Mark Tame QTeQ - Quantum Technology at Queen’s Queen’s University, Belfast Fault-tolerant One-way quantum computation using minimal resources

2/21 The one-way model for quantum computation – Brief introduction 1) Preparation of |+> 2) Application of CZ ’s |  > = |+> |+> |+> |+> |  > = 1/4(|+> |+> |+> |+> + |+> |-> |+> |-> + |-> |+> |-> |+> - |-> |-> |-> |-> ) S ac : |0> |0> --> |0> |0> |0> |1> --> |0> |1> |1> |0> --> |1> |0> |1> |1> --> - |1> |1> - R. Raussendorf & H.-J. Briegel, PRL Raussendorf, Browne & Briegel, PRA 2003 just type “one-way” or “cluster state” on the archive.

3/21 The one-way model for quantum computation – Brief introduction 3) Measurement process ?

4/21 The one-way model for quantum computation – Brief introduction 3) Measurement process (i) |  > = (  |0> |+> +  |1> |-> ) | Q 1 > = (  |0> +  |1>)

5/21 The one-way model for quantum computation – Brief introduction 3) Measurement process (ii) |  > = (  |0> |0> +  |0> |1>+  |0> |1> -  |0> |1> ) | Q 1 > = (  |0> +  |1>) | Q 2 > = (  |0> +  |1>)

6/21 The one-way model for quantum computation – Brief introduction 3) Measurement process (iii)

7/21 The one-way model for quantum computation – Brief introduction 3) Measurement process

8/21 The one-way model for quantum computation – Brief introduction Algorithms: Grover’s Algorithm Deutsch’s Algorithm Quantum Games M. S. Tame et al., PRL (2007) P. Walther et al., PRL (2005) M. Paternostro et al., NJP (2005)

9/21 Noise in the one-way model for quantum computation Environment effects during time evolution – Decoherence Pauli error General error Loss Local/Global noise: Pauli error General error Loss Preparation of |+> controlled phase gate error controlled unitary gate error Loss from non-deterministic gates Application of CZ ’s Measurement process error in measurement of qubits propagates into the remaining cluster Stage 1 Stage 2

10/21 Work on Fault-tolerance in the one-way model -Raussendorf, PhD Thesis (2003) ( -Nielsen and Dawson, PRA 71, (2005) -Aliferis and Leung, PRA 73, (2006) Proved that an Error Threshold existed, which could be determined by mapping noise in the cluster state to noise in a corresponding circuit model. -Dawson, Haselgrove and Nielsen, PRL 96, (2006) PRA 73, (2006) Error correcting schemes and associated error threshold values for optical setups STEANE 7 qubit and GOLAY 23 qubit codes -Ralph, Hayes and Gilchrist PRL, 95, (2005) -Varnava, Browne and Rudolph PRL 97, (2006) Loss tolerant schemes for linear optics setups -Raussendorf, Harrington and Goyal, Ann. Phys. 321, 2242 (2006) -Raussendorf and Harrington, quant-ph/ (2006) Fault-tolerant using topological error correction and surface codes -Silva et al., quant-ph/ (2006) -Fujii and Yamamoto, quant-ph/ (2006) Most Recently: -Dawson, Haselgrove and Nielsen, PRL 96, (2006). -Silva et al., quant-ph/ (2006).

11/21 Problems with Fault-tolerant schemes in the one-way model Large resource overheads: - A minimum of 7 qubits for an encoded qubit (STEANE code) Complicated structure for the encoded qubit: - Underlying graph to encode qubit is complex Error syndrome extraction techniques lead to additional overheads “One-buffered”, “two-at-a-time” and “fully-parallel” approaches complicate the model: - They modify the measurement patterns and entangling steps Off-line preparation of ancilla qubits can also be a cumbersome process: - setup dependent Q: Is there a way to achieve fault-tolerance using less resources?

12/21 Minimal-resource Fault-tolerance in the one-way model Local Collective noise 4-qubit collective noise 2-qubit collective noise 3-qubit collective noise Universal resource for one-way QC -Van den Nest, Miyake, Dür, Briegel PRL 97, (2006)

13/21 Decoherence-free subspace one-way model - Simple protection from collective noise G. M. Palma et al., Proc. Roy. Soc. London A 452, (1996) Basic 1-bit teleportation unit: 4 physical qubits

14/21 Decoherence-free subspace one-way model - Protection from all types of collective noise (I) Theory: Kempe et al., PRA (2001) Experiment: Bourenanne et al., PRL (2004)

15/21 Decoherence-free subspace one-way model - Protection from all types of collective noise (II) Knill, Laflamme and Viola PRL 84, 2525 (2000) (Decoherence-free subsystems) Basic 1-bit teleportation unit: 6 physical qubits

16/21 Performance of Decoherence-free subspace one-way model - Theoretical (I) M. S. Tame, M. Paternostro, M. S. Kim -submitted (2007) Probe states: QPT techniques: H H H H

17/21 Performance of Decoherence-free subspace one-way model - Theoretical (I)

18/21 Performance of Decoherence-free subspace one-way model - Experimental (II) R. Prevedel, M. S. Tame, A. Stefanov, M. Paternostro, M. S. Kim and A. Zeilinger -submitted (2007) Standard DFS encoded Information transfer protocol: 4 physical qubits Linear optical setup See also: Kwiat et al., Science 290, (2000) for single qubit DFS encoding.

19/21 Outlook M. S. Tame et al., work in progress (2007) 1) Investigating the performance of the fault-tolerance, for asymmetries in the collective approximation How does the performance of the 2- and 3-qubit Codes with asymmetries compare to standard cluster state Quantum Error Correcting Codes (QECC). 2) Most resourceful method for the 3-qubit code

20/21 Special thanks to Collaborators Queen’s, UK : Mauro Paternostro and Myungshik Kim Vienna, Austria : Robert Prevedel, André Stefanov, Pascal Böhi, Anton Zeilinger Leeds, UK : Vlatko Vedral London, UK : Chris Hadley, Sougato Bose Palermo, Italy : Massimo Palma

21/21 References DFS one-way QC -Hein et al., Proceedings of the International School of Physics "Enrico Fermi" on "Quantum Computers, Algorithms and Chaos", Varenna, Italy, July, 2005; also at quant-ph/ Raussendorf, Browne and Briegel, PRA 68, (2003). -Dawson, Haselgrove and Nielsen, PRL 96, (2006) PRA 73, (2006) -Lidar and Birgitta Whaley, "Irreversible Quantum Dynamics", F. Benatti and R. Floreanini (Eds.), pp (Springer Lecture Notes in Physics vol. 622, Berlin, 2003); also at quant-ph/ Introduction to graph states and one-way QC using cluster states Fault-tolerant one-way QC using QECC Introduction to DFS -M. S. Tame, M. Paternostro, M. S. Kim -submitted (2007) -R. Prevedel, M. S. Tame, A. Stefanov, M. Paternostro, M. S. Kim and A. Zeilinger -submitted (2007) *Thanks for your attention*

 t=0.15  t=0.5  t=1  t=5