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Canadian Journal of Physics

Bounds on the quantity of entanglement in parallel quantum computing of a single ensemble quantum computer

Manuél Ávila, Guo-Hua Sun, Shi‐Hai Dong

Published Sep 16, 2013• 0 citations
Publisher

Abstract

Speeding up of the processing of quantum algorithms has been focused on from the point of view of an ensemble quantum computer (EQC) working in a parallel mode. As a consequence of such efforts, additional speed up has been achieved for processing both Shor’s and Grover’s algorithms. On the other hand, in the literature there is scarce concern about the quantity of entanglement contained in EQC approaches, for this reason in the present work we study such a quantity. As a first result, an upper bound on the quantity of entanglement contained in EQC is imposed. As a main result we prove that equally weighted states are not appropriate for EQC working in parallel mode. In order that our results are not exclusively purely theoretical, we exemplify the situation by discussing the entanglement on an ensemble of n1 = 3 diamond quantum computers.

DOI
10.1139/cjp-2013-0083

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Speeding up of the processing of quantum algorithms has been focused on from the point of view of an ensemble quantum computer (EQC) working in a parallel mode. As a consequence of such efforts, additional speed up has been achieved for processing both Shor’s and Grover’s algorithms. On the other hand, in the literature there is scarce concern about the quantity of entanglement contained in EQC approaches, for this reason in the present work we study such a quantity. As a first result, an upper bound on the quantity of entanglement contained in EQC is imposed. As a main result we prove that equally weighted states are not appropriate for EQC working in parallel mode. In order that our results are not exclusively purely theoretical, we exemplify the situation by discussing the entanglement on an ensemble of n1 = 3 diamond quantum computers.
We propose a parallel quantum computing mode for ensemble quantum computer. In this mode, some qubits are in pure states while other qubits are in mixed states. It enables a single ensemble quantum computer to perform ``single-instruction-multidata'' type of parallel computation. Parallel quantum computing can provide additional speedup in Grover's algorithm and Shor's algorithm. In addition, it also makes a fuller use of qubit resources in an ensemble quantum computer. As a result, some qubits discarded in the preparation of an effective pure state in the Schulman-Varizani and the Cleve-DiVincenzo algorithms can be reutilized.
For any quantum algorithm operating on pure states, we prove that the presence of multi‐partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed‐up over classical computation. Furthermore, we prove that the algorithm can be efficiently simulated classically to within a prescribed tolerance η even if a suitably small amount of global entanglement is present. We explicitly identify the occurrence of increasing multi‐partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed‐up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure‐state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum‐computational power.
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.
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.
Quantum computation is the effective and inherent parallelized way of solving real world problems over quantum computer. The real world demands for a parallelization just to achieve the performance speed-up and for obtaining optimized computational results. This paper investigates various quantum computational approaches to achieve intrinsic parallelization, focusing on superposition and entanglement to perform multiple computations simultaneously. Based on a comparative classical-quantum aspect, the emergence and significance of the quantum computing is justified. The key quantum algorithms, including Shor’s and Grover’s, are analyzed for their parallel processing potential and impact on solving complex problem. The potential of quantum computation in the cryptography, cybersecurity and machine learning generate benefits and enhances performance using the quantum approaches. This paper aims to advance the understanding and the development of quantum computational methods as highly efficient and it contributes to near future of high-performance computing through intrinsic parallelization.