Understanding domain-wall encoding theoretically and experimentally

Abstract

We analyze the performance of encoding pairwise interactions of higher-than-binary discrete variables (these models are sometimes referred to as discrete quadratic models) into binary variables based on domain walls on one dimensional Ising chains. We discuss how this is relevant to quantum annealing, but also many gate model algorithms such as VQE and QAOA. We theoretically show that for problems of practical interest for quantum computing and assuming only quadratic interactions are available between the binary variables, it is not possible to have a more efficient general encoding in terms of number of binary variables per discrete variable. We furthermore use a D-Wave Advantage 1.1 flux qubit quantum annealing computer to show that the dynamics effectively freeze later for a domain-wall encoding compared to a traditional one-hot encoding. This second result could help explain the dramatic performance improvement of domain wall over one hot which has been seen in a recent experiment on D-Wave hardware. This is an important result because usually problem encoding and the underlying physics are considered separately, our work suggests that considering them together may be a more useful paradigm. We argue that this experimental result is also likely to carry over to a number of other settings, we discuss how this has implications for gate-model and quantum-inspired algorithms.

Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond

“…In particular, this raises the question of binary encodings, where bit strings are used directly to represent configurations, or encoding individual variables using unary (qubits required scale linearly with the number of configurations) encodings. In the case of annealing, the inability to efficiently engineer higher order terms means that it can be shown mathematically that a kind of unary encoding (domain wall) is the most efficient method in terms of qubit usage for completely general interactions [102]. However, for gate model approaches, the tradeoff is far more complicated and encoding strategies remain an area of active research [103][104][105][106] with many open questions.…”

Controller-Based Energy-Aware Wireless Sensor Network Routing Using Quantum Algorithms

“…Motivated by the enhanced performance seen in [30,31] (although not yet shown on a real engineering problem on a quantum annealer), we employ the domain-wall encoding scheme first proposed in [19], translate to this encoding we first replace one-hot constraints with:…”

“…For example [30] shows a dramatic improvement in performance in solving colouring problems. Meanwhile, [31] explains this improvement both in terms of entropic arguments related to the size of the solution space, and a physical mechanism related to perturbative dynamics of bit flips within the encoding.…”

Controller-based Energy-Aware Wireless Sensor Network Routing using Quantum Algorithms

“…Motivated by the enhanced performance seen in [30,31] (although not yet shown on a real engineering problem), we employ the domain-wall encoding scheme first proposed in [19], translate to this encoding we first replace one-hot constraints with:…”