The focal point of the week was the September 15 panel discussion titled “AI in Mathematical Research,” moderated by Tony Feng of UC Berkeley and Google DeepMind. The panelists included Fields Medalists Peter Scholze and Jacob Tsimerman, along with Michael Harris of Columbia University and Geordie Williamson of the University of Sydney. Rather than presenting a united front, the panel surfaced sharply divergent positions on how the mathematical community should respond to the rapid encroachment of machine intelligence into a discipline that has, for centuries, prized human insight, proof and understanding above all else. The onstage discussion, organizers noted, became the starting point for countless conversations among participants throughout the remainder of the week.

Scholze delivered one of the most pointed assessments of the entire Forum. “AI wants to accelerate everything, and I don’t see any inherent value in acceleration at all,” he said. His argument cut against the prevailing narrative that faster discovery is automatically better discovery. Scholze contended that even in the absence of AI, mathematics is already advancing so quickly that genuinely understanding its progress has become a bottleneck in its own right. In a field where a single landmark proof, such as his own work on perfectoid spaces, can take the community years to absorb and verify, the prospect of machine-generated conjectures and proofs arriving at an even greater pace raises an uncomfortable possibility: a mathematics that grows faster than any human, or even the collective of humans, can comprehend it.

Jessica Fintzen and Albrecht Schmidt, Scientific Co-Chairs of the Heidelberg Laureate Forum Foundation, reflected on the range of views the panel exposed. According to the co-chairs, the discussion spanned everything from embracing the use of AI, to cautioning against supporting companies whose values might not be aligned with those of the mathematical community, to trying to pinpoint what mathematics should really be about. They emphasized that the onstage debate addressed many questions that the mathematical and broader community must face, and that it served as a catalyst for conversations among HLF participants throughout the week. That framing captured the spirit of the Forum itself: not a search for consensus, but a structured confrontation with a technological shift whose consequences no one in the room claimed to fully predict.

Those questions extended well beyond the single panel. In a Spark Talk on “AI Safety and Mathematics,” Tsimerman, who received this year’s Fields Medal, argued that societies manage real risks by looking at them clearly and steering away from bad outcomes, not by ignoring them. His position framed AI safety not as a reason for disengagement but as a discipline in its own right, one in which mathematicians have a distinctive role to play. Torsten Hoefler, recipient of the 2024 ACM Prize in Computing, posed the question “Can We Build an AI Scientist?” and offered a prediction with sweeping implications: “everything that is verifiable will be automated.” For a discipline built on verification, that forecast lands with particular force, suggesting that the parts of mathematical work reducible to formal checking may eventually belong entirely to machines, leaving humans the tasks of conjecture, taste and interpretation.

Fields Medalist Martin Hairer presented “First Proof,” an initiative designed to measure how useful AI genuinely is to mathematical research by giving it the kinds of problems that talented mathematicians would actually work on. The approach addresses a persistent weakness in public claims about AI capability, where benchmarks often fail to correspond to the difficulty and texture of real research mathematics. By testing systems against problems of genuine current interest, Hairer’s framework aims to replace anecdote and hype with calibrated evidence, offering the community a way to track whether machine assistance is becoming a practical tool for working mathematicians or remains confined to well-trodden territory.

While AI dominated the conversation, the Forum also confronted a second technological threat with a very different character. Charles H. Bennett and Gilles Brassard, recipients of the 2025 ACM A.M. Turing Award for their pioneering role in the quantum revolution in computer science, delivered a joint lecture titled “The Quantum Information Revolution from Wiesner to Turing.” Brassard issued a caution that the quantum threat to today’s public-key cryptography is widely misunderstood. The danger, he explained, does not begin when a large quantum computer finally appears. Because encrypted data can be harvested now and decrypted later, adversaries need only record today’s traffic and wait. As he put it, “if a quantum computer arrives tomorrow, then my data is gone today.” The warning underscores why the migration to post-quantum cryptographic standards is treated by security researchers as an urgent present-day task rather than a distant concern.

Questions of trust in computation also ran through the program. Yael Tauman Kalai, recipient of the 2022 ACM Prize in Computing, took up a problem that grows more pressing as computation is increasingly outsourced to untrusted servers and, increasingly, to AI systems: when a massive computation is performed remotely, how can anyone be sure the result is correct? Her lecture presented succinct cryptographic proofs, compact certificates that are quick to check even for computations that are prohibitively expensive to rerun, as one answer. The technique connects directly to the week’s AI debates, since verifiable computation offers a potential safeguard for a future in which machine-generated results must be trusted at scale.

