Hybrid Quantum-Classical Algorithms: Limited Speedup in Search Problem, Study Reveals

The article explores the role of classical computation in hybrid quantum-classical algorithms, particularly in the context of the Search problem. It explains that while quantum computing has made significant progress, it still lacks the speed and precision of classical computers, leading to the exploration of hybrid quantum-classical computation. The article reveals that no amount of classical computation can reduce the number of quantum operations needed for the Search problem. It also discusses the limitations of hybrid quantum-classical algorithms, the techniques used in quantum query lower bound techniques, and the future of hybrid quantum-classical algorithms.

What is the Role of Classical Computation in Hybrid Quantum-Classical Algorithms?

Quantum computing has been making strides towards practicality in recent years. However, the speed and precision of its operations are still not on par with what classical computers can do. This has led to the exploration of hybrid quantum-classical computation, where some parts of the computation are delegated to classical processing while others are essentially quantum. This approach is especially important for tasks that are omnipresent in quantum computation.

One of the most fundamental and most studied problems in quantum computing is the Search problem. The famous Grover’s algorithm achieves a quadratic speedup over purely classical computation. However, it is still not fully understood under what conditions this quadratic speedup can be preserved. This work investigates if one can save on the amount of quantum computation necessary for Search by employing classical computation. The answer to this question is essentially “no” – asymptotically, no amount of classical computation can reduce the number of quantum operations needed unless this classical computation on its own can solve the Search problem.

How Do Hybrid Quantum-Classical Algorithms Work?

In a hybrid quantum-classical algorithm, the search is conducted over a domain of size n, with the goal of finding a marked element. Whether an element is marked or not is given by a black-box function f, which is the input of the problem. The algorithm is given both access to a quantum oracle for f that can evaluate f in a superposition, as well as a classical access to f, which delegates the computation of f to a classical oracle.

The question then arises: Can most of the evaluations be classical? For example, could it be possible to construct a hybrid quantum-classical search algorithm that uses 1/3 quantum queries and 1/2 classical queries to f? The answer to this question is negative – any hybrid quantum-classical algorithm for Search needs to perform at least Ω(√n) quantum queries or at least Ω(n) classical queries.

What are the Limitations of Hybrid Quantum-Classical Algorithms?

While a hybrid algorithm that uses 1/3 quantum queries and 1/2 classical queries cannot solve the Search problem, it does not tell us what are the best possible chances of success of such an algorithm. The success probability of an algorithm can be boosted by running it multiple times. However, the success probability of the algorithm is at most O(1/√n), which for our example would be O(1/6). If there were only quantum queries available, a better bound on the success probability could be proven using the amplitude amplification, but the presence of classical queries thwarts such an approach.

Even in scenarios where one is concerned with small success probabilities, as for example for cryptographic applications, a hybrid algorithm cannot substantially benefit from classical queries. The probability that the algorithm finds the unique marked element is at most O(1/√n).

What Techniques are Used in Quantum Query Lower Bound Techniques?

Quantum query lower-bound techniques can be broadly divided into two groups. One group is that of polynomial methods, which utilize the fact that the output probability of the algorithm can be represented by a polynomial in input values. The maximum possible degree of this polynomial is related to the number of queries performed, and the lower bounds are obtained by showing that polynomials approximating the value of the problem must have a large degree.

The other group is progress-based methods, which define a certain progress function that is based on comparing the execution of the algorithm on various inputs. The lower bounds are proven by showing that for the algorithm to achieve the desired success guarantees, the final value of the progress function must differ significantly from its initial value, and that a single oracle call cannot change this value by much.

What is the Future of Hybrid Quantum-Classical Algorithms?

The study of hybrid quantum-classical algorithms is still in its early stages, and there is much to learn about the interplay between quantum and classical computation. While this work shows that classical computation cannot assist quantum computation in solving the Search problem, it opens up new avenues for exploring how classical computation can be used in conjunction with quantum computation in other problem domains. As quantum computing continues to advance, the role of classical computation in hybrid quantum-classical algorithms will undoubtedly continue to be a topic of intense research and debate.