Experimental one-way quantum computing

Student presentation by

Andreas Reinhard

Outline

1. Introduction

2. Theory about OWQC

3. Experimental realization

4. Outlook

Introduction• Standard model:

– Computation is an unitary (reversible) evolution on the input qubits

– Balance between closed system and accessibility of qubits=> decoherence, errors

– Scalability is a problem

Introduction• A One-Way Quantum Computer1

proposed for a lattice with Ising-type next-neighbour interaction

– Hope that OWQM is more easlily scalable– Error threshold between 0.11% and 1.4% depending

on the source of the error2 (depolarizing, preparation, gate, storage and measurement errors)

– Start computation from initial "cluster" state of a large number of engangled qubits

– Processing = measurements on qubits => one-way, irreversible

1R. Raussendorf, H. J. Briegel, A One-Way Quantum Computer, PhysRevLett.86.5188, 20012R. Raussendorf, et al., A fault-tolerant one-way quantum computer, ph/050135v1, 2005

Cluster states• Start from highly entangled configuration of "physical"

qubits.Information is encoded in the structure: "encoded" qubits

• quantum processing = measurements on physical qubits• Measure "result" in output qubits

• How to entangle the qubits?

Entanglement of qubits with CPhase operations

• Computational basis: • Notation:

• Prepare "physical" 2-qubit state (not entangled)

• CPhase operation =>highly entangled state

1 1 2 2

1 1 10 1 0 1 00 10 01 11

22 2

0 , 1

1 2

100 10 01 11

2 –

Cluster states

• Prepare the 4-qubit state

• and connect "neighbouring" qubits with CPhase operations.The final state is highly entangled:

• Nearest neighbour interaction sufficient for full entanglement!

1+ 0 1

2 where

10 1

2

0 0

0 111 02

1 1

Cluster state

Operations on qubits

• Prepare cluster state

• We can measure the state of qubit j in an arbitrarily chosen basis

• Consecutive measurements on qubits 1, 2, 3 disentangle the state and completely determine the state of qubit 4.

• The state of "output" qubit 4 isdependent on the choses bases.

• That‘s the way a OWQC works!

1+ , where 0 1

2i

j j j j j jB e

A Rotation• Disentangle qubit 1 from qubits 2, 3, 4

• and project the state on => post selection

2 22 3 4

2 22 3 4

2 3 41 1

2 3 4 2 22 3 4

2 22 3 4

cos sin2 2

cos sin2 20

0 01

sin cos2 2

sin cos2 2

i i

i i

i i

i i

e e i

e e i

e i e

e i e

( ) ( )

1 2 3 4 0 R R other 3 termsx z

2 3

Single qubit rotation

SU(2) rotation & gates• A general SU(2) rotation and 2-qubit gates

• CPhase operations + single qubit rotations = universal quantum computer!

A one-way Quantum Computer

• Initial cluster structure <=> algorithm

• The computation is performed with consecutive measurements in the proper bases on the physical qubits.

• Classical feedforward makesa OWQC deterministic

Clusters are subunits of larger clusters.

Experimental realization1

• A OWQC using 4 entangled photons

• Polarization states of photons = physical qubits

• Measurements easily performable. Difficulty: Preperation of the cluster state

1P. Walther, et al, Experimental one-way quantum computing, Nature, 434, 169 (2005)

Experimental setup• Parametric down-conversion with

a nonlinear crystal

• PBS transmits H photons and reflects V photons

• 4-photon events:

• => Highly entangled state

• Entanglement achieved through post-selection

• Equivalent to proposed cluster state under unitary transformations on single qubits

HHHH HHVV VVHH VVVV

State tomography• Prove successful generation of cluster state => density matrix

• Measure expectation values

in order to determine all elements

• Fidelity:

2

H ,

V ,

1A B C D with A , B , C , D H V ,2

1H V

2

Cluster

i

0.63 0.02Cluster ClusterF

Realization of a rotationand a 2-qubit gate

• Output characterized by state tomography

• Rotation:

• 2-qubit CPhase gate:

2 0.86 0.03

0.85 0.044 20.83 0.030

F

0

0.84 0.03F

Problems of this experiment

• Noise due to imperfect phase stability in the setup (and other reasons). => low fidelity

• Scalability: probability of n-photon coincidence decreases exponentially with n

• No feedforward• No storage• Post selection

=> proof of principle experiment

Outlook• 3D optical lattices with Ising-type interacting

atoms

• Realization of cluster states on demand with a large number of qubits

• Cluster states of Rb-atoms realized in an optical lattice1

– Filling factor a problem– Single qubit measurements not realized

(adressability)

1O. Mandel, I. Bloch, et al., Controlled collisions for multi-particle entanglement of optically trapped atoms, Nature 425, 937 (2003)