Stochastic OneWay Quantum Computing with Ultracold Atoms in

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Stochastic One-Way Quantum Computing with Ultracold Atoms in Optical Lattices Michael C. Garrett David

Stochastic One-Way Quantum Computing with Ultracold Atoms in Optical Lattices Michael C. Garrett David L. Feder (supervisor) CQISC August 16, 2006.

OUTLINE The Quantum Circuit Model One-Way Quantum Computing Cluster States from Ultracold Atoms in

OUTLINE The Quantum Circuit Model One-Way Quantum Computing Cluster States from Ultracold Atoms in Optical Lattices PROBLEM: Imperfect Cluster States SOLUTION: Stochastic Protocol SUMMARY

The Quantum Circuit Model (the standard) U 1 m 1 U U m 2

The Quantum Circuit Model (the standard) U 1 m 1 U U m 2 2 U U 3 m 3 U U U 4 m 4 Time {U} Universal set of gates (eg. {CZ, R(x, h, z) } )

One-Way Quantum Computing 1) Initialize qubits:

One-Way Quantum Computing 1) Initialize qubits:

One-Way Quantum Computing 2) Entangle qubits: Apply CZ gates to nearest neighbors “cluster state”

One-Way Quantum Computing 2) Entangle qubits: Apply CZ gates to nearest neighbors “cluster state”

One-Way Quantum Computing 3) Remove unwanted qubits: Z-basis measurements “real-space quantum circuit”

One-Way Quantum Computing 3) Remove unwanted qubits: Z-basis measurements “real-space quantum circuit”

One-Way Quantum Computing 4) Computation via XY measurements & feedforward: horizontal chains = logical

One-Way Quantum Computing 4) Computation via XY measurements & feedforward: horizontal chains = logical qubits vertical links = 2 -qubit gates

One-Way Quantum Computing “one-bit teleportation” (the key identity) x-basis measurement HRZ(x) x m =

One-Way Quantum Computing “one-bit teleportation” (the key identity) x-basis measurement HRZ(x) x m = By-product operator “Classical feedforward” Sufficient for arbitrary single-qubit rotations

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing Single qubit rotation:

One-Way Quantum Computing x z h j Universal set of operators

One-Way Quantum Computing x z h j Universal set of operators

Cluster States from Ultracold Atoms in Optical Lattices JILA Bose-Einstein Condensate Superfluid UHH Optical

Cluster States from Ultracold Atoms in Optical Lattices JILA Bose-Einstein Condensate Superfluid UHH Optical Lattice Mott Insulator Goal: One atom per lattice site IQO

Cluster States from Ultracold Atoms in Optical Lattices • Ising interactions • Heisenberg interactions

Cluster States from Ultracold Atoms in Optical Lattices • Ising interactions • Heisenberg interactions • collisional phase shifts Ideally, j f j+1 In practice, imperfect cluster states

PROBLEM: Imperfect cluster states x q HRZ(x) m = S(p +q) Fidelity loss is

PROBLEM: Imperfect cluster states x q HRZ(x) m = S(p +q) Fidelity loss is small if q «p Over a series of teleportations, fidelity losses add up M. S. Tame, et al. , PRA 72, 012319 (2005).

SOLUTION: Stochastic protocol HRZ(x) x q 0 q = S(p +q) m m' H

SOLUTION: Stochastic protocol HRZ(x) x q 0 q = S(p +q) m m' H X S(p +q) X ; m' = 0 (failure) ; m' = 1 (success) q = 0 : max entangled q = p : unentangled

SOLUTION: Stochastic protocol j j+1 S(f) X D. Jaksch, et al. , PRL 82,

SOLUTION: Stochastic protocol j j+1 S(f) X D. Jaksch, et al. , PRL 82, 1975 (1999).

SOLUTION: Stochastic protocol HRZ(x) x q 0 q = S(p +q) m m' H

SOLUTION: Stochastic protocol HRZ(x) x q 0 q = S(p +q) m m' H X S(p +q) X ; m' = 0 (failure) ; m' = 1 (success) H? q = 0 : max entangled q = p : unentangled

SOLUTION: Stochastic protocol ; m' = 0 (failure) ; m' = 1 (success) H

SOLUTION: Stochastic protocol ; m' = 0 (failure) ; m' = 1 (success) H can be inserted manually (single atom addressing) H

SOLUTION: Stochastic protocol Repair via concatenation x q 0 q 0 q , •

SOLUTION: Stochastic protocol Repair via concatenation x q 0 q 0 q , • flag success in advance (Clifford measurements) • physically rearrange good/bad chains Improved success rates

SUMMARY ØSystematic phase errors expected (imperfect cluster states) ØStochastic protocol can perform perfect teleportation

SUMMARY ØSystematic phase errors expected (imperfect cluster states) ØStochastic protocol can perform perfect teleportation ØSuccess determined by X-basis measurements (Clifford) ØSuccess increased via concatenation and physical manipulation Can prepare error-free algorithm-specific graph states in advance!

Referee’s Report… (recent update) ; m' = 0 (failure) ; m' = 1 (success)

Referee’s Report… (recent update) ; m' = 0 (failure) ; m' = 1 (success) H can be inserted manually (single atom addressing) cannot H S(p +q) X

Referee’s Report… (recent update) q 0 q Distillation perspective Z H Not universal…?

Referee’s Report… (recent update) q 0 q Distillation perspective Z H Not universal…?

Thank You! David Feder (supervisor) Peter Hoyer (co-supervisor) Nathan Babcock (CQISC organizer)

Thank You! David Feder (supervisor) Peter Hoyer (co-supervisor) Nathan Babcock (CQISC organizer)