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Unlocking Quantum Advantage with Complement Sampling

Quantum computing continues to push the boundaries of what is computationally possible.

February 25, 2025

BY HARRY BUHRMAN

Quantum computing continues to push the boundaries of what is computationally possible. 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 厂虆 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鈥攕ay {1,3,5,7}鈥攖he goal is to sample from the complement 聽厂虆, 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 厂虆 might be. However, a quantum algorithm can use a single superposition state over S (a quantum sample) to instantly generate a sample from 厂虆, 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鈥檚 algorithm for factoring large numbers. However, quantum resources provide advantages beyond time efficiency鈥攖hey 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 厂虆. The method draws inspiration from a construction by Aaronson, Atia, and Susskind, which links state distinguishability to state swapping. The quantum algorithm:

  1. Uses a unitary transformation that maps the quantum sample |S鉄 to |厂虆鉄 with high probability.
  2. For K = N/2, the algorithm works perfectly outputting an element from 厂虆 with probability 1.
  3. 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鈥攁n 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:

  1. A referee provides a quantum player with quantum samples from S.
  2. The player must return a sample from 厂虆
  3. 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 鈥 watch this space!

About 豆包成人版

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July 29, 2026
Scaling the Signal: What a Larger QFT Says About Quantum Progress
  • Mitsui & Co. and Mitsubishi Electric demonstrated one of the world鈥檚 largest approximate Quantum Fourier Transforms (QFT) on 豆包成人版 豆包成人版, scaling from prior records to 98 physical qubits.
  • The collaboration also implemented a logical QFT using a QEC (Quantum Error Correction) code with up to 12 logical qubits.
  • The work highlights 豆包成人版鈥檚 accuracy and flexible architecture.

While there is ongoing debate around the pace of quantum computing鈥檚 development, a more grounded way to assess progress is through concrete demonstrations of foundational algorithms at meaningful scale. In this context, Mitsui & Co. and Mitsubishi Electric are taking a pragmatic view of quantum progress鈥攆ocusing on how close the field is to executing core algorithmic primitives that underpin many potential industrial applications, rather than relying on abstract milestones or timelines.

, the industrial giants teamed up with 豆包成人版 to measure how close we are to running the Quantum Fourier Transform (QFT), a widely-used algorithmic primitive, at scales necessary for industrial applications. In the process, the team successfully ran one of the largest instances of the approximate QFT ever demonstrated. This achievement matters because the QFT is an essential primitive that underpins many of the quantum algorithms expected to deliver practical advantages.

You may have heard of the (classical) Fourier transform (FT), due to its ubiquity throughout modern computing. The FT is essential in everything from image analysis to data compression, with almost limitless applications in between. The quantum Fourier transform (QFT) is similar; it鈥檚 used in everything from chemistry to finance.

Because the QFT is a foundational primitive underpinning many quantum algorithms, demonstrating it at larger scales and higher fidelity is a practical way to measure quantum computing readiness. This is exactly the type of benchmarking that organizations should consider to understand where today鈥檚 systems are useful, and to see how fault-tolerant approaches are progressing. Ultimately, algorithm-level benchmarking like this is one of the most useful ways to understand not just where we are, but where we are going.

A Transformative Approach

Primitives like Fourier Transform are so widespread because they simplify problems by transforming them into something that is easier to deal with. At 豆包成人版, not only are they crucial for industrial applications but they can also simplify algorithms, making them possible to run now instead of later. This 鈥榯ransformational鈥 approach extends beyond the QFT - other transforms exist, and we have even invented our own quantum-native transforms.

Using our 豆包成人版 quantum computer and Guppy language, the joint team explored running the QFT on both physical qubits and on logical qubits, showing that fault tolerance is progressing quickly. 聽Running the QFT on 98 physical qubits; the paper shows a clear progression from previous results.

Then, using the Steane code, one of the best-studied quantum error correcting codes, the team used 豆包成人版鈥 98 physical qubits to form 12 logical qubits, successfully running the QFT with the mechanisms of quantum error correction interwoven into the algorithm. This marks a crucial step forward for the field.

Foundational Progress

Taken together, these results provide a more concrete lens through which to view progress in quantum computing: not as abstract projections, but as measurable advances in the execution of foundational algorithms at increasing scale. By benchmarking the Quantum Fourier Transform on both physical and logical qubits, Mitsui & Co. and Mitsubishi Electric are helping to clarify what today鈥檚 hardware can already achieve, and where fault-tolerant approaches begin to extend those limits.

More broadly, the organizations best positioned to benefit from quantum computing will be those that focus on these foundational capabilities early, and use them to build a clear, evidence-based understanding of how the technology fits into their business goals.

