All-to-all connections are required in general quantum annealing machines to solve various combinatorial optimization problems. The Lechner, Hauke, and Zoller method, which is used to realize the all-to-all connections, requires many-body interactions in locally connected qubits. Because most of the qubit interactions are two-body interactions, Lechner also proposed the construction of each four-body interaction by six controlled-NOT (CNOT) gates between two qubits. However, it is difficult to construct many CNOT gates. Herein, we show more concrete sequences to produce four-body and three-body interactions based on a two-dimensional solid-state qubit system. We show that the number of operations needed to construct the many-body interactions can be reduced using appropriate pulse sequences. These findings will help reduce quantum computation costs for solving combinatorial problems.
All-to-all connections are required in general quantum annealing machines to solve various combinatorial optimization problems. The Lechner, Hauke, and Zoller method, which is used to realize the all-to-all connections, requires many-body interactions in locally connected qubits. Because most of the qubit interactions are two-body interactions, Lechner also proposed the construction of each four-body interaction by six controlled-NOT (CNOT) gates between two qubits. However, it is difficult to construct many CNOT gates. Herein, we show more concrete sequences to produce four-body and three-body interactions based on a two-dimensional solid-state qubit system. We show that the number of operations needed to construct the many-body interactions can be reduced using appropriate pulse sequences. These findings will help reduce quantum computation costs for solving combinatorial problems.
Quantum annealing aims at solving optimization problems efficiently by preparing the ground state of an Ising spin-Hamiltonian quantum mechanically.A prerequisite of building a quantum annealer is the implementation of programmable long-range two-, three-, or multispin Ising interactions.We discuss an architecture, where the required spin interactions are implemented via two-port or in general multiport quantum Ising wires connecting the spins of interest.This quantum annealing architecture of spins connected by Ising quantum wires can be realized by exploiting the three-dimensional (3D) character of atomic platforms, including atoms in optical lattices and Rydberg tweezer arrays.The realization only requires engineering on-site terms and two-body interactions between nearest neighboring qubits.The locally coupled spin model on a 3D cubic lattice is sufficient to effectively produce arbitrary all-to-all coupled Ising Hamiltonians.We illustrate the approach for few-spin devices solving Max-Cut and prime factorization problems, and discuss the potential scaling to large atom-based systems.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
We describe an architecture to map the Ising model problem onto the dynamics of two-dimensional neutral trapped atom arrays. Specifically, we show that the NP-complete Ising model decision problem on n variables can be reduced to deciding whether, for O(n2) Rubidium atoms in two dimensions, interacting via Rydberg excitations, there exists a state with energy below a certain threshold. This provides an alternative proof of the NP-hardness of two dimensional Rydberg dynamics with a single species in the regime where long-range interactions are negligible. Therefore, we can encode NP-complete problems onto the ground state of Rydberg atom arrays and use quantum optimization techniques such as quantum adiabatic algorithms and variational algorithms to estimate the ground state. Furthermore, as the Ising model describes many-body phenomena in statistical physics, we can probe a plethora of emergent phenomena via quantum simulators. Our result is based on the Lechner-Hauke-Zoller (LHZ) scheme that enables the encoding of an all-to-all connected Ising model onto systems with local-only interactions. We improve the scheme in the following ways to make the it feasible for near-term Rydberg simulators: (1) Implementations of the scheme in literature that use qubits suffer from a double-counting error causing inconsistencies in the strength of the interactions. Identical interactions must have different strengths based on their spatial location. We describe and resolve this error without impacting the O(n2) scaling of the atom array by introducing additional ancillary qubits. (2) We introduce new formalism and simpler proofs via novel approaches regarding the validity of the LHZ scheme and its variants. These proofs lend more easily to complexity-theoretic arguments regarding the hardness of Rydberg dynamics. (3) Finally, we circumvent the LHZ scheme constraint of requiring four-body interactions, qutrits, or multiple species of atoms by providing an implementation that uses only Rubidium atoms to encode both physical and ancillary qubits. To engineer the required interactions, the atoms are coupled to different Rydberg energy levels. We also provide a brief review of classical and quantum computation, quantum computing platforms, near-term quantum experiments, and Rydberg physics.
One of the current major challenges surrounding the use of quantum annealers for solving practical optimization problems is their inability to encode even moderately sized problems, the main reason for this being the rigid layout of their quantum bits as well as their sparse connectivity. In particular, the implementation of constraints has become a major bottleneck in the embedding of practical problems, because the latter is typically achieved by adding harmful penalty terms to the problem Hamiltonian, a technique that often requires an all-to-all connectivity between the qubits. Recently, a novel technique designed to obviate the need for penalty terms was suggested; it is based on the construction of driver Hamiltonians that commute with the constraints of the problem, rendering the latter constants of motion. In this work we propose general guidelines for the construction of such driver Hamiltonians given an arbitrary set of constraints. We illustrate the broad applicability of our method by analyzing several diverse examples, namely, graph isomorphism, not-all-equal three-satisfiability, and the so-called Lechner-Hauke-Zoller constraints. We also discuss the significance of our approach in the context of current and future experimental quantum annealers.
There is a significant ongoing effort in realizing quantum annealing with different physical platforms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabatic quantum dynamics. Here we show that combining the well-developed quantum simulation toolbox for Rydberg atoms with the recently proposed Lechner-Hauke-Zoller (LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactions and, therefore, a platform for quantum annealing. In LHZ an infinite-range spin-glass is mapped onto the low energy subspace of a spin-1/2 lattice gauge model with quasi-local four-body parity constraints. This spin model can be emulated in a natural way with Rubidium and Caesium atoms in a bipartite optical lattice involving laser-dressed Rydberg-Rydberg interactions, which are several orders of magnitude larger than the relevant decoherence rates. This makes the exploration of coherent quantum enhanced optimization protocols accessible with state-of-the-art atomic physics experiments.
Abstract Quantum annealing provides a way of solving optimization problems by encoding them as Ising spin models which are implemented using physical qubits. The solution of the optimization problem then corresponds to the ground state of the system. Quantum tunneling is harnessed to enable the system to move to the ground state in a potentially high non-convex energy landscape. A major difficulty in encoding optimization problems in physical quantum annealing devices is the fact that many real world optimization problems require interactions of higher connectivity, as well as multi-body terms beyond the limitations of the physical hardware. In this work we address the question of how to implement multi-body interactions using hardware which natively only provides two-body interactions. The main result is an efficient circuit design of such multi-body terms using superconducting flux qubits in which effective N -body interactions are implemented using N ancilla qubits and only two inductive couplers. It is then shown how this circuit can be used as the unit cell of a scalable architecture by applying it to a recently proposed embedding technique for constructing an architecture of logical qubits with arbitrary connectivity using physical qubits which have nearest-neighbor four-body interactions. It is further shown that this design is robust to non-linear effects in the coupling loops, as well as mismatches in some of the circuit parameters.