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 Stabilizer  Cluster state is the +1 eigenstate of all stabilizers  Massively entangled (in every sense of the word)

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 int n s  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?: 1. effective many-body couplings 2. 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. Interactions between sites  Interact spins with a different Hamiltonian Ground state is Hamiltonian for lattice is

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

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

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 4 th qubit measure 4 th 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: Perturbation energy Related to order of perturbation

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

19. 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? has 2-d singlet ground state manifold

20. Conclusions/Discussion