Abstract
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.
Highlights
A Quantum Simulation Logic (QSL) simulation of this will not behave as the quantum algorithm, and we have found that the behavior is too far from the quantum behavior to give the quadratic speed-up of GROVER’S algorithm
If we look at the 3-round Feistel network as a block cipher running in the Electronic Code Book (ECB) mode, it permutes a block of the plaintext into a block of the ciphertext
We have looked at the QSL simulation framework, which is efficiently simulatable on a classical Turing machine and can at the same time reproduce many quantum phenomena
Summary
Quantum computers are said to be more powerful than classical computers This is usually quantified in the following way: a quantum computer needs a smaller number of function calls of some mathematical function under investigation than a classical computer would. To answer this question we construct a simulation framework, Quantum Simulation Logic (QSL), that contains one (and only one) property that a quantum computer has, but a classical computer does not: an extra degree of freedom of each bit that the computation is performed on. We continue to show in that for some problems, a classical system equipped with an extra phase degree of freedom (QSL), needs the same number of function calls of the mathematical function under investigation, as a quantum computer would.
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