Quantifying computational advantage of Grover’s algorithm with the trace speed

Abstract

Despite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover’s search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the polarization of the pseudo-pure state, which can be connected to a wide class of quantum statistical speeds. For time-dependent partial depolarization and for interrupted Grover searches, the speed-up is specifically bounded by the maximal trace speed that occurs during the algorithm operations. Our results quantify the quantum speed-up with a physical resource that is experimentally measurable and related to multipartite entanglement and quantum coherence.

Entanglement in Quantum Search Database: Periodicity Variations and Counting

“…While there are examples where such a claim is corroborated, other cases are known where this is not evident [24]. As alternatives to such counterexamples other resources beyond quantum entanglement have been considered e.g., coherence, distinguish-ability, contextuality, interference etc (see [25] and references therein).…”

“…Can the accelerated period of entanglement is valid for other measures of correlation like the ones mentioned previously? How the fast counting procedure is modified in the case of open, dissipative extensions of quantum search as those analysed in [25] and the more complicated ones as in [15,17] as well?…”

A Quantum Planner for Robot Motion

“…More recently, Hlembotskyi et al [41] investigated the performance of Grover's algorithm on real quantum devices equipped with ion-trap technology, presenting encouraging data that demonstrate the actual improvements of the quantum search algorithm, which exceeds any possible classical approach even in the case of a single oracle call. Other studies have been made towards the exploration of the effectiveness of Grover's algorithm, such as [42][43][44], and even a hardware efficient implementation has been proposed in [45], reducing the total depth and gate count of the circuit for the quantum search algorithm. All such experimental clues tilt the scales in favor of the actual, practical usefulness of Grover's algorithm for real-world applications once more powerful and stabler quantum technologies are built [46,47], which, in the case of IBM's quantum systems, should occur no longer than a handful of years, according to their 2025 quantum roadmap [48].…”

Quantum Cryptography for Securing Personal Health Information in Hospitals

“…80 The two most popular quantum algorithms are Shor's algorithm and Grover's algorithm. 81,82 Shor's algorithm is an algorithm for finding the prime factors of an integer using a specific unitary operator. Unfortunately, this algorithm can undermine the security of RSA and ECC due to program-related issues.…”

“…Grover's algorithm can improve these searches as it is based on the properties of superposition, entanglement, and interference. 82 There is a special qubit gate called oracle which takes the input state and flips the phase of the chosen ket we are looking for and another gate which inverts the amplitudes of all the component kets about the mean of all the associated amplitudes. 85 However, all problems are still not resolved, and some limitations might appear when fullfledged quantum computers become a reality.…”