Resource-Efficient Quantum Optimization via Higher-Order Encoding

TL;DR

This work tackles resource bottlenecks in quantum optimization by replacing conventional QUBO encodings with HUBO encodings for multi-valued variables. It presents a general framework to construct HUBO Hamiltonians, derives efficient circuit implementations for HUBO terms (utilizing Gray-code traversal and Walsh-Hadamard-based coefficient computation), and demonstrates substantial resource savings across three representative COPs: GAP, MkCS, and IP. Empirical results show HUBO reduces qubit counts exponentially and lowers CNOT and RZ gate counts by up to 89.6–100% relative to QUBO, while achieving equal or better approximation ratios at comparable or lower quantum resources. The work also provides an open-source Python library, PyHUBO, to automate HUBO-Hamiltonian construction, promoting practical adoption on near-term devices and enabling broader exploration of resource-efficient quantum optimization.

Abstract

Quantum approaches to combinatorial optimization problems (COPs) are often limited by the resource demands of Quadratic Unconstrained Binary Optimization (QUBO) encodings, which enlarge circuits through penalty terms and increase qubit and gate counts. We show that Higher-Order Unconstrained Binary Optimization (HUBO) enables a more resource-efficient formulation. Our method systematically constructs HUBO Hamiltonians and, compared to QUBO in benchmarks on Gate Assignment (GAP), Maximum k-Colorable Subgraph (MkCS), and Integer Programming (IP) problems, exponentially reduces qubit requirements and decreases CNOT gate counts by at least 89.6% after compilation to single- and two-qubit gates for all tested instances. These results highlight HUBO as a practical alternative for current and near-term devices. To promote adoption, we release an open-source Python library that automates HUBO model construction, broadening access to resource-efficient quantum optimization.

Figures (18)

• Figure 1: Number of CNOT gates required to reach solution threshold for three COPs, optimized by QAOA with QUBO (yellow marker) and HUBO (blue marker) encodings as a function of problem size $n\times m$. The GAP optimizes the assignment of $n$ aircraft to $m$ airport gates. In the MkCS problem, $n$ vertices are colored with $m$ colors, IP optimizes a polynomial over $n$ variables, each taking one of $m$ possible values. Further details and quantum encodings are provided in the main text.
• Figure 2: Illustration of the GAP with $m=4$ airport-gates and $n=3$ flights. Two of the flights have passengers who require a transfer connection, which is represented by the two-way symbol. A third flight has only arriving and departing passengers, illustrated by the luggage symbol. The goal is to assign airport-gates to flight such that the total passenger walking time is minimized.
• Figure 3: Example instance for the MkCS problem: a properly 4-colored graph with $|V|=7$. All adjacent vertices have distinct colors, so the objective value (number of monochromatic edges) is zero.
• Figure 4: Schematic representation of the QAOA algorithm showing the initialization in the superposition state, the alternating application of cost and mixer unitaries across layers, followed by measurement and classical optimization of variational parameters $\boldsymbol{\gamma}$ and $\boldsymbol{\beta}$.
• Figure 5: Circuit implementation of the cost Hamiltonian for a three-qubit term, i.e. $\exp(i \gamma J_{1,2,3} \hat{Z}_1 \hat{Z}_2 \hat{Z}_3)$. The circuit consists of a sequence of CNOT gates to entangle the qubits, such that the initial state $\ket{b_1}_1 \ket{b_2}_2 \ket{b_3}_3$ in the computational basis becomes $\ket{b_1}_1 \ket{b_1 \oplus b_2}_2 \ket{b_1 \oplus b_2 \oplus b_3}_3$ through the first chain of CNOT gates. The single qubit rotation $R_z(2\gamma J_{1,2,3})$ acts on the last qubit. After the rotation, another sequence of CNOT gates disentangles the qubits, resulting in the final state $\exp((-1)^{b_1 \oplus b_2 \oplus b_3}i \gamma J_{1,2,3})\ket{b_1}_1 \ket{b_2}_2 \ket{b_3}_3$.