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 ł§Ě… ‍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  ł§Ě…, 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 ł§Ě… might be. However, a quantum algorithm can use a single superposition state over S (a quantum sample) to instantly generate a sample from ł§Ě…, 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 ł§Ě…. 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 |Sâź© to |ł§Ě…âź© with high probability.
- For K = N/2, the algorithm works perfectly outputting an element from ł§Ě… 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 ł§Ě…
- 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!
