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Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can check whether quantum computers are indeed producing correct results. This task, known as quantum verification, has been highlighted as a significant challenge on the road to scalable quantum computing technology. We review the most significant approaches to quantum verification and compare them in terms of structure, complexity and required resources. We also comment on the use of cryptographic techniques which, for many of the presented protocols, has proven extremely useful in performing verification. Finally, we discuss issues related to fault tolerance, experimental implementations and the outlook for future protocols.

Quantum mechanics, the subfield of physics that describes the behavior of very small (quantum) particles, provides the basis for a new paradigm of computing. First proposed in the 1980s as a way to improve computational modeling of quantum systems, the field of quantum computing has recently garnered significant attention due to progress in building small-scale devices. However, significant technical advances will be required before a large-scale, practical quantum computer can be achieved. Quantum Computing: Progress and Prospects provides an introduction to the field, including the unique characteristics and constraints of the technology, and assesses the feasibility and implications of creating a functional quantum computer capable of addressing real-world problems. This report considers hardware and software requirements, quantum algorithms, drivers of advances in quantum computing and quantum devices, benchmarks associated with relevant use cases, the time and resources required, and how to assess the probability of success.

We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their projected entangled pair state representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.

The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has a treewidth $d$ can be simulated deterministically in $T^{O(1)}\\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if $d=O(\\log T)$. Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Phys. Rev. Lett., 86 (2001), pp. 5188-5191), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph with a constant maximum degree. (The requirement on the maximum degree was removed in [I. L. Markov and Y. Shi, preprint:quant-ph/0511069].)

Much enhanced contents to tell the basic to applications of quantum computing. So, the book provides the theory of quantum computing in its actual status starting from the basic algorithms.

Blind quantum computation (BQC) allows a client with limited quantum power to delegate his quantum computational task to a powerful server and still keep his input, output, and algorithm private. There are mainly two kinds of models about BQC, namely circuit-based and measurement-based models. In addition, a hybrid model called ancilla-driven universal blind quantum computation (ADBQC) was proposed by combining the properties of both circuit-based and measurement-based models, where all unitary operations on the register qubits can be realized with the aid of single ancilla coupled to the register qubits. However, in the ADBQC model, the quantum capability of the client is strictly limited to preparing single qubits. If a client can only perform single-qubit measurements or a few simple quantum gates, he will not be able to perform ADBQC. This paper solves the problem and extends the existing model by proposing two types of ADBQC protocols for clients with different quantum capabilities, such as performing single-qubit measurements or single-qubit gates. Furthermore, in the two proposed ADBQC protocols, clients can detect whether servers are honest or not with a high probability by using corresponding verifiable techniques.

A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave funct- ions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.

Adiabatic quantum computation is a paradigmatic model aiming to solve a computational problem by finding the many-body ground state encapsulating the solution. However, its use of an adiabatic evolution depending on the spectral gap of an intricate many-body Hamiltonian makes its analysis daunting. While it is plausible to directly cool the final gapped system of the adiabatic evolution instead, the analysis of such a scheme on a general ground is missing. Here, we propose a specific Hamiltonian model for this purpose. The scheme is inspired by cavity cooling, involving the emulation of a zero-temperature reservoir. Repeated discarding of ancilla reservoir qubits extracts the entropy of the system, driving the system toward its ground state. At the same time, the measurement of the discarded qubits hints at the energy-level structure of the system as a return. We show that quantum computation based on this cooling procedure is equivalent in its computational power to the one based on quantum circuits. We then exemplify the scheme with a few illustrative use cases for combinatorial optimization problems. To circumvent the issue of local energy minima, we implant a mechanism in the Hamiltonian that allows the population trapped in the local minima to tunnel out via high-order transitions, and support the idea with numerical simulations. We also discuss its application to preparing quantum many-body ground states, arguing that the spectral gap is a crucial factor in determining the time scale of the cooling.

Quantum computing is an emerging technology with the potential to have a significant impact on science and technology. Organised into four parts, this comprehensive second edition covers topics such as the basic concepts of quantum computing alongside quantum implementation of different circuits; the fault tolerant concepts of quantum computing; the concept of QCA alongside the design processes of different QCA circuits; and an overview of QCA fault-tolerant circuits and their design procedures. In addition to updates to first edition chapters to reflect developments in recent years, this new edition sees the inclusion of problems to every chapter and eight new chapters. This book will be a great help for quantum computing researchers, faculty members and students who can develop a working understanding of circuit-based quantum computing.

Recent advances in theoretical and experimental quantum computing raise the problem of verifying the outcome of these quantum computations. The recent verification protocols using blind quantum computing are fruitful for addressing this problem. Unfortunately, all known schemes have relatively high overhead. Here we present a novel construction for the resource state of verifiable blind quantum computation. This approach achieves a better verifiability of 0.866 in the case of classical output. In addition, the number of required qubits is 2N+4cN, where N and c are the number of vertices and the maximal degree in the original computation graph, respectively. In other words, our overhead is less linear in the size of the computational scale. Finally, we utilize the method of repetition and fault-tolerant code to optimise the verifiability.