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Stephen Bartlett

A simple nearest-neighbour two-body Hamiltonian system for which the ground state is a universal resource for quantum computation. Stephen Bartlett. Terry Rudolph. Phys. Rev. A 74 040302(R) (2006). Quantum computing with a cluster state.

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Stephen Bartlett

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  1. A simple nearest-neighbour two-body Hamiltonian system for which the ground state is a universal resource for quantum computation Stephen Bartlett Terry Rudolph Phys. Rev. A 74 040302(R) (2006)

  2. Quantum computing with a cluster state Quantum computing can proceed through measurements rather than unitary evolution Measurements are strong and incoherent: easier Uses a cluster state: • a universal circuit board • a 2-d lattice of spins in a specific entangled state

  3. So what is a cluster state? • Describe via the eigenvalues of a complete set of commuting observables • Cluster state is the +1 eigenstate of all stabilizers • Massively entangled (in every sense of the word) Stabilizer

  4. “State of the art” -Making cluster states Optical approaches Cold atom approaches

  5. Can Nature do the work? • Is the cluster state the ground state of some system? • If it was (and system is gapped), we could cool the system to the ground state and get the cluster state for free! • Has 5-body interactions • Nature: only 2-body intns • Nielsen 2005 – gives proof: no 2-body nearest-neighbour H has the cluster state as its exact ground state

  6. Some insight from research in quantum complexity classes • Kitaev (’02): Local Hamiltonian is QMA-complete • Original proof required 5-body terms in Hamiltonian • Kempe, Kitaev, Regev (‘04), then Oliviera and Terhal (‘05): 2-Local Hamiltonian is QMA-complete • Use ancilla systems to mediate an effective 5-body interaction using 2-body Hamiltonians • Approximate cluster state as ground state • Energy gap ! 0 for large lattice • Requires precision on Hams that grows with lattice size • Not so useful... M. Van den Nest, K. Luttmer, W. Dür, H. J. Briegel quant-ph/0612186

  7. Some insight from research in classical simulation of q. systems • Projected entangled pair states (PEPS) – a powerful representation of quantum states of lattices For any lattice/graph: • place a Bell state on every edge, with a virtual qubit on each of the two verticies • project all virtual qubits at a vertex down to a 2-D subspace Cluster state can be expressed as a PEPS state: F. Verstraete and J. I. Cirac PRA 70, 060302(R) (2004)

  8. Can we make use of these ideas?: • effective many-body couplings • encoding logical qubits in a larger number of physical qubits

  9. Encoding a cluster state • KEY IDEA: Encode a qubit in four spins at a site • Ground state manifold is a qubit code space

  10. Hamiltonian for lattice is Interactions between sites • Interact spins with a different Hamiltonian Ground state is

  11. Perturbation theory • Intuition: “strong” site Hamiltonian effectively implements PEPS projection on “weak” bond Hamiltonian’s ground state • Degenerate perturbation theory in First order: directly break ground-state degeneracy? All excited states of HS “Illogical states” Ground state manifold of HS “Logical states”

  12. Perturbation theory • Intuition: “strong” site Hamiltonian effectively implements PEPS projection on “weak” bond Hamiltonian’s ground state • Degenerate perturbation theory in Second order: use an excited state to break ground-state degeneracy? All excited states of HS “Illogical states” Ground state manifold of HS “Logical states”

  13. Perturbation theory • Look at how Pauli terms in bond Hamiltonian act

  14. Is it what we want? • Basically, yes. • Low energy behaviour of this system, for small , is described by the Hamiltonian • Ground state is a cluster state with first-order correction • System is gapped:

  15. Can we perform 1-way QC? • 1-way QC on an encoded cluster state would require single logical qubit measurements in a basis • Encoding is redundant ! decode • measure 3 physical qubits in |§i basis • if an odd number of |–i outcomes occurred, apply z to the 4th qubit • measure 4th in basis • Note: results of Walgate et al (’00) ensure this “trick” works for any encoding

  16. The low-T thermal state • Consider the low-temperature thermal state Is it useful for 1-way QC? • Two types of errors: • Thermal • Perturbative corrections

  17. Thermal logical-Z errors • Thermal state: cluster state with logical-Z errors occurring independently at each site with probability • Raussendorf, Bravyi, Harrington (’05): correctable if • Energy scales: Related to order of perturbation Perturbation energy

  18. Perturbative corrections • Ground state is a cluster state with first-order correction • Treat as incoherentxz errors occurring with probability • x-error ! out of code space • appears as measurement error in computation

  19. has 2-d singlet ground state manifold Conclusions/Discussion • Simple proof-of-principle model – Can it be made practical? • Energy gap scales as where n is the perturbation order at which the degeneracy is broken ! use hexagonal rather than square lattice • Generalize this method to other PEPS states? • Use entirely Heisenberg interactions?

  20. Conclusions/Discussion

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