Evaluating the Effect of the Order of Optimization Passes in Quantum Circuit Optimization

Quantum computers promise to solve problems that would take classical machines thousands of years to crack, from designing new medicines to modeling complex climate systems. However, these machines are incredibly fragile. The slightest disturbance from the environment can cause them to make errors, a phenomenon known as noise. To fight this, scientists must translate high-level computer programs into instructions that the specific quantum hardware can actually understand. This translation process, called compilation, often adds extra steps and makes the circuit—the path the information travels—longer and more complex. A longer circuit means more time for errors to creep in, so the goal is to make these circuits as short and simple as possible before they run. This is where optimization comes in: a set of rules designed to strip away unnecessary steps and clean up the circuit.

For years, engineers have known that applying these cleaning rules helps, but they have largely assumed that the order in which they apply them does not matter much. It was thought that as long as the rules were applied, the result would be roughly the same. A new study challenges this assumption, revealing that the sequence of these rules is actually critical. Just as a chef might get a different result by chopping vegetables before sautéing them versus sautéing first, the researchers found that the order of optimization steps in quantum circuits can drastically change the final quality of the program.

The researchers, working with the popular Qiskit software used to program quantum computers, set out to test this idea systematically. They selected sixteen different optimization techniques, each designed to fix a specific type of inefficiency. Some techniques merge small steps into one, others cancel out pairs of operations that undo each other, and some rearrange the order of steps to make them more efficient. The team did not just look at how well each technique worked on its own; they tested every possible pair of these sixteen techniques. They ran the same quantum circuits through the compiler, first applying technique A then B, and then applying B then A. They repeated this for all one hundred and twenty possible pairs, measuring the length of the circuit and the total number of steps in each result.

The results were clear and surprising. In twelve out of the one hundred and twenty pairs, the order made a significant difference. In some cases, one sequence produced a circuit that was nearly half the size of the other. The study showed that certain techniques create a better foundation for others to work on. For instance, removing useless steps early on can reveal new patterns that a later technique can then simplify, but if the later technique is run first, it might miss those patterns entirely. The researchers found that the best order depended heavily on the specific type of quantum hardware being targeted, as different machines speak different "languages" of basic operations. For example, a sequence that worked best for one type of machine might be ineffective for another.

The team also investigated whether a poor choice of order could be fixed later. They asked if running a technique a second time could repair a suboptimal sequence. In seven out of the twelve cases where the order mattered, reapplying the first technique did indeed improve the result, sometimes even making it better than the originally best sequence. However, this was not a universal fix; in other cases, repeating the steps added unnecessary work without improving the outcome. This suggests that while a second pass can sometimes save a bad start, it is far better to get the order right the first time to avoid wasting time and resources.

When the researchers expanded their tests to sequences of three, four, or five techniques, they found that the interactions became even more complex. The best order for a pair of techniques did not always hold true when a third technique was added. Sometimes, a technique that was best placed at the beginning of a two-step process needed to be moved to the middle when a third step was introduced. This indicates that the relationship between these rules is not just a simple chain of cause and effect, but a web of interactions where the presence of one rule changes how the others behave.

Based on their findings, the authors proposed a new recommended order for applying these optimization rules. Their suggested sequence starts by removing gates that have almost no effect, then rearranges the remaining steps to allow for more simplification, followed by pattern matching and cancellation of opposing operations. This recommended order differs from the default settings currently used in standard quantum compilers. When they tested this new order against the standard one, they found that in a controlled environment with only optimization steps, the difference was minimal. However, when the full compilation process was included, the new order performed slightly better in many cases, particularly for reducing the total number of steps.

The study concludes that there is no single "perfect" order that works for every situation. The best sequence depends on the specific circuit being run, the size of that circuit, and most importantly, the native gate set of the target hardware. The researchers emphasize that while their work provides a strong foundation and a better default starting point, the optimal path for any given quantum program is likely unique. Their work shifts the focus from simply applying a fixed list of rules to understanding the dynamic relationships between them, ensuring that quantum programs are as efficient and error-resistant as possible before they ever touch the hardware.