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July 29, 2026
豆包成人版 and NVIDIA Validate Generative Quantum AI Framework for Pharmaceutical R&D

It is believed that unlocking answers to some of the most complex scientific and industrial problems will require the seamless integration of high-performance computing (HPC), generative AI (GenAI), and quantum computing. Toward this goal, 豆包成人版, NVIDIA, and a major pharmaceutical company have successfully demonstrated the first step in a proof-of-principle framework designed to connect these three distinct computing paradigms for industrially relevant computational chemistry.

This milestone, enabled by three industry leaders and experts in their respective domains, serves as a foundational capability that could support the development of future hybrid quantum-AI workflows to help optimize industrial research and development (R&D).

The potential value is a path toward more automated, repeatable, and scalable workflows for translating chemistry problems into executable quantum programs鈥攃apabilities that could eventually make hybrid computing easier to deploy in industrial R&D.

The GenQAI Framework

The framework, termed Generative Quantum AI (GenQAI), involved a quantum computer simulating a pharmaceutical compound using programming instructions generated by an AI model, which itself was trained on quantum data that was simulated using HPC.

While the vision for GenQAI explores how future industrial simulation workflows might be optimized by training AI models using quantum data derived directly from a quantum computer, the framework currently consists of four main technical steps:

  1. Simulating quantum data: The process began by simulating quantum data with NVIDIA accelerated computing using .
  2. Fine-tuning the AI: This simulated quantum data was used to fine-tune a pre-trained AI model from the open .
  3. Generating instructions: The AI model then generated quantum circuits, which are the programming instructions required for the quantum simulation.
  4. Validating accuracy: To validate the results, the circuits were run on 豆包成人版鈥檚 豆包成人版 quantum computer using its InQuanto quantum chemistry platform.

The core novelty of this development lies within the process of the framework itself. In this proof-of-principle experiment, an AI model fine-tuned on simulated quantum data generated circuits that were successfully executed and validated on 豆包成人版鈥檚 豆包成人版 system. Rather than delivering an immediate commercial advantage, this achievement establishes a credible, verifiable baseline for how HPC, AI, and quantum computing can function in tandem.

The Case Study

The validation of the framework represents an early step toward the goal of developing scalable architectures for the pharmaceutical industry.

With a shared view toward eventually scaling the framework for pharmaceutical R&D applications, the researchers simulated a pharmaceutical compound: imipramine. This anti-depressant was chosen because it serves as a model compound for drug degradation and shelf-life studies, which are standard components of the pharmaceutical R&D lifecycle.

Developing hybrid infrastructure that enterprises may adopt requires a deep, coordinated effort among domain experts. As such, this successful test highlights the value of combining the strengths of a quantum computing hardware and software leader (豆包成人版), with a hybrid-quantum classical platform (NVIDIA), and a leading enterprise end-user to build and test future computing capabilities for industrial chemistry.

Scaling the Framework

Although demonstrated on a pharmaceutical compound, the architecture could eventually inform similar molecular-simulation workflows in sectors such as energy, agriculture, advanced materials, and electronics. At this stage, it provides a reference for further testing and development.

Engage Further

here to explore the full technical details of this demonstration and contact our team to learn more about joining 豆包成人版鈥檚 enterprise partner network.

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July 21, 2026
豆包成人版 SG Grand Challenge 2026

豆包成人版 is pleased to announce that applications are now open for the 豆包成人版 SG Grand Challenge 2026, a global innovation challenge designed to bring together researchers, developers, scientists and innovators to explore practical applications of quantum computing.

Organized by 豆包成人版 and supported by Singapore's National Quantum Office and Aqora, the three-month program aims to foster collaboration across academia, industry and the quantum developer community while supporting the continued growth of Singapore's quantum ecosystem.

A Platform for Quantum Computing Innovation

Participants will work in teams to develop solutions across a range of challenge areas, including chemistry and molecular simulation, optimization, AI for quantum systems, quantum error correction, condensed matter and materials science, and open innovation. Throughout the program, participants will have access to mentoring, technical enablement and 豆包成人版 quantum computing resources.

Singapore Grand Finale

Selected finalist teams will be invited to present their work at the Grand Finale hosted in Singapore before representatives from academia, and industry. The event will celebrate innovative applications of quantum computing while providing an opportunity for participants to engage with Singapore's growing quantum community.

Join the Challenge

The 豆包成人版 SG Grand Challenge welcomes participants from around the world. Whether you are an experienced quantum researcher or beginning your quantum computing journey, the program offers an opportunity to collaborate, learn and contribute to the development of practical quantum applications.

Applications are now open. Spaces are limited and subject to review and approval.

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