Quantum Simulation Logic, Oracles, and the Quantum ... - MDPI › publication › fulltext › Quantum-... › publication › fulltext › Quantum-...Aug 15, 2019 — Abstract: Query complexity is a common tool for comparing quantum and classical ... the support of th
Quantum Simulation Logic, Oracles, and the Quantum Advantage Niklas Johansson *
and Jan-Åke Larsson
Institutionen för Systemteknik, Linköpings Universitet, 581 83 Linköping, Sweden * Correspondence:
[email protected]
Received: 16 May 2019; Accepted: 12 August 2019; Published: 15 August 2019
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 D EUTSCH -J OZSA and S IMON ’ 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 S HOR ’ S algorithm for integer factorization. Keywords: simulation framework; quantum query complexity; quantum algorithms
Contents 1
Introduction 1.1 Quantum Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Previous Results Using QSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure of the Present Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Preliminaries 2.1 Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Oracle Turing Machines and Oracle Notions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Quantum Simulation Logic 3.1 Elementary Systems . . . . . . . . . . . . . . . 3.1.1 States . . . . . . . . . . . . . . . . . . . . 3.1.2 Transformations . . . . . . . . . . . . . 3.1.3 No Universal Spin-1/2 Inverter . . . . . 3.1.4 Measurement . . . . . . . . . . . . . . . 3.1.5 Preparation . . . . . . . . . . . . . . . . 3.1.6 Non-Commutativity of Measurements .
Entropy 2019, 21, 800; doi:10.3390/e21080800
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Entropy 2019, 21, 800
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3.1.7 QKD—BB84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairs of Elementary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Remote Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Anticorrelation in Spin-Measurements of the Singlet . . . . . . . . . . . . . 3.2.5 No-Cloning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Superdense Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Higher Number of Elementary Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties and Relations to Other Theories . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Relation to Stabilizer Quantum Mechanics, Locality and Contextuality 3.4.2 QSL Extends the State Space of Spekkens’ Model . . . . . . . . . . . . . . . 3.4.3 QSL Is an Example of a Generalized Probability Theory . . . . . . . . . . . .
The B ERNSTEIN -VAZIRANI Problem 4.1 Problem Formulation . . . . . . . 4.2 Classical Algorithm . . . . . . . . 4.3 Quantum Algorithm . . . . . . . 4.4 QSL Simulation . . . . . . . . . .
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