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Register a new userOverlapped grouping measurement: A unified framework for measuring quantum states
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Quantum algorithms designed for noisy intermediate-scale quantum devices usually require repeatedly perform a large number of quantum measurements in estimating observable expectation values of a many-qubit quantum state. Exploiting the ideas of importance sampling, observable compatibility, and classical shadows of quantum states, different advanced quantum measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct to each other, and how to systematically find the optimal scheme remains a critical theoretical challenge. Here, we address this challenge by firstly proposing a unified framework of quantum measurements, incorporating the advanced measurement methods as special cases. Our framework further allows us to introduce a general scheme -- overlapped grouping measurement, which simultaneously exploits the advantages of the existing methods. We show that an optimal measurement scheme corresponds to partitioning the observables into overlapped groups with each group consisting of compatible ones. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians. Our numerical results show great improvements to the overall existing measurement schemes. Our work paves the way for efficient quantum measurement with near-term quantum devices.
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The main results on quantum walk search are scattered over different, incomparable frameworks, most notably the hitting time framework, originally by Szegedy, the electric network framework by Belovs, and the MNRS framework by Magniez, Nayak, Roland and Santha. As a result, a number of pieces are currently missing. For instance, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. In recent work by Ambainis et al., this problem was resolved for the more restricted hitting time framework, in which quantum walks must start from the stationary distribution. We present a new quantum walk search framework that unifies and strengthens these frameworks. This leads to a number of new results. For instance, the new framework not only detects, but finds marked elements in the electric network setting. The new framework also allows one to interpolate between the hitting time framework, which minimizes the number of walk steps, and the MNRS framework, which minimizes the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. Whereas the original frameworks only rely on quantum walks and phase estimation, our new algorithm makes use of a technique called quantum fast-forwarding, similar to the recent results by Ambainis et al. As a final result we show how in certain cases we can simplify this more involved algorithm to merely applying the quantum walk operator some number of times. This answers an open question of Ambainis et al.
We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes used to perform the simulation of a given measurement. This general approach also allows us to identify connections between different types of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.
Experimental determination of an unknown quantum state usually requires several incompatible measurements. However, it is also possible to determine the full quantum state from a single, repeated measurement. For this purpose, the quantum system whose state is to be determined is first coupled to a second quantum system (the assistant) in such a way that part of the information in the quantum state is transferred to the assistant. The actual measurement is then performed on the enlarged system including the original system and the assistant. We discuss in detail the requirements of this procedure and experimentally implement it on a simple quantum system consisting of nuclear spins.
We present unified, systematic derivations of schemes in the two known measurement-based models of quantum computation. The first model (introduced by Raussendorf and Briegel [Phys. Rev. Lett., 86, 5188 (2001)]) uses a fixed entangled state, adaptive measurements on single qubits, and feedforward of the measurement results. The second model (proposed by Nielsen [Phys. Lett. A, 308, 96 (2003)] and further simplified by Leung [Int. J. Quant. Inf., 2, 33 (2004)]) uses adaptive two-qubit measurements that can be applied to arbitrary pairs of qubits, and feedforward of the measurement results. The underlying principle of our derivations is a variant of teleportation introduced by Zhou, Leung, and Chuang [Phys. Rev. A, 62, 052316 (2000)]. Our derivations unify these two measurement-based models of quantum computation and provide significantly simpler schemes.
Wide area measurement system (WAMS) is one of the essential components in the future power system. To make WAMS construction plans, practical models of the power network observability, reliability, and underlying communication infrastructures need to be considered. To address this challenging problem, in this paper we propose a unified framework for WAMS planning to cover most realistic concerns in the construction process. The framework jointly optimizes the system construction cost, measurement reliability, and volume of synchrophasor data traffic resulting in a multi-objective optimization problem, which provides multiple Pareto optimal solutions to suit different requirements by the utilities. The framework is verified on two IEEE test systems. The simulation results demonstrate the trade-off relationships among the proposed objectives. Moreover, the proposed framework can develop optimal WAMS plans for full observability with minimal cost. This work develops a comprehensive framework for most practical WAMS construction designs.
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