Universal Quantum Computation with AKLT States and Spectral Gap of - - PowerPoint PPT Presentation
Universal Quantum Computation with AKLT States and Spectral Gap of AKLT Models Tzu-Chieh Wei ( ) C.N. Yang Institute for Theoretical Physics Workshop on Statistical Physics of Quantum Matter, Taipei, July 2013 Outline I.
Single-qubit measurements on the 2D cluster state
2D cluster state
[Raussendorf &Briegel, PRL01’] QC = pattern
Key points:
Equivalent to circuit model: Universal gates can be implemented
Via stabilizer generators:
X Z Z Z
Graph states: defined on any graph [Hein, Eisert & Briegel 04’] [Raussendorf &Briegel 01’]
Via controlled-Z gates:
Cluster states: special case of graph states on
(X,Y,Z: Pauli matrices)
[These eqs. uniquely define |G>.]
[Raussendorf &Briegel PRL 01’] Cluster-state QC = a set of measurement patterns
Can universal resource states be unique ground state?
Any other 2D graph states on regular lattices (≡cluster states):
MPS & PEPS framework: alternative view & further examples
[Van den Nest et al. ‘06] [Verstraete & Cirac ‘04] [Gross & Eisert ‘07, Gross, Eisert, Schuch & Perez-Garcia ‘10]
Other known examples: Can other states beyond the 2D cluster state be used
[Nielsen ‘04]
Spin-1 chain: two virtual qubits per site
singlet Project into symmetric subspace
[AKLT ’87,’88] [Brennen & Miyake, PRL ‘09] [Gross & Eisert, PRL ‘07]
In a large system, cluster state has length 2/3 of AKLT
Our approach uses a POVM: (outcome: x, y, z)
gives rise to a cluster state (a logical qubit is a domain of connected sites with same outcome)
Any outcome preserves a two- dimensional subspace
[Wei, Affleck & Raussendorf ’12]
Each site contains three virtual qubits Two virtual qubits on an edge form a singlet
singlet
Projector onto symmetric subspace Addition of angular momenta of 3 spin-1/2’s The four basis states in the symmetric subspace
Symmetric subspace
Effective 2 levels
Each site contains three virtual qubits Two virtual qubits on an edge form a singlet
singlet
Each site contains three virtual qubits Two virtual qubits on an edge form a singlet
singlet
Projection (PS,v) onto symmetric subspace of 3 qubits at each site
POVM outcome (x,y, or z) is random (av ={x,y,z} ϵ A for all sites v)
[Wei,Affleck & Raussendorf ’11; Miyake ‘11]
v: site index
Sufficient number of wires if graph is in supercritical phase (percolation)
Follow the same procedure Bond Percolation Threshold ≈ 0.6768 > 2/3
Neighboring sites with same POVM outcome
Two domains connected by even edges = no edge
Identify new “backbone” (may not exist on original graph)
Typical graphs are in percolated (or supercritical) phase
Site percolation by deletion (Honeycomb)
supercritical subcritical
Threshold = 1- Pdelete* ≈1-0.33=0.67
Sufficient (macroscopic) number of traversing paths exist (supercritical) These AKLT states (also that on ‘cross’) are universal for QC
(Square-octagon)
threshold ≈1-0.26=0.74
supercritical subcritical
Site percolation by deletion
[Wei,Affleck & Raussendorf ’11] [Wei ’13]
Pspan
Edges in triangles are removed with 50% (occupied with 50%) Edges connecting triangles never removed 50% is smaller than bond percolation threshold (≈0.5244) of Kagome No connected path AKLT not universal
Whether such spin-2 state is universal remains open
Technical problem: trivial extension of POVM does NOT work!
Leakage out of logical subspace (error)
Hamiltonian Known to have exponential decaying
We use tensor network methods to show the existence
[AKLT ’87,’88]
See Artur Garica’s poster for details
By applying an external field, can probe the gap Ground state is a spin singlet state;
Schematic energy response
h E
0.05 0.1 0.15 0.2 0.4 0.6 0.8 1 1.2
h
singlet
1D AKLT with N=8
A, B, C, … traces lowest energy
First cross and the slope infer E1 - E0 Slope = Magnetization Plateau finite gap
Hamiltonian:
E
0.5 1 1.5 2 2.5 3 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2
h M
−5.5 −5 −4.5 −4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 log(h − hc) log(M)
0.2 0.4 0.6 0.8 1 1.2 1.4 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1
h e0
Energy per spin Magnetic moment per spin Gap ∆ ≈ 0.350
Hamiltonian:
Energy per spin Magnetic moment per spin Gap ∆ ≈ 0.10
0.5 1 1.5 2 2.5 3 3.5 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0.2
h
M
0.05 0.1 −0.2 −0.15 −0.1 −0.05 D=2 D=3 D=4 D=5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
h e0
Several AKLT states on 2D lattices provide resources for
AKLT Hamiltonians on the honeycomb (and square)
Spin-2 AKLT state on square lattice universal?
Wei, arXiv:1306.1420 Wei, Affleck & Raussendorf, Phys. Rev. A 86, 032328 (2012) Garcia-Saez, Murg & Wei, in preparation Raussendorf & Wei, Annual Review of Cond. Mat. Phys., vol.3, 239 (2012) Wei, Affleck & Raussendorf, Phys. Rev. Lett. 106, 070501 (2011)