Truncated-Binary Encoding: Spectral Degree Reduction of Combinatorial Optimization Problems for Quantum Hardware

TL;DR

This paper introduces truncated-binary encoding (TBE), a method to reduce spectral degree in combinatorial optimization problems for quantum hardware, balancing accuracy and hardware constraints.

Contribution

It proposes TBE as a modification of exact-binary encoding, providing bounds, conditions, and criteria for effective spectral degree reduction in quantum optimization.

Findings

TBE drops high-degree monomials to reduce circuit complexity.

Bounds on truncation error and conditions for preserving minima are established.

A criterion for selecting the cutoff degree based on Walsh transform decay is derived.

Abstract

Exact-binary encoding compiles a discrete cost function network (CFN) into a higher-order unconstrained binary optimization (HUBO) problem whose maximum monomial degree grows with the cardinalities of the underlying CFN variables. Given that quantum optimization hardware generally favours quadratic unconstrained binary optimization or low-degree HUBO Hamiltonians, high-cardinality CFNs therefore incur substantial overhead in the form of circuit depth, or ancilla qubits when degree-reduction techniques are employed. To ameliorate these issues, we propose \textit{truncated-binary encoding} (TBE): a modification of exact-binary encoding in which Ising-basis monomials exceeding a chosen cutoff $k_\text{max}$ are dropped from the encoded cost. We establish a tight $L^\infty$ bound on the truncation error in terms of the omitted couplings, derive sufficient conditions on the energy gap and on the single bit-flip basin barrier under which TBE preserves the global minimum and its local-minimum structure, and characterize a noise floor condition on the spectral profile under which the truncation residual acts as a perturbative correction to the underlying landscape. We then express the encoded coefficients directly as Walsh transforms of the underlying CFN cost tables, and prove a bound under which smoothness of each cost table implies rapid decay of its high-degree Walsh mass. Together these results yield a principled \textit{a priori} criterion for selecting $k_\text{max}$ and for judging whether a given CFN admits a small-$k_\text{max}$ TBE.