Perils of embedding for sampling problems

Abstract

Sampling from certain distributions is a prohibitively challenging task. Special-purpose hardware such as quantum annealers may be able to more efficiently sample from such distributions, which could find application in optimization, sampling tasks, and machine learning. Current quantum annealers impose certain constraints on the structure of the cost Hamiltonian due to the connectivity of the individual processing units. This means that in order to solve many problems of interest, one is required to embed the native graph into the hardware graph. The effect of embedding for sampling is more pronounced than for optimization; for optimization one is just concerned with mapping ground states to ground states, whereas for sampling one needs to consider states at all energies. We argue that as the problem size grows, or the embedding size grows, the chance of a sample being in the logical subspace is exponentially suppressed. It is therefore necessary to construct post-processing techniques to evade this exponential sampling overhead, techniques that project from the embedded distribution one is physically sampling from, back to the logical space of the native problem. We show that the most naive (and most common) projection technique, known as majority vote, can fail quite spectacularly at preserving distribution properties. We show that, even with care, one cannot avoid bias in the general case. Specifically we prove through a simple and generic (counter) example that no post-processing technique, under reasonable assumptions, can avoid biasing the statistics in the general case. On the positive side, we demonstrate a new resampling technique for performing this projection which substantially out-performs majority vote and may allow one to much more effectively sample from graphs of interest.

Ferromagnetically Shifting the Power of Pausing

“…All of these possibilities should be explored on a variety of optimization problems as well as on the BD-MST problem class investigated here. Embedding affects sampling problems even more than optimization problems [19], so a study of the interplay between embedding parameters and annealing parameters should be done in that context as well. Experiments at other temperatures and with the ability to do quick quenches at arbitrary points in the anneal would give further insight in the the underlying physics.…”

Quantum-assisted associative adversarial network: applying quantum annealing in deep learning

“…Firstly, the cost of embedding a logical graph in a quantum annealer may be large in terms of the performance of the model (Marshall et al 2020). Specifically, it is not clear whether how close the model of the distribution sampled by the quantum annealer is to the real distribution for an embedded graph.…”

3-Regular 3-XORSAT Planted Solutions Benchmark of Classical and Quantum Heuristic Optimizers

“…In minor embedding, multiple qubits are chained together via strong ferromagnetic couplings [64,65,86]. For sufficiently strong ferromagnetic couplings this ensures that the ground state of the Ising Hamiltonian is unaffected by the minor embedding, but it affects the spectrum along the anneal in unpredictable ways, making the strength of these connections critical: too high, and the logical qubit will not flip readily enough to solve the Ising problem, too low and the logical qubit will "break" as its member physical qubits resolve into different final states that do not align [87]. With large problems and good tuning, a small number of chains still break, and it is necessary to choose how to map a broken chain to a logical state.…”