The scientific program showcased the breadth of both disciplines. Shayan Oveis Gharan, recipient of the 2026 IMU Abacus Medal, lectured on the polynomial paradigm and its applications in algorithms. Yu Deng, a 2026 Fields Medalist, gave a Spark Talk addressing critical problems in statistical physics and partial differential equations. Efim Zelmanov, Fields Medalist in 1994, opened the final day with a sweeping look at 200 years of abstract algebra. Nobel laureate Brian P. Schmidt, whose Nobel Prize in Physics recognized the discovery of the universe’s accelerating expansion, delivered this year’s Lindau Lecture, “The State of the Universe,” tracing how scientists came to understand that the cosmos is expanding. Attendees also watched livestreams marking the 100th birthday of Jean-Pierre Serre, the first recipient of the Abel Prize, including a lecture by Pierre Deligne in Serre’s honor and a talk by Serre himself from the celebrations in Paris.

The next generation made its own mark on the week. Dorothee Bär, Germany’s Federal Minister of Research, Technology and Space, addressed the Opening Ceremony by video, praising the Forum as a place where knowledge that cannot be learned from a paper passes in person from one generation to the next, and closing with an appeal: “Mathematics has outlived every border ever drawn around it. Please keep it that way.” The young researchers carried that spirit into the Forum’s first-ever Un-Conference on the final day, a bottom-up format for which participants had pitched and voted on discussion topics throughout the week. The results confirmed that the laureates’ concerns were widely shared: many sessions dealt with AI, including its impact on academia, how to reshape education in the age of AI, and technofascism and Silicon Valley ideology, while others took up the role of mathematics and computer science for society and the value of pure mathematics. The entire scientific program is available on the HLF’s YouTube channel, with in-depth coverage on the HLFF Blog, and journalists can register for accreditation for the 14th Forum in 2027 from March onwards. What the week made plain is that the debate over AI in mathematics is no longer a speculative exercise; it is a live argument being conducted at the very top of the discipline, and the outcome will shape how human knowledge itself is made.

Subject of Research: Debates on artificial intelligence in mathematical research at the 13th Heidelberg Laureate Forum

Article Title: Leading mathematicians and computer scientists weigh AI's growing role in mathematics at the 13th Heidelberg Laureate Forum

Article References: Leading mathematicians and computer scientists weigh AI's growing role in mathematics at the 13th Heidelberg Laureate Forum. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: Heidelberg Laureate Forum, artificial intelligence, mathematics, Fields Medal, Peter Scholze, quantum cryptography, post-quantum security, verifiable computation, AI safety, Turing Award, young researchers, Un-Conference

Blind quantum computation is the umbrella term for protocols that allow a relatively weak client to delegate a computation to a powerful but potentially untrusted quantum server without revealing the input, the output, or the algorithm being executed. The circuit model, in which a computation is expressed as a sequence of quantum gates acting on qubits, is the natural framework for today’s gate-based quantum processors, and much of the earlier work in this setting followed a familiar pattern: the client encrypts the quantum state, the server operates on the encrypted data, and the client decrypts the result. The trouble, as the researchers emphasize, is that in many conventional schemes the decryption information is not static. As the circuit deepens, the corrections that the client must apply to interpret the final result change from gate to gate, so the decryption key effectively propagates through the computation one layer at a time. Associate Professor Xiaoqian Zhang of Jinan University describes this as a major limitation: decryption information may need to be updated gate by gate, increasing the client-side burden as the circuit becomes deeper. For a client with limited quantum capabilities, that burden can become prohibitive exactly where it matters most, in long, deep circuits.

The new protocol breaks this dependency by decoupling the keys. Instead of allowing one key to evolve through the circuit, the client independently generates two separate keys, one for encryption and one for decryption. Before sending the quantum state to the server, the client applies a Pauli encryption operation; after receiving the output, it applies a Pauli decryption operation. The elegance of the construction lies in what happens in between: the effects of the two independent keys are directly incorporated into the blinded quantum operation that the server executes. In other words, the server is not simply running the client’s raw circuit on encrypted data and hoping the bookkeeping works out. The circuit it runs has already absorbed the key structure, so the client never needs to track how its decryption information changes as gates accumulate. This eliminates gate-by-gate key propagation entirely, and it means the client’s quantum workload collapses to a single encryption step at the beginning and a single decryption step at the end, regardless of how deep the delegated circuit is.

Hiding the client’s quantum state, however, is only half of the privacy problem. As Associate Professor Bingwen Feng of Jinan University points out, protecting the quantum state alone is not enough, because the target computation itself may also contain sensitive information. A proprietary optimization algorithm, a cryptographic subroutine, or a simulation of a commercially valuable material could be revealed by the pattern of gates even if the input state is perfectly disguised. The theoretical core of the new paper addresses this concern head-on. The team proved that, under their protocol, the server cannot determine which specific operation within the client’s Pauli equivalence class is actually being performed. The Pauli equivalence class is a family of quantum operations that are related to one another by Pauli operators, the fundamental single-qubit transformations that include bit flips, phase flips, and their combinations. By concealing the target operation within this class, the protocol presents the server with an ambiguity that is information-theoretically built in: from the server’s perspective, every member of the class looks like a plausible candidate for the true computation. This provides what Feng calls an additional layer of privacy for delegated quantum computing, one that protects the algorithm rather than only the data.

