Quantum rotations shrink optimization circuits in benchmark study

Quantum search algorithms rely on an oracle, a circuit that identifies or evaluates candidate solutions. Building that circuit can be expensive. For optimization problems, conventional designs may calculate a candidate’s score, store it across several qubits and compare it with a threshold that changes between searches.

Sonia Yang, Ali Al-Bayaty, Shanyan Chen and Marek Perkowski propose a different representation: accumulate the score as rotations of a single qubit. Their Angular Rotational Counter, or ARC, substantially reduced circuit resources in the comparisons they conducted. The work was submitted to arXiv on November 9, 2026, as a preprint; the supplied record does not establish peer review.

Turning a score into a rotation

ARC applies to objective functions—the scores being minimized or maximized—that can be written as weighted sums of Boolean functions. Each Boolean function returns either zero or one, and each has a nonnegative weight.

Rather than adding those contributions into a binary register, ARC uses each Boolean result to control a rotation. A result of one triggers a rotation proportional to its weight; zero triggers none. Rotations around the same axis add together, making the accumulated angle proportional to the candidate’s total score. The authors prove this relationship for individual candidates and for a quantum superposition of candidates.

The output is not a binary score that can simply be read from one measurement. Instead, the score affects the probability of measuring the counter qubit as one. With a suitable scaling constant, that probability increases with score when the counter starts at zero, or decreases with score when it starts at one. These alternatives support favoring higher or lower scores, respectively.

The scaling matters because rotation probabilities repeat periodically. The constant must be at least as large as the maximum objective value to preserve the intended ordering over the relevant range.

Testing state-machine encodings

The researchers applied ARC to state assignment: choosing distinct binary labels for the states of a finite automaton or state machine. Those labels influence the complexity of the eventual implementation.

Their objective was a weighted Hamming-distance cost. Hamming distance counts the bit positions where two labels differ. Here, each pair’s distance was weighted by the total number of transitions between its states. Minimizing this score is a heuristic for finding economical encodings, rather than a direct measurement of the final implementation’s physical cost.

The ARC oracle combined a rotation-based cost calculator with a checker ensuring that every state received a distinct label. Its Boolean-based counterpart used the same encoding checker, but calculated costs with arithmetic circuits and compared them with a threshold stored in another register.

For a four-state example called dk15, a Qiskit simulation using 2,048 measurement shots produced higher counts for minimum-cost valid encodings than for higher-cost alternatives. This illustrated the intended preference, not a guarantee that every measurement returns an optimum.

What the resource comparison found

The main evaluation covered 26 benchmarks from the LGSynth’91 library, containing between four and 27 states. Both oracle designs were transpiled—translated and optimized into a specified elementary gate set—under identical settings using a Qiskit Aer simulator target. Each oracle was transpiled 200 times, and the version with the lowest Transpilation Quantum Cost was selected.

Across the benchmarks, ARC delivered arithmetic mean reductions of:

• 20.8% in qubit count;
• 89.3% in gate count;
• 92.4% in circuit depth;
• 90.6% in Transpilation Quantum Cost.

Circuit depth measures the longest sequence of dependent operations. The study’s combined cost metric adds single-qubit gates, two-qubit gates, SWAP gates and depth. It is a circuit-resource measure, not measured execution time.

Smaller circuits are not a demonstrated speedup

The savings arise from replacing arithmetic increments with controlled rotations and removing the threshold comparator and its register. Only the score register becomes a single qubit; candidate labels, validity checks and working qubits remain necessary.

ARC alone also does not produce valid or optimal answers. It needs a validity-marking circuit and an appropriate quantum algorithm. Hardware precision is another concern: larger scaling constants require smaller rotation angles, which the authors identify as increasingly vulnerable to noise and practical control limits.

The demonstrated result is therefore narrower than faster quantum optimization: substantially smaller oracle circuits for these benchmarks and this baseline. Whether those savings improve complete algorithm performance, physical execution time or larger applications remains a question for further investigation.