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

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

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)

“State of the art” -Making cluster states

Optical approaches

Cold atom approaches

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

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

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)

Can we make use of these ideas?:

1. effective many-body couplings

2. encoding logical qubits in a larger number of physical qubits

Encoding a cluster state KEY IDEA: Encode a qubit in four

spins at a site

Ground state manifold is a qubit code space

Interactions between sites Interact spins with a different

Hamiltonian

Ground state is

Hamiltonian for lattice is

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 HS

“Logical states”

All excited states of HS

“Illogical states”

First order: directly break ground-state degeneracy?

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 HS

“Logical states”

All excited states of HS

“Illogical states”

Second order: use an excited state to break ground-state degeneracy?

Perturbation theory Look at how Pauli terms in

bond Hamiltonian act

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:

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

The low-T thermal state Consider the low-

temperature thermal stateIs it useful for 1-way QC?

Two types of errors: Thermal Perturbative corrections

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

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

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

Conclusions/Discussion