Understanding domain-wall encoding theoretically and experimentally

Abstract

We analyse the method 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 with 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. This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.

NP-hard but no longer hard to solve? Using quantum computing to tackle optimization problems

“…Importantly, arbitrary pairwise interactions between discrete variables can be mapped to quadratic interactions between qubits. It is suitable for devices with limited connectivity (Chancellor, 2019) and uses the minimum number of qubits where such a guarantee is possible (Berwald et al, 2023). It has been explored for networking applications has (Chen et al, 2022).…”

Spin-Variable Reduction Method for Handling Linear Equality Constraints in Ising Machines

“…By contrast, the penalty method gives the QUBO with |L (r) ||L (c) | spins. The domain-wall method formulates the QUBO with (|L (r) | − 1)|L (c) | spins when k (r) = 1 [25] (see also Appendix B). The SVR method at |L (r) | = |L (c) | and k (r) = k (c) = 1 is equivalent to the inserted method in Ref.…”

Quantum annealing and computation: challenges and perspectives

“…The paper entitled 'Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model' [26] [29] the authors consider domain-wall qubit encodings using integer valued variables and map the original optimization problem into a QUBO of binary variables.…”

“…In the paper by Dridi et al . entitled ‘Understanding domain-wall encoding theoretically and experimentally’ [ 29 ] the authors consider domain-wall qubit encodings using integer valued variables and map the original optimization problem into a QUBO of binary variables.…”