DecoherenceFree Subspaces Quantum Repeaters

1
Decoherence-Free Subspaces Quantum Repeaters
Carolina Moura Alves (University of Oxford,
UK) 1st December 2004
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
2
Entanglement
Long-distance quantum communication

• A and B are arbitrarily far apart (distance L).
• A and B want to store shared maximally entangled
  pairs of qubits, to be used as, e.g. a resource
  in quantum key distribution and quantum
  teleportation.
• In (realistic) noisy channels, the amount of
  entanglement in each pair decreases exponentially
  with L.
• Efficient long-distance distribution of
  entanglement is essential in the realization of
  quantum networks.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
3
Entanglement
Quantum repeaters (I)

• Pairs of entangled states distributed over N1
  nodes.
• Distance between adjacent nodes is d.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
4
Entanglement
Quantum repeaters (II)

• Entanglement swapping between mth nearest nodes,
  such that the fidelity of pairs distributed over
  remaining N/m nodes is greater than Fmin.
• At each node, perform entanglement purification
  protocol with the M copies available.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
5
Entanglement
Quantum repeaters (III)

• Repeating the process iteratively
• Number of resources scales polynomially with L

A
B
distance between adjacent nodes
number of nodes
purification copies
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
6
Entanglement
Quantum memory

• High quality quantum memory at each node is
  essential for the success of the quantum
  repeaters protocol.
• After the protocol, pairs of nearly maximally
  entangled qubits will be stored by A and B, to be
  used as a resource.
• Greatest obstacle to successfully storing and
  processing quantum information decoherence!

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
7
Entanglement
Optical lattices as quantum memory

• Neutral atoms couple weakly to the environment.
• Qubits encoded in two internal states of neutral
  atoms can be efficiently stored in large numbers
  in optical lattices.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
8
Entanglement
Decoherence in optical lattices

• Main source of decoherence unwanted coupling
  with external fluctuating magnetic fields.
• Possible solutions
• Encoding logical qubits in protected states
  require many atoms in order to efficiently
  protect the qubit.
• Encoding logical qubits in decoherence free
  subspaces hard to implement but fully protect
  the qubit, as long as the sources of decoherence
  are well characterized.
• Quantum error correction highly entangled
  states, difficult to generate.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
9
Entanglement
Decoherence in optical lattices

• Main source of decoherence unwanted coupling
  with external fluctuating magnetic fields.
• Possible solutions
• Encoding logical qubits in protected states
  require many atoms in order to efficiently
  protect the qubit.
• Encoding logical qubits in decoherence free
  subspaces hard to implement but fully protect
  the qubit, as long as the sources of decoherence
  are well characterized.
• Quantum error correction highly entangled
  states, difficult to generate.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
10
Entanglement
Decoherence-free subspaces

• Coupling of memory qubits with external
  fluctuating magnetic field.
• Decoherence-free subspace (zero angular momentum
  subspace)

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
11
Entanglement
DFS quantum memory (I)

• Initializing the memory qubits in a well-defined
  state
• Pairs of singlets can be generated in a
  controlled way in optical lattices via e.g.
  controlled collisions or 1D entangling pipelines.
  

Neutral atom (qubit)
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
12
Entanglement
DFS quantum memory (II)

• Transfer the information between flying qubits
  (photons) and stationary qubits (neutral atoms)
• State transfer between photon and single atom in
  a cavity.
• Entangling operation between the DFS qubits and
  the flying qubit.
• Measurement of the flying/memory qubits state.

q
a
a


J
J
J
J
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
13
Entanglement
DFS quantum memory (III)

• Entangling operation between the DFS qubits and
  the flying qubit
• In the logical subspace

• Presence of atom q prevents atoms a from
  tunnelling between the sites -1 and 1, but not
  atoms b.
• In the absence of q, atoms swap between sites -1
  and 1 for .

C-SWAP, with q acting as the control qubit!
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
14
Entanglement
DFS quantum memory (IV)

• Swapping atoms 1 and 2, or atoms 3 and 4, of the
  DFS memory qubit corresponds to applying a DFS Z
  gate
• C-SWAP corresponds to C-Z gate in the logical
  subspace.
• Information is transferred between flying and
  memory qubit by

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
15
Entanglement
DFS quantum memory (V)

• Applying Hadamard gate to memory (flying) qubit
• DFS Hadamard via judicious choice of lattice
  parameters and interaction times (all the DFS
  operations are implemented from swap
  Hamiltonians ).
• Entangle memory an flying qubit via C-Z gate.
• Apply Hadamard gate to flying (memory) qubit,
  after which the state of the two logical qubits
  is

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
16
Entanglement
DFS quantum memory (VI)

• Measurement of the flying/memory qubits state
• Projective measurement on DFS logical subspace
  exploiting the logical states symmetry
  properties.
• is anti-symmetrical with respect to
  permutations between atoms 1 and 2, or 3 and 4
  is symmetrical with respect to same
  permutations.

• Beam-splitter transformation effectively
  projects state in symmetric and anti-symmetric
  subspaces.
• Implemented in optical lattices via a pure
  hopping Hamiltonian for

J
J
Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
17
Entanglement
Integration in repeaters protocol

• Entanglement purification/swapping C-NOT gate
  measurement in computational basis.
• C-Z gate between flying qubit in state and
  pair of DFS qubits in the same node.
• Measure flying qubit in Hadamard basis.
• Apply DFS Hadamard to each DFS qubit.
• Measure DFS qubits in logical basis.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.
18
Entanglement
Summary

• Quantum repeaters.
• DFS quantum memory in optical lattices.
• Robust quantum repeaters using DFS quantum
  memory.

Centre for Quantum Computation, Clarendon
Laboratory, University of Oxford.