H 7.H 8.H 6.H 7.HRz() 8.HRz(’) 4.H 4.H 2.H 3.H 5.H Step 5&6: propagate the quantum information Step 7&8: perform second unitary 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?