Smaller two-qubit circuits for quantum communication and computation

We provide an analytic way to implement any arbitrary two-qubit unitary operation, given an entangling two-qubit gate together with local gates. This is shown to provide explicit construction of a universal quantum circuit that exactly simulates arbitrary two-qubit operations in SU(4). Each block in this circuit is given in a closed form solution. We also provide a uniform upper bound of the applications of the given entangling gates, and find that exactly half of all the controlled-unitary gates satisfy the same upper bound as the CNOT gate. These results allow for the efficient implementation of operations in SU(4) required for both quantum computation and quantum simulation.

The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases. realizations of controlled unitary gates (pp139-155) G. Song and A. Klappenecker The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases.

The canonical decomposition for two-qubit operators has proven very useful for applications in quantum computing. This decomposition generates equivalence classes up to local quantum gates. We provide a variety of complete, explicit decompositions of given two-qubit operators in terms of single, double, and triple controlled-NOT (CNOT) gates. By analytically addressing the needed pre- and post-tensor product factors, we demonstrate that exact results are possible, even when a parameter is included. The examples given are of interest to superconducting qubit, spin-based, dipolar molecule, and other quantum information processing systems.

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

An experimental scheme, its hardware implementation, and procedure for experimental verification of tripartite quantum state sharing and perfect teleportation of the two-qubit photonic state using four different multipartite quantum channels is presented. The four multipartite channels considered are (1) a pair of Greenberger–Horne–Zeilinger triplets, (2) a six-particle cluster state, (3) a highly entangled five-particle Brown state, and (4) a highly entangled six-particle Borras state. In this experiment, optical pulses are delivered by a high repetition rate mode-locked pulsed laser and arbitrary photonic states, in the two-qubit Bloch sphere, are created by the spontaneous parametric down-conversion of optical pulses in a nonlinear crystal. The experimental implementation of the scheme is based on both single-particle and two-particle complete Bell-state measurements, and the unitary transformation required for reconstruction of the qubit is performed using single- and two-qubit operations. Two-qubit operations are implemented using the quantum controlled phase and C-NOT gates. Relative comparison among different multipartite channels along with experimental difficulties and their solutions are also discussed.

Quantum information science has leaped forward with the exploration of high-dimensional quantum systems, offering greater potential than traditional qubits in quantum communication and quantum computing. To advance the field of high-dimensional quantum technology, a significant effort is underway to progressively enhance the entanglement dimension between two particles. An alternative effective strategy involves not only increasing the dimensionality but also expanding the number of particles that are entangled. We present an experimental study demonstrating multi-partite quantum non-locality beyond qubit constraints, thus moving into the realm of strongly entangled high-dimensional multi-particle quantum systems. In the experiment, quantum states were encoded in the path degree of freedom (DoF) and controlled via polarization, enabling efficient operations in a two-dimensional plane to prepare three- and four-particle Greenberger-Horne-Zeilinger (GHZ) states in three-level systems. Our experimental results reveal ways in which high-dimensional systems can surpass qubits in terms of violating local-hidden-variable theories. Our realization of multiple complex and high-quality entanglement technologies is an important primary step for more complex quantum computing and communication protocols.

Rapid development has taken place in quantum algorithms, but few of them can be implemented directly on the quantum computer because of the coupling constraint and high error rates of quantum computer operations. Transforming a circuit into an executable one on the quantum computer with the highest fidelity is an NP problem. Therefore, the algorithm tackling the mapping and routing problem of qubits is needed. Following the policy of VQA, a heuristic algorithm to solve the problem is proposed. The algorithm uses the error score derived from quantum operation’s error rate as the heuristic cost. Combining SWAP gates, physical CNOT gates, remote CNOT gates and inverse CNOT gates, the algorithm transforms the input circuit into an executable output circuit on the quantum computer with lower error score in acceptable execution time and moves forward from local optimal to global optimal.

This text offers an introduction to quantum computing, with a special emphasis on basic quantum physics, experiment, and quantum devices. Unlike many other texts, which tend to emphasize algorithms, Quantum Computing