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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

Diagrammatic representation of direct measurement systems of quantum states, processes, and measurements

Kazuhisa Ogawa, H. Kobayashi, Akihisa Tomita

Published Jan 1, 2019• 0 citations
Publisher

Abstract

A diagrammatic representation that intuitively shows the relation between measurement systems using qubit probes and their outcomes is introduced. This representation enables us to design direct measurement systems of quantum states, processes, and measurements systematically.
DOI
10.1364/cqo.2019.tu5a.3

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A classical state specifies the quantitative values of all the state variables relevant to a given system. In some situations, we can know the overall state of the system without being sure about the individual states of the constituent parts. On top of the problems inherent in picking out the crucial details of a system, there is another issue with trying to specify an initial state precisely: there is always some uncertainty connected with measurement. The quantum state of a system contains a series of complex numbers related to the probability that the system will collapse into a new state when a measurement takes place. Each of these new states represents a possible result of the measurement, which might be a path or a specific quantitative value of some physical variable.
Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis.In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for fixed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry.We provide several new ZX-diagram rewrite rules and generalizations for this setting.In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coefficients can be effectively ignored.This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules.We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance by considering several ansätze from the literature including realizations of hardware-efficient ansätze and QAOA.We find that by employing a diagrammatic representation, calculations across different ansätze can become more intuitive and potentially easier to approach systematically than by alternative means.Finally, we outline how diagrammatic approaches may aid in the design and study of new and more effective quantum circuit ansätze.
Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for the analysis of quantum protocols, especially non-Gaussian processes. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, which enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations for various quantum protocols. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.