Unlocking Quantum Advantage with Complement Sampling
Quantum computing continues to push the boundaries of what is computationally possible.
BY HARRY BUHRMAN
Quantum computing continues to push the boundaries of what is computationally possible. A new study by Marcello Benedetti, Harry Buhrman, and Jordi Weggemans introduces Complement Sampling, a problem that highlights a dramatic separation between quantum and classical sample complexity. This work provides a robust demonstration of quantum advantage in a way that is not only provable but also feasible on near-term quantum devices.
The Complement Sampling Problem
Imagine a universe of N = 2n elements, from which a subset S of size K is drawn uniformly at random. The challenge is to sample from the complement S̅ without explicitly knowing S, but having access to samples of S. Classically, solving this problem requires roughly K samples, as the best a classical algorithm can do is guess at random after observing only some of the elements of S.
To better understand this, consider a small example. Suppose N = 8, meaning our universe consists of the numbers {0,1,2,3,4,5,6,7}. If a subset S of size K = 4 is drawn at random—say {1,3,5,7}—the goal is to sample from the complement S̅, which consists of {0,2,4,6}. A classical algorithm would need to collect and verify enough samples from S before it could infer what S̅ might be. However, a quantum algorithm can use a single superposition state over S (a quantum sample) to instantly generate a sample from S̅, eliminating the need for iterative searching.
Why This Matters: Quantum Advantage in Sample Complexity
Quantum advantage is often discussed in terms of computational speedups, such as those achieved by Shor’s algorithm for factoring large numbers. However, quantum resources provide advantages beyond time efficiency—they also affect how data is accessed, stored, and processed.
Complement Sampling fits into the category of sample complexity problems, where the goal is to minimize the number of samples needed to solve a problem. The authors prove that their quantum approach not only outperforms classical methods but does so in a way that is:
- Provable: It provides rigorous lower bounds on classical sample complexity, demonstrating an exponential separation.
- Verifiable: The correctness of the output of the sampler can be efficiently checked classically.
- NISQable: The quantum circuit required is shallow and feasible for Noisy Intermediate-Scale Quantum (NISQ) devices.
How the Quantum Algorithm Works
At its core, the quantum approach to Complement Sampling relies on the ability to perform a perfect swap between a subset S and its complement S̅. The method draws inspiration from a construction by Aaronson, Atia, and Susskind, which links state distinguishability to state swapping. The quantum algorithm:
- Uses a unitary transformation that maps the quantum sample |S⟩ to |S̅⟩ with high probability.
- For K = N/2, the algorithm works perfectly outputting an element from S̅ with probability 1.
- For other values of K, a probabilistic zero-error approach is used, ensuring correctness while reducing success probability.
This is made possible by quantum interference and superposition, allowing a quantum computer to manipulate distributions in ways that classical systems fundamentally cannot.
Classical Hardness and Cryptographic Implications
A crucial aspect of this work is its robustness. The authors prove that even for subsets generated using strong pseudorandom permutations, the problem remains hard for classical algorithms. This means that classical computers cannot efficiently solve Complement Sampling even with structured input distributions—an important consideration for real-world applications.
This robustness suggests potential applications in cryptography, where generating samples from complements could be useful in privacy-preserving protocols and quantum-secure verification methods.
Towards an Experimental Demonstration
Unlike some quantum advantage demonstrations that are difficult to verify classically (such as the random circuit sampling experiment), Complement Sampling is designed to be verifiable. The authors propose an interactive quantum versus classical game:
- A referee provides a quantum player with quantum samples from S.
- The player must return a sample from S̅
- A classical player, given the same number of classical samples, attempts to do the same.
While the classical player must resort to random guessing, the quantum player can leverage the swap algorithm to succeed with near certainty. Running such an experiment on NISQ hardware could serve as a practical demonstration of quantum advantage in a sample complexity setting.
Future Directions
This research raises exciting new questions:
- Can Complement Sampling be extended to more general probability distributions?
- Are there cryptographic protocols that can directly leverage this advantage?
- How well does the quantum algorithm perform in real-world noisy conditions?
With its blend of theoretical depth and experimental feasibility, Complement Sampling provides a compelling new frontier for demonstrating the power of quantum computing.
Conclusion
Complement Sampling represents one of the cleanest demonstrations of quantum advantage in a practical, verifiable, and NISQ-friendly setting. By leveraging quantum information processing in ways that classical computers fundamentally cannot, this work strengthens the case for near-term quantum technologies and their impact on computational complexity, cryptography, and beyond.
For those interested in the full details, the paper provides rigorous proofs, circuit designs, and further insights into the nature of quantum sample complexity. As quantum computing continues to evolve, Complement Sampling may serve as a cornerstone for future experimental demonstrations of quantum supremacy.
We have commenced work on the experiment – watch this space!
About Quantinuum
Quantinuum, the world’s largest integrated quantum company, pioneers powerful quantum computers and advanced software solutions. Quantinuum’s technology drives breakthroughs in materials discovery, cybersecurity, and next-gen quantum AI. With over 500 employees, including 370+ scientists and engineers, Quantinuum leads the quantum computing revolution across continents.
