In a quantum search algorithm, the initial state which is a linear superposition of all possible basis states is a separable state. At each iteration, the state becomes more and more entangled until eventually it disentangles and reverts to a separable state consisting of the marked state. It is therefore interesting to study how entanglement changes in a Grover search algorithm.

Laser PhysicsFeb 1, 20061 citation

Coherence and Entanglement Monogamy in the Discrete Analogue of Analog Grover Search

Namit Anand, Arun Kumar Pati

arXiv:1611.04542Nov 14, 201624 citations

arXiv:quant-ph/0008018

Speed-up and entanglement in quantum searching

Samuel L. Braunstein, Arun Kumar Pati

Published Aug 3, 2000• 2 citations

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Abstract

DOI: 10.48550/arxiv.quant-ph/0008018

Related papers

Speed-up and entanglement in quantum Searching

Samuel L. Braunstein, Arun Kumar Pati

We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `charged for' then in existing machines speed-up without entanglement is not possible.We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `charged for' then in existing machines speed-up without entanglement is not possible.

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Speed-up and entanglement in quantum searching

Samuel L. Braunstein, Arun Kumar Pati

Grover's algorithm on pure states naturally generates entanglement during computation. For pseudo-pure state implementations we show that not only is entanglement necessary to achieve a speed-up, but it must be present throughout the computation. Despite the non-asymptotic character of this result we find that it only unambiguously applies to ensemble implementations, such as in liquid-state NMR, for asymptotically large search spaces. This ambiguity and its implications for interpreting a claimed speed-up in existing NMR based quantum searches is discussed.Grover's algorithm on pure states naturally generates entanglement during computation. For pseudo-pure state implementations we show that not only is entanglement necessary to achieve a speed-up, but it must be present throughout the computation. Despite the non-asymptotic character of this result we find that it only unambiguously applies to ensemble implementations, such as in liquid-state NMR, for asymptotically large search spaces. This ambiguity and its implications for interpreting a claimed speed-up in existing NMR based quantum searches is discussed.

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

Valentin Gebhart, Luca Pezzè, Augusto Smerzi

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.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.

arXiv:2001.04486Scientific ReportsJan 14, 202111 citations

Entanglement guides quantum computation

Arun Kumar Pati

Origin of the speed‐up in quantum algorithms is not yet fully understood. But there are indications that entanglement does play an important role in quantum computation. Algorithms that do not involve entanglement can be efficiently simulated on a classical computer. Here I make a simple observation. I show that infinitesimal change in multi‐particle pure product state always gives rise to an entangled state. During quantum computation even though at each time instant the state is not entangled, it does pass through entangled states at infinitesimal time steps. This applies to multi‐particle pure, pseudo‐pure and general mixed states as well. This suggests that even though unitary operators do not produce any entanglement, quantum entanglement guides the process of quantum computation.Origin of the speed‐up in quantum algorithms is not yet fully understood. But there are indications that entanglement does play an important role in quantum computation. Algorithms that do not involve entanglement can be efficiently simulated on a classical computer. Here I make a simple observation. I show that infinitesimal change in multi‐particle pure product state always gives rise to an entangled state. During quantum computation even though at each time instant the state is not entangled, it does pass through entangled states at infinitesimal time steps. This applies to multi‐particle pure, pseudo‐pure and general mixed states as well. This suggests that even though unitary operators do not produce any entanglement, quantum entanglement guides the process of quantum computation.

AIP conference proceedingsJan 1, 20061 citation

Entropy of a needle in a haystack

Yiyuan Fang, L. C. Kwek, Keng Wah Choo +2 more

Laser PhysicsFeb 1, 20061 citation

Coherence and Entanglement Monogamy in the Discrete Analogue of Analog Grover Search

Namit Anand, Arun Kumar Pati

Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the $\textit{analog analogue of Grover search algorithm}$ . The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a $n$ -party monogamy inequality for arbitrary times, hence bounding the amount of $n$ -qubit entanglement during the quantum search.Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the $\textit{analog analogue of Grover search algorithm}$. The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a $n$-party monogamy inequality for arbitrary times, hence bounding the amount of $n$-qubit entanglement during the quantum search.

arXiv:1611.04542Nov 14, 201624 citations

\textit{analog analogue of Grover search algorithm}

. The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a

n

-party monogamy inequality for arbitrary times, hence bounding the amount of

n

-qubit entanglement during the quantum search.