Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon's algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor's algorithm for integer factorization.
Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon's algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor's algorithm for integer factorization.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Simon's problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch-Jozsa and Simon's problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms, and calls for further study of oracle separation between quantum and classical computation.
Abstract The main promise of quantum computing is to efficiently solve certain problems that are prohibitively expensive for a classical computer. Most problems with a proven quantum advantage involve the repeated use of a black box, or oracle, whose structure encodes the solution. One measure of the algorithmic performance is the query complexity, i.e., the scaling of the number of oracle calls needed to find the solution with a given probability. Few-qubit demonstrations of quantum algorithms, such as Deutsch–Jozsa and Grover, have been implemented across diverse physical systems such as nuclear magnetic resonance, trapped ions, optical systems, and superconducting circuits. However, at the small scale, these problems can already be solved classically with a few oracle queries, limiting the obtained advantage. Here we solve an oracle-based problem, known as learning parity with noise, on a five-qubit superconducting processor. Executing classical and quantum algorithms using the same oracle, we observe a large gap in query count in favor of quantum processing. We find that this gap grows by orders of magnitude as a function of the error rates and the problem size. This result demonstrates that, while complex fault-tolerant architectures will be required for universal quantum computing, a significant quantum advantage already emerges in existing noisy systems.
Quantum computation transformed our thinking about computational complexity by showing that quantum mechanical principles can be used to reduce the number of queries needed to solve some problems. The work constitutes a critical analysis of the evolution of the query complexity through three fundamental quantum algo rithms: the Deutsch algorithm, the Deutsch–Jozsa algorithm and Simon’s algorithm. Together, these algorithms illustrate the emergence of quantum advantage where quantum systems outperform classical deterministic and probabilistic approaches by exploiting superposition, interference, entanglement and unitary evolution.The study begins with the Deutsch algorithm, the first demonstration that a quantum computer can compute a global property of a Boolean function with fewer oracle queries than any deterministic classical algorithm. We further extend the analysis to the Deutsch–Jozsa algorithm, which generalizes this speedup to the exponential level by distinguishing between balanced and constant Boolean functions with one quantum query. Then we study Simon’s Algorithm, a milestone that achieved an exponential separation between classical randomized and quantum query complexity. The groundwork for many later advances, such as Shor’s algorithm for factoring integers, was established by this body of work. The discussion continues with an emphasis on Linear Algebra and the use of Hilbert spaces, Tensor Products, Unitary Transformations, and Fourier based interference methods. As the report describes, significant advancements toward reducing the time it takes for quantum systems to perform calculations are made possible through use of Oracle construction, Spectral Decomposition, and Mea surement Theory. This comparative analysis explores the use of computational queries in deterministic, probabilistic, and quan tum models while examining the transition from classical decision tree complexities into quantum computation paradigms. Ultimately, the results of this analysis show that the early quantum algorithms must be recognized not merely as isolated computational oddities, but rather as crucial components of the development of quantum information sci ence. They provided a conceptual and mathematical framework for the modern understanding of what constitutes a quantum advantage; Moreover, they also provide valuable insight into how quantum mechanical systems influ ence both computational efficiency and complexity theory, specifically as well as future information processing system design.
In this thesis we address the question, what is the resource, or property, that enables the advantage of quantum computers?The theory of quantum computers dates back to the eighties, so one would think there already is an answer to this question.There are several proposed solutions, but to this date, there is no consensus on an answer.Primarily, the advantage of quantum computers is characterized by a speedup for certain computational problems.This speedup is measured by comparing quantum algorithms with the bestknown classical algorithms.For some algorithms we assume access to an object called oracle.The oracle computes a function, and the complexity of the oracle is of no concern.Instead, we count the number of queries to the oracle needed to solve the problem.Informally, the question we ask using an oracle is: if we can compute this function efficiently, what else could we then compute.However, using oracles while measuring a quantum speedup, we assume access to vastly different oracles residing in different models of computation.For our investigation of the speedup, we introduce a classical simulation framework that imitates quantum algorithms.The simulation suggests that the property enabling the potential quantum speedup is the ability to store, process, and retrieve information in an additional degree of freedom.We then theoretically verified that this is true for all problems that can be efficiently solved with a quantum computer.In parallel to this, we also see that quantum oracles sharply specify the information we can retrieve from the additional degree of freedom, while regular oracles do not.A regular oracle does not even allow for an extra degree of freedom.We conclude that comparing quantum with classical oracle query complexity bounds does not provide conclusive evidence for a quantum advantage.I want to thank Jan-Åke Larsson, without whom this research would not have been possible.If there is any magic in the world, it shows through Jan-Åke's ability to simultaneously be a good supervisor, colleague, and friend.I would also like to thank my co-supervisor Peter Jonsson, for many valuable discussions -not only restricted to theoretical computer science.To my friends and collogues that have challenged me to become better, especially Jonathan Jogenfors
This presentation gives a comprehensive overview of quantum query algorithms, beginning with an introduction to the query model of computation where input is accessed via an oracle function. It then details several foundational quantum algorithms that demonstrate a quantum advantage over classical approaches: Deutsch's problem is solved with a single query using phase kickback to determine if a function is constant or balanced; the Deutsch-Jozsa algorithm extends this to n-bit inputs, efficiently distinguishing constant from balanced functions with one query versus an exponential need classically; the Bernstein-Vazirani algorithm identifies a hidden string with one query, outperforming classical linear requirements; and Simon's algorithm, which finds a hidden period in a two-to-one function with a linear number of queries, showcases an exponential speedup over the best classical probabilistic algorithms.