Qubit-efficient embedding of parity-encoded Hamiltonians in quantum annealers

Abstract

The Sourlas-Lechner-Hauke-Zoller (SLHZ) scheme for quantum annealing uses the parity to encode logical variables and has several advantages, but it has not been implemented for large-scale quantum annealers. If the SLHZ-based approach can be implemented on currently available quantum annealers, we can evaluate its performance. An efficient method to embed the parity-encoded model into the hardware graphs of available quantum annealers is one of the key elements for this approach. We propose a qubit-efficient embedding scheme for parity-encoded Hamiltonians on quantum annealers with the Zephyr connectivity. We give an explicit constructive embedding of the interaction graph of an intermediate Hamiltonian, which contains only one- and two-body interactions, into the Zephyr graph. Our embedding maps each spin to a two-qubit chain using systematic chain-assignment rules. Its validity is verified via the resulting chain-to-chain connectivity. Our embedding also offers practical flexibility. Chains assigned to ancillary spins allow reduction to a single physical qubit, leading to options to avoid inactive qubits. The number of required qubits per spin in the parity Hamiltonian is three, which is fewer than that for a known embedding scheme for the Pegasus graph.

Figures (7)

• Figure 1: (a) Lattice for the Parity Hamiltonian for $3 \times 3$ spins depicted as circles. Squares represent four-body interactions between spins located at their vertices. (b) Lattice for the corresponding intermediate Hamiltonian. Each vertex on the lattice is parameterized by a coordinate $(x_P, y_P, a)$, where $a=0$ for parity spins and $a=1$ for ancillary spins. A parity spin interacts with up to eight parity spins and four ancillary spins via two-body interactions depicted as edges. Ancillary spins interact only with parity spins.
• Figure 2: Zephyr graph $Z_m$ for $m = 2$. Each vertex (qubit) is parametrized by $(u, w, k, j, z)$. $u$ indicates the orientation, $w$ ($z$) the index of the qubit's tile in the orientation perpendicular (parallel) to $u$, $k$ the index of a qubit within a tile. Each qubit is shifted to the left/top for $j = 0$ and the right/bottom for $j = 1$. Parameters of qubits A--J are shown as examples. See the reference K.Boothby2021 for the detailed description.
• Figure 3: Examples of chains specified by Eq. (\ref{['eq:chain']}). Each chain consists of two vertices for $u = 0$ and 1 [Fig. \ref{['fig:Zephyr']}] and the edge connecting them. Vertices and edges for the labelled chains are highlighted.
• Figure 4: Couplings to a chain, say, chain 0, used in our embedding in the Zephyr graph. The couplings of chains are based on physical couplers between qubits belonging to different chains. We only consider the internal couplers K.Boothby2021, i.e., the couplers between physical qubits for $u=0$ and $1$ [Fig. \ref{['fig:Zephyr']}]. Chain 0 and chains which couple to it are labelled and highlighted. Edges representing the couplings to chain 0 via the internal couplers are also highlighted. Parameters of the labelled chains when chain 0 is parameterized as $(x_Z, y_Z, \alpha, \beta)$ are shown in Table \ref{['table:chain_couplings']}. Two examples for parameters $(x_Z, y_Z, \alpha, \beta)$ of chain 0 are shown: (a) $(1, 1, 0, 0)$ and (b) $(1, 1, 1, 0)$. In these examples, all the forth parameters of chains 2--11, which are represented as $\beta'$ in Table \ref{['table:chain_couplings']}, are set to 0 $(= \beta)$, but each of them can also be 1, since whether the coupling between two chains exists does not depend on the value of their forth parameters.
• Figure 5: Mapping of a spin and its coupled spins in the intermediate Hamiltonian [Eq. (\ref{['eq:H_intermediate']})] to two-qubit chains in the Zephyr graph. (a) Lattice for the intermediate Hamiltonian for $3\times 3$ parity spins and $2\times 2$ ancillary spins. Edges for couplings to spin 0 are depicted as solid lines, while those for the other couplings are depicted as dashed lines. See Fig. \ref{['fig:LHZ-intermediate_lattice']}(b) for the coordinated description. The spins are labelled. For example, spin $(1, 1, 0)$ is labelled as 0. (b), (c) Chains in the Zephyr graph assigned to the spins in (a) as (b) Eq. (\ref{['eq:mapping']}) and (c) Eq. (\ref{['eq:mapping_2']}), provided that spins 0 and 12 are mapped to (b) chains $(1,1,0,0)$ and $(1,1,0,1)$ and (c) chains $(1,1,0,0)$ and $(1,1,1,1)$, respectively. See, Fig. \ref{['fig:chains']} for the coordinate description. The chains have the same labels as their corresponding spins in (a). Parameters of the labelled chains when chain 0 is parameterized as $(x_Z, y_Z, \alpha, \beta)$ are shown in (b) Table \ref{['table:correspondence']} and (c) Table \ref{['table:correspondence_2']}.