AI system proves long-standing quantum computing conjecture

Solving a decades-old mystery

A US startup called Physical Superintelligence (PSI) recently announced that its AI system has proved the Aaronson-Ambainis conjecture. This mathematical problem has remained open since 2008 and sits at the heart of quantum complexity theory. This field of computer science sorts problems based on how much effort different types of machines need to solve them.

The startup utilized an internal research system running an unreleased Anthropic model to produce the proof. The process took just three days on the company's own computing infrastructure. To ensure the logic was sound, the team used Lean, which is a proof assistant that verifies every individual step of a mathematical argument.

The final version of the proof was accepted by Lean and has been published on GitHub. PSI even gave the system related open problems to solve, and it successfully improved its own result twice. This milestone provides a formal proof for a warning that quantum computers won't solve hard problems instantly by trying all solutions in parallel.

How quantum speedups work

To understand the proof, one must first understand what a quantum speedup is. It represents the efficiency gain a quantum machine provides compared to the best standard computer for a specific task. Complexity theorists measure this by looking at how the work grows as the problem size increases. A modest speedup is polynomial, while the big prize is an exponential speedup.

An exponential speedup means the effort for an ordinary computer doubles with every small increase in input, while the quantum machine's effort climbs much more slowly. The most famous example is Shor’s algorithm, which can factor large numbers in polynomial time. This is why quantum computing is seen as a threat to the RSA encryption that protects much of the internet today.

However, every known exponential speedup depends on the input having a specific structure. Shor’s algorithm relies on hidden periodicity, and other algorithms use secret strings concealed inside functions. The Aaronson-Ambainis conjecture specifically addressed what happens when there is no such promise and the data is effectively random.

The mechanics of the proof

The conjecture explores the amount of time a quantum computer can save on typical random inputs for any yes-or-no question. PSI’s write-up explains that these speedups are counted in queries, or the number of times a computer reads part of its input. While an ordinary computer reads one bit per look, a quantum computer can spread its look across many positions using superposition.

The proof involves viewing the quantum algorithm's process as a polynomial, which is a sum of terms where each term multiplies some input bits together. The degree of this polynomial is determined by the number of bits multiplied in any one term. The conjecture suggested that if the reading varies at all, at least one bit has a significant influence on the result.

The theorem confirms that for any fixed margin of error, an ordinary computer can predict the quantum computer's answer by reading a number of bits that grows only as a fixed power of the number of looks. This means that for almost every random input, the standard computer can match the performance of the quantum one without needing exponentially more looks. It proves that the quantum machine cannot achieve an exponential advantage on messy, unstructured data without a pattern to exploit.

Impact on the quantum field

This result has significant implications for the commercial promise of quantum computing. Many people hope for a machine that can solve any hard problem quickly, but this theorem shows that such gains only turn up when the data has a pattern to exploit. Anyone promising exponential gains on unstructured data now has to specify exactly where that pattern is located.

The proof also validates the skepticism of prominent figures like Scott Aaronson. He has long argued that quantum computers get their power from amplitudes interfering with one another rather than just holding every answer at once. This new theorem puts a formal mathematical proof behind part of his long-standing warnings about the limits of quantum hardware.

For the broader tech industry, this clarifies the roadmap for useful quantum applications. It suggests that the most viable near-term uses will continue to be those that rely on specific mathematical structures. It moves the conversation away from general-purpose magic and toward specialized, structured advantages that can be mathematically verified.

Next steps for the research

While the proof is complete and published, it has not yet undergone formal peer review. The researchers at PSI have made the full formal proof available for the community to examine on GitHub. This allows other scientists to verify the logic and the specific power of the bound worked out by the AI.

The discovery also opens the door for further exploration into the specific types of structures that do allow for exponential speedups. Now that the Aaronson-Ambainis conjecture is settled, researchers can focus on identifying which types of structured data will yield the most significant practical advantages for quantum hardware.

The work also highlights the growing role of AI in high-level mathematics. By using an unreleased model to navigate complex logic and then verifying that logic with a tool like Lean, PSI has demonstrated a new workflow for tackling some of the most difficult problems in theoretical computer science.