Research ArticleQUANTUM INFORMATION

Stabilizers as a design tool for new forms of the Lechner-Hauke-Zoller annealer

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Science Advances  21 Oct 2016:
Vol. 2, no. 10, e1601246
DOI: 10.1126/sciadv.1601246

Figures

  • Fig. 1 Illustration of the fully connected architecture in the stabilizer formalism.

    The lattice of N(N + 1)/2 physical spins for encoding a fully connected Ising Hamiltonian, here shown for the case of N = 5 logical spins. (A) The larger triangle represents the formal scheme. Black and white dots are the physical spins; the spin at each black dot corresponds to a two-body coupling between logical spins, whereas that at each white dot corresponds to the local field acting on a logical spin. In the small triangle (top left), we show the chains of logical x operators (one color for each logical spin). The intersection of two such chains determines the meaning of the individual physical spin at that location. Rounded brackets (…) indicate a physical spin, whereas square brackets […] identify a plaquette; indices i and j are consistent with those in the main text, for example, in Eq. 3. (B) A physical implementation using ancilla spins to realize the stabilizer constraints. In the insets, the lines indicate the Ising couplings between the spins; all solid lines have a common strength, as do all dashed lines. (C) A schematic indicating that common resonators might mediate the interactions, similar to the proposal in Fig. 2 of Chancellor et al. (20).

  • Fig. 2 Energy levels at the end of the adiabatic sweep.

    Deviation between the lowest energy gap in the logical Hamiltonian Hlogic and the physical Hamiltonian Hphys as a function of the constraint’s strength. These graphs demonstrate that the even- and odd-parity variants perform almost identically.

  • Fig. 3 Scaling of the minimum gap.

    Minimum gap ratio between the physical and logical systems as a function of the constraint’s strength. The even- and odd-parity variants perform differently, especially in the N = 3 case, as discussed in the text.

  • Fig. 4 Lattice variants.

    (A) The lattice for encoding all-to-all connected logical spins using three-spin stabilizers. Here, the number of physical spins is NP = 15 and the number of logical spins is N = 5. The number of stabilizers must then be NS = NPN = 10, and we choose these to be triangular stabilizers (green triangles). Each circle denotes a physical spin. Empty circles are vertex spins (0, j), and solid circles are edge spins (i ≠ 0, j). Spins (0, i), (0, j), and (i, j) always form an isosceles triangle (marked with the dashed blue line). (B) Stabilizer code of the tree graph logical model. The top figure depicts the logical model, the bottom left figure depicts the physical lattice used to encode the logical model, and the bottom right figure depicts the chains of logical x operators. Each empty circle denotes a vertex physical spin (or a logical spin in the tree graph), and each solid circle denotes an edge spin corresponding to the two-body Ising interaction between two logical spins. Each plaquette corresponds to a stabilizer, as in Fig. 1A.

  • Fig. 5 Lattices with bespoke connectivity.

    (A) Strategic disruptions to the stabilizer lattice can control the routing of logical x operators. Starting from a regular lattice section (i) and replacing the central four stabilizers A, B, C, and D with three stabilizers E, F, and G (where F is a six-body stabilizer) and removing a physical spin results in a new lattice (ii) where the paths of the x operators for logical spins 2 and 6 are reflected. (B) Designing nontrivial connectivities via such reflections: Diagram (i) is a connectivity graph showing a highly connected (but not all-to-all) relationship between 22 nodes. Diagram (ii) is a lattice formed from three-, four-, and six-body stabilizers, arranged so as to realize that connectivity; for every linked pair in (i), there is at least one physical spin in (ii) representing the relative orientation of the logical spins.

  • Fig. 6 Lattices with higher-order interactions.

    (A and B) Methods for representing multibody interactions. In (A), this is achieved by a ladder of stabilizers; on the left figure, filled circles represent increasingly high-order correlations. The top right figure shows how the chains of logical x operators intersect, whereas the bottom right figure shows, for clarity, the same chains but without overlapping. In (B), the lattice notation follows that of Fig. 1A. Green circles denote additional physical spins representing multibody interactions. Purple links denote stabilizers. In case I, the stabilizer of physical spins (i, j) and (i + 1, j + 1) and of the additional physical spin corresponds to the four-body term Embedded Image; in case II, the stabilizer of physical spins (i − 1, j − 1), (i, j), and (i − 1, j + 1) and of the additional physical spin corresponds to the four-body term Embedded Image; and in case III, the stabilizer of physical spins (i − 1, j − 1), (i, j), and (i + 1, j + 1) and of the additional physical spin corresponds to the six-body term Embedded Image.

Tables

  • Table 1 Scaling of the computational space.

    The size of the computational space scales quadratically with the number of logical spins N. Left column, size of the logical system; middle column, size of the computational space for the ancillary qutrit (even parity) architecture; right column, size of the computational space for the ancillary qubit (odd-parity) architecture.

    NQutritQubit
    3263329
    421036216
    5215310225
    6221315236
    7228321249

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