Privacy alone, though, is not the whole story in a delegated setting. A malicious or faulty server could run the wrong circuit, introduce errors deliberately, or quietly substitute a cheaper approximation, and an encrypted output would give the client no obvious way to notice. Delegated computation therefore demands verifiability, the ability to detect when the server has deviated from its instructions. The researchers tackled this with a statistical verification strategy built on randomly sampled Pauli observables. The idea is deliberately low-overhead. Rather than running the full computation many times and comparing complete output distributions, which would multiply the cost of the protocol, the client interleaves multiple interaction rounds in which only one round carries the actual target computation. The remaining rounds serve as verification rounds, in which the server is asked to measure randomly chosen Pauli observables whose expected outcomes the client can predict. Because Pauli measurements are among the simplest and cheapest operations a quantum processor can perform, this verification layer adds minimal burden while creating a statistical web that abnormal server behavior struggles to escape.

The detection logic is probabilistic but powerful. If the server performs the correct blinded operation in every round, the sampled Pauli observables will return outcomes consistent with the client’s predictions, and the verification rounds will look statistically unremarkable. If, however, the server deviates, the deliberately introduced wrong operations will perturb the statistics of the sampled observables in a way that accumulates across the verification rounds. With enough rounds, sufficiently large deviations become detectable with high confidence, and the client can reject the result or flag the server, all without ever learning more about the server’s internal behavior than the statistics themselves reveal. The trade-off between security and overhead, which has long been a sticking point in verifiable blind quantum computation, is thereby softened: verification is achieved not by replicating the whole computation but by scattering cheap, randomized checks throughout the interaction.

To show that the scheme is more than a paper construction, the team ran proof-of-principle experiments on IBM’s 133-qubit Torino superconducting quantum processor. In a representative test, the researchers orchestrated 16 interaction rounds: a single round carrying the target computation and 15 rounds devoted to verification. The experiment successfully detected sufficiently large deviations that had been deliberately introduced, confirming that the statistical test behaves as designed on real, noisy quantum hardware. The choice of a commercially available cloud-accessed processor is itself significant, because it places the demonstration in precisely the environment the protocol is meant to protect: a user sending a task to a quantum server they do not control.

Beyond the single representative test, the team ran additional experiments on several single-qubit gates and carried out numerical simulations for representative two-qubit and three-qubit gates. These simulations confirmed that the verification strategy extends naturally to multi-qubit systems, an essential requirement for any protocol hoping to keep pace with the growing size of practical quantum computations. Scaling verifiability from one qubit to several is not a trivial matter, since the space of possible deviations grows combinatorially with system size, but the sampled-observable framework is designed so that the client’s cost grows modestly even as the server’s space for misbehavior expands. Together, the hardware demonstrations and the simulations support the claim that the approach is both feasible today and scalable toward the multi-qubit regime where quantum cloud computing will eventually operate.

The broader significance of the work lies in what it removes from the client’s shoulders. Many earlier proposals for secure delegated quantum computation implicitly assumed a client sophisticated enough to track evolving keys, prepare complex resource states, or perform interactive corrections mid-computation. By reducing the client’s quantum responsibilities to a Pauli encryption before transmission and a Pauli decryption upon return, the key-decoupled design opens the door to clients that are little more than ordinary users with minimal quantum hardware, which is exactly the profile of a typical quantum cloud customer. As Zhang notes, the work provides a simple and verifiable route toward privacy-preserving delegated quantum computation, and it may help support the development of secure quantum cloud computing and networked quantum information processing. In a future where quantum processors are concentrated in a handful of data centers and accessed remotely by businesses, hospitals, and research laboratories, protocols of this kind could form the trust layer that makes such services viable.

Challenges remain, of course. The experimental demonstration took place on today’s noisy intermediate-scale hardware, and bridging from proof-of-principle circuits to industrially relevant computations will require further engineering, deeper statistical analysis of detection thresholds, and integration with error-correction and networking infrastructure. The statistical verification method detects sufficiently large deviations, and characterizing exactly what a modestly malicious server could slip past a given number of rounds will be an important direction for follow-up work. Nevertheless, the combination of key decoupling, Pauli-class concealment, and low-overhead sampled verification represents a coherent and practical architectural statement: that privacy and verifiability in delegated quantum computing need not come at the price of an impossibly demanding client. As quantum clouds move from speculation to deployment, the ability to hand a server a computation it cannot read and cannot cheat on may prove to be one of the foundational technologies of the quantum internet era, and this protocol offers a concrete, experimentally tested blueprint for how that handover can work.

Subject of Research: Privacy-preserving delegated quantum computing through key-decoupled circuit-model blind quantum computation with statistical verification

Article Title: Circuit-model blind quantum computation with key decoupling

Article References: Circuit-model blind quantum computation with key decoupling. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: blind quantum computation, quantum cloud computing, key decoupling, Pauli equivalence class, statistical verification, delegated computation, superconducting quantum processor, privacy-preserving computing, IBM quantum processor, quantum cryptography, verifiable computation, client-server quantum protocols