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Institute of Physics eBooks

Creating genuine quantum algorithms for quantum energy-based computing

Koji Nagata, Do Ngoc Diep, Ahmed Farouk, Tadao Nakamura

Published Jul 1, 2022• 0 citations
Publisher
DOI
10.1088/978-0-7503-4700-6ch13

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Abstract Any quantum resource theory is based on free states and free operations, i.e. states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. We propose here the framework of genuine quantum coherence. This approach is based on a simple principle: we demand that a genuinely incoherent operation preserves all incoherent states. This framework captures coherence under additional constrains such as energy preservation and all genuinely incoherent operations are incoherent regardless of their particular experimental realization. We also introduce the full class of operations with this property, which we call fully incoherent. We analyze in detail the mathematical structure of these classes and also study possible state transformations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e. there is no single state from which all other states can be created deterministically with the free operations. This suggests that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.

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The execution cost of quantum algorithms is typically quantified through asymptotic gate counts and qubit register sizes, yet these metrics do not directly capture which genuinely quantum resources, and in what amount, must be created and maintained for the computation to succeed. The systematic quantification of such information-theoretic requirements in quantum computing protocols remains an extremely challenging open problem, despite their direct role in establishing quantum advantage. To address this gap, we investigate the generation of non-stabilizerness (or magic), one of the key resources, in the paradigmatic Shor's factoring algorithm, revealing a deep connection between intrinsic quantum complexity and the computational hardness of the underlying number-theoretic problem. By developing an explicit analytic theory, we demonstrate the fundamental role of magic in the successful execution of the algorithm, and show that Shor's routine maximally exploits the quantum resource in practically relevant regimes. Our findings create a concise conceptual link between the classical algorithmic difficulty of a task and the non-stabilizer price to solve it on quantum hardware, complementing standard circuit-cost analyses with a resource-based metric that is naturally aligned with the real bottlenecks of fault-tolerant quantum computing.
In the discussion about the quantumness of NMR computation a conclusion is done that computational states are separable and therefore can not be entangled. This conclusion is based on the assumption that the initial density matrix of an individual molecule coincides with whole sample molecules distribution over single molecule energy levels. This means that quantum stochasticity is replaced by classical stochasticity. In the present paper it is shown, that quantum NMR computation can create genuine entangled states if initial system states are thermodynamical equilibrium ones. A separability analysis problem can arise when one interprets the readout signal from whole sample.
Abstract Any quantum resource theory is based on free states and free operations, i.e. states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. We propose here the framework of genuine quantum coherence. This approach is based on a simple principle: we demand that a genuinely incoherent operation preserves all incoherent states. This framework captures coherence under additional constrains such as energy preservation and all genuinely incoherent operations are incoherent regardless of their particular experimental realization. We also introduce the full class of operations with this property, which we call fully incoherent. We analyze in detail the mathematical structure of these classes and also study possible state transformations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e. there is no single state from which all other states can be created deterministically with the free operations. This suggests that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.