Quantum annealing is a general strategy for solving difficult optimization problems with the aid of quantum adiabatic evolution. Both analytical and numerical evidence suggests that under idealized, closed system conditions, quantum annealing can outperform classical thermalization-based algorithms such as simulated annealing. Current engineered quantum annealing devices have a decoherence timescale which is orders of magnitude shorter than the adiabatic evolution time. Do they effectively perform classical thermalization when coupled to a decohering thermal environment? Here we present an experimental signature which is consistent with quantum annealing, and at the same time inconsistent with classical thermalization. Our experiment uses groups of eight superconducting flux qubits with programmable spin–spin couplings, embedded on a commercially available chip with >100 functional qubits. This suggests that programmable quantum devices, scalable with current superconducting technology, implement quantum annealing with a surprising robustness against noise and imperfections.

Hybrid quantum-classical algorithms are central to much of the current research in quantum computing, particularly when considering the noisy intermediate-scale quantum (NISQ) era, with a number of experimental demonstrations having already been performed. In this perspective, we discuss in a very broad sense what it means for an algorithm to be hybrid quantum-classical. We first explore this concept very directly, by building a definition based on previous work in abstraction-representation theory, arguing that what makes an algorithm hybrid is not directly how it is run (or how many classical resources it consumes), but whether classical components are crucial to an underlying model of the computation. We then take a broader view of this question, reviewing a number of hybrid algorithms and discussing what makes them hybrid, as well as the history of how they emerged and considerations related to hardware. This leads into a natural discussion of what the future holds for these algorithms. To answer this question, we turn to the use of specialized processors in classical computing. The classical trend is not for new technology to completely replace the old, but to augment it. We argue that the evolution of quantum computing is unlikely to be different: Hybrid algorithms are likely here to stay well past the NISQ era and even into full fault tolerance, with the quantum processors augmenting the already powerful classical processors which exist by performing specialized tasks.

In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimisation. The new method is based on the physics of domain walls in one-dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms Following on from similar results for the ‘one hot’ method of encoding discrete variables (Hadfieldet al2019Algorithms1234) I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz. I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.

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.

We demonstrate a direct mapping of max k-SAT problems (and weighted max k-SAT) to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc. We further show that this mapping can be used to map a similar class of maximum satisfiability problems where the clauses are replaced by parity checks over potentially large numbers of bits. The latter is of specific interest for applications in decoding for communication. We discuss an example in which the decoding of a turbo code, which has been demonstrated to perform near the Shannon limit, can be mapped to a Chimera graph. The weighted max k-SAT problem is the most general class of satisfiability problems, so our result effectively demonstrates how any satisfiability problem may be directly mapped to a Chimera graph. Our methods faithfully reproduce the low energy spectrum of the target problems, so therefore may also be used for maximum entropy inference.