Mark TameMark Tame

QTeQQTeQ - Quantum Technology at Queen’s - Quantum Technology at Queen’s

Queen’s University, BelfastQueen’s University, Belfast

Fault-tolerantFault-tolerant One-way quantum One-way quantum computation using minimal resourcescomputation using minimal resources

2/21The one-way model for quantum computation – Brief introduction

1) Preparation of |+>

2) Application of CZ ’s| > = |+> |+> |+> |+>

| > = 1/4(|+> |+> |+> |+> + |+> |-> |+> |-> + |-> |+> |-> |+> - |-> |-> |-> |-> )

Sac: |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.

3/21The one-way model for quantum computation – Brief introduction

3) Measurement process

?

4/21The one-way model for quantum computation – Brief introduction

3) Measurement process

(i)

| > = ( |0> |+> + |1> |-> )

| Q1> = ( |0> + |1>)

5/21The one-way model for quantum computation – Brief introduction

3) Measurement process

(ii)

| > = ( |0> |0> + |0> |1>+ |0> |1> - |0> |1> )

| Q1> = ( |0> + |1>)

| Q2> = ( |0> + |1>)

6/21The one-way model for quantum computation – Brief introduction

3) Measurement process

(iii)

7/21The one-way model for quantum computation – Brief introduction

3) Measurement process

8/21The 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)

9/21Noise 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

10/21Work 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 schemesfor 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).

11/21Problems 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?

12/21Minimal-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)

13/21Decoherence-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

14/21Decoherence-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)

15/21Decoherence-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

16/21Performance 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

17/21Performance of Decoherence-free subspace one-way model

- Theoretical (I)

18/21Performance 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.

19/21Outlook

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

20/21Special 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

21/21References

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*

t=0.15 t=0.5

t=1 t=5