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

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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 2001 - Raussendorf, Browne & Briegel, PRA 2003 just type “one-way” or “cluster state” on the archive.

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

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

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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>)

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

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

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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)

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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

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10/21 Work on Fault-tolerance in the one-way model -Raussendorf, PhD Thesis (2003) (http://edoc.ub.unimuenchen.de/archive/00001367) -Nielsen and Dawson, PRA 71, 042323 (2005) -Aliferis and Leung, PRA 73, 032308 (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, 020501 (2006) PRA 73, 052306 (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, 100501 (2005) -Varnava, Browne and Rudolph PRL 97, 120501 (2006) Loss tolerant schemes for linear optics setups -Raussendorf, Harrington and Goyal, Ann. Phys. 321, 2242 (2006) -Raussendorf and Harrington, quant-ph/0610082 (2006) Fault-tolerant using topological error correction and surface codes -Silva et al., quant-ph/0611273 (2006) -Fujii and Yamamoto, quant-ph/0611160 (2006) Most Recently: -Dawson, Haselgrove and Nielsen, PRL 96, 020501 (2006). -Silva et al., quant-ph/0611273 (2006).

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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?

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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, 150504 (2006)

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13/21 Decoherence-free subspace one-way model - Simple protection from collective noise G. M. Palma et al., Proc. Roy. Soc. London A 452, 567-584 (1996) Basic 1-bit teleportation unit: 4 physical qubits

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14/21 Decoherence-free subspace one-way model - Protection from all types of collective noise (I) Theory: Kempe et al., PRA 63 042307 (2001) Experiment: Bourenanne et al., PRL 92 107901 (2004)

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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

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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

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

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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, 498-501 (2000) for single qubit DFS encoding.

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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

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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 QUINFO @ London, UK : Chris Hadley, Sougato Bose Palermo, Italy : Massimo Palma

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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/0602096 -Raussendorf, Browne and Briegel, PRA 68, 022312 (2003). -Dawson, Haselgrove and Nielsen, PRL 96, 020501 (2006) PRA 73, 052306 (2006) -Lidar and Birgitta Whaley, "Irreversible Quantum Dynamics", F. Benatti and R. Floreanini (Eds.), pp. 83-120 (Springer Lecture Notes in Physics vol. 622, Berlin, 2003); also at quant-ph/0301032 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*

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t=0.15 t=0.5 t=1 t=5

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