- Quantinuum continues its progress toward fault-tolerant quantum computing, with a series of peer-reviewed breakthroughs in fault-tolerant operations.
- Our progress is not only scientific; it is commercial. By improving logical-qubit reliability and encoding efficiency, Quantinuum is reducing the resource overhead required to scale its quantum computers toward commercially useful workloads.
- These results were achieved on commercial Quantinuum hardware, reinforcing that our architecture is not just setting new standards, but building a practical foundation for customers, partners, and researchers preparing for the fault-tolerant era.
Fault-tolerant quantum computing is the threshold the industry must cross before quantum computers can solve the hardest, highest-value problems with confidence. To be commercially useful at scale, the question is not simply who can build more qubits. It is who can build reliable, efficient, scalable systems that reduce technical risk and accelerate the path to commercial usefulness.
Quantinuum is progressing on that path.
Last year, in partnership with Microsoft, we published a breakthrough in logical computing, demonstrating logical qubits that outperformed their physical counterparts by a factor of 800. We are proud to announce that this work is now being published in Nature, one of the most highly regarded scientific journals in the world.
This work highlights our leading fidelities, as shown in Table 1:
Since then, we’ve accelerated our efforts to reach large-scale fault tolerance and advanced what we believe to be the core building blocks of fault-tolerant quantum computing, from logical-qubit teleportation and multiple error-correction breakthroughs to one of the first meaningful computations using logical qubits. Importantly, these results were achieved on commercial Quantinuum hardware, demonstrating not just scientific progress, but a practical and efficient path toward scalable, customer-ready fault tolerance.
A Recap of our Recent Technical Progress
Since the work with Microsoft, we achieved a milestone years ahead of schedule, demonstrating high-fidelity teleportation of a logical qubit, which was published in Science, one of the world’s most prestigious journals. Later, we beat our own record in this crucial fault tolerance milestone, thanks to continued improvements to our System Model H2’s fidelity.
Then, a series of results demonstrating more error-correcting milestones (and codes):
- Better than physical results in a high-rate non-local code,
- First-ever demonstration of a single-shot error correcting code (which significantly reduces resource requirements) in 4 dimensions
- Extending qubit lifetimes by 10x with a concatenated code
- Observed fault-tolerance thresholds with concatenated codes
- High fidelity magic states and a fully fault tolerant universal gate set in two different papers
Recently, we topped ourselves yet again by performing one of the first meaningful computations with logical qubits – exploring key questions in materials and magnetism, using logical qubits with better error rates than their physical counterparts. This result also includes a leading “encoding rate” squeezing 48 logical qubits out of just 98 physical qubits, emphasizing how our architecture helps to support large scale fault tolerance without enormous resource costs.
It is worth noting that all these results were achieved on our commercial hardware, not on one-off laboratory test-stands – reflecting the performance that we are able to deliver to our customers.
We also did crucial theoretical work, exploring new options for error correction that can reduce resource requirements, time to solution, and shorten the timeline to large scale fault tolerance.
Commercial Implications and the Road Ahead
We believe the commercial implication is clear: Quantinuum is reducing the uncertainty around the path to fault-tolerant quantum computing. Our architecture, hardware fidelity, full-stack control, and error-correction progress are converging into a practical roadmap for systems that can support valuable scientific and commercial workloads.
For those evaluating when quantum computing will become strategically relevant, we believe the signal is also increasingly clear: the fault-tolerant era is no longer a distant concept. It is becoming an engineering reality, and Quantinuum is leading the way.
- University of Southern Denmark (SDU) to use Quantinuum Helios, supported by the Danish e-Infrastructure Consortium (DeiC)
- Access to Helios enables SDU to test and refine fault-tolerant algorithms and error-correction codes under realistic hardware conditions
- The collaboration supports at a scale of 48 logical qubits, positioning Denmark at the forefront of scalable, practical quantum computing
- Researchers exploring the scientific foundations for future development of applications in fields including pharmaceuticals, finance, and defense
Progress in quantum computing is measured by hardware advances plus the algorithms and quantum error-correction codes that turn quantum systems into useful computational tools.
Thanks to recent hardware advances, researchers are increasingly sharpening their tools to probe the performance of quantum algorithms and understand how they behave in realistic conditions – where stability, system architecture and algorithm design all shape performance.
A new Denmark-based collaboration between the University of Southern Denmark (SDU), Quantinuum, and the Danish e-Infrastructure Consortium (DeiC) will utilize Quantinuum Helios. Researchers at the SDU’s Centre for Quantum Mathematics, led by Jørgen Ellegaard Andersen, will use Helios to pursue research into topological quantum computing.
Their work could help explain how and why successful quantum algorithms perform as they do, informing the development of high-performance algorithms suited to emerging quantum systems. They’re exploring the scientific foundations that support future quantum applications across areas including pharmaceuticals, finance, and defense.
