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