“We are thrilled to gain access to Quantinuum’s high-fidelity Helios system. This collaboration gives us a unique opportunity to test the limits of our algorithms and evaluate system performance, while advancing fundamental research and laying the foundation for future applications.”
— Professor Jørgen Ellegaard Andersen, Director of the Centre for Quantum Mathematics at University of Southern Denmark
Why topological methods matter
Topological quantum computing is an area of research that connects quantum computation with deep mathematical structures. It includes the study of error correcting codes known as surface codes that encode quantum information in the global properties of systems of logical qubits.
The research team will explore how these codes behave, and how they may support the development of fault-tolerant quantum algorithms in practical implementations under realistic conditions.
This distinction between theory and practical implementation matters. In theory, topological approaches offer a rich framework for designing algorithms and error-correcting codes. In practice, researchers need to understand how those ideas perform when implemented on real systems, where questions of noise, stability, overhead, and scaling become central. The collaboration will allow the SDU team to investigate these questions directly.
New ways to benchmark quantum processors
Beyond individual algorithms and codes, the research will also develop tools for benchmarking quantum processors. The goal is to develop new ways to characterize fidelity and stability in regimes that can be difficult to access.
The team will also explore hybrid quantum–classical approaches, including machine-learning techniques assisted by quantum hardware, to study the mathematical structures at the heart of topological quantum computing. This work reflects a broader field of research in which quantum and classical methods are used together, each contributing to parts of a computational problem.
Strengthening Denmark’s quantum ecosystem
The collaboration reflects the growing role of national quantum infrastructure in supporting research and talent development. Denmark has a long tradition of scientific innovation, and this collaboration is intended to support the country’s continued development in quantum technology.
The initiative is supported by DeiC, which played a central role in securing funding and enabling access to Quantinuum’s systems. DeiC has been assigned a particular role in developing and coordinating quantum infrastructure initiatives for the benefit of universities and industry, operating without its own commercial, sectoral, or geographical interests. This includes securing dedicated access to quantum computers, producing advisory services and supporting the development of new talent in the Danish quantum sector.
“DeiC’s special effort to secure funding and access for this research initiative is rooted in our organization’s role in relation to the Danish Government’s strategy for quantum technology.”
— Henrik Navntoft Sønderskov, Head of Quantum at Danish e-Infrastructure Consortium
This collaboration promises to accelerate the development of practical algorithms. It is grounded in fundamental science – but its focus is practical: discovering and testing mathematical approaches to topological quantum computing that can be implemented, evaluated, and improved on real quantum hardware.
That work requires both theoretical insight and access to a system such as Helios capable of supporting meaningful scientific work.
This month, Quantinuum welcomed its global user community to the first-ever Q-Net Connect, an annual forum designed to spark collaboration, share insights, and accelerate innovation across our full-stack quantum computing platforms. Over two days, users came together not only to learn from one another, but to build the relationships and momentum that we believe will help define the next chapter of quantum computing.
Q-Net Connect 2026 drew over 170 attendees from around the world to Denver, Colorado, including representatives from commercial enterprises and startups, academia and research institutions, and the public sector and non-profits - all users of Quantinuum systems.
The program was packed with inspiring keynotes, technical tracks, and customer presentations. Attendees heard from leaders at Quantinuum, as well as our partners at NVIDIA, JPMorganChase and BlueQubit; professors from the University of New Mexico, the University of Nottingham and Harvard University; national labs, including NIST, Oak Ridge National Laboratory, Sandia National Laboratories and Los Alamos National Laboratory; and other distinguished guests from across the global quantum ecosystem.
Congratulations to Q-Net Connect 2026 Award Recipients!
The mission of the Quantinuum Q-Net user community is to create a space for shared learning, collaboration and connection for those who adopt Quantinuum’s hardware, software and middleware platform. At this year’s Q-Net Connect, we awarded four organizations who made notable efforts to champion this effort.
- JPMorganChase received the ‘Guppy Adopter Award’ for their exemplary adoption of our quantum programming language, Guppy, in their research workflows.
- Phasecraft, a UK and US-based quantum algorithms startup, received the ‘Rising Star’ award for demonstrating exceptional early impact and advancing science using Quantinuum hardware, which they published in a December 2025 paper.
- Qedma, a quantum software startup, received the ‘Startup Partner Engagement’ award for their sustained engagement with Quantinuum platforms dating back to our first commercially deployed quantum computer, H1.
- Anna Dalmasso from the University of Nottingham received our ‘New Student Award’ for her impressive debut project on Quantinuum hardware and for delivering outstanding results as a new Q-Net student user.
Congratulations, again, and thank you to everyone who contributed to the success of the first Q-Net Connect!
Become a Q-Net Member
Q-Net offers year‑round support through user access, developer tools, documentation, trainings, webinars, and events. Members enjoy many exclusive benefits, including being the first to hear about exclusive content, publications and promotional offers.
By joining the community, you will be invited to exclusive gatherings to hear about the latest breakthroughs and connect with industry experts driving quantum innovation. Members also get access to Q‑Net Connect recordings and stay connected for future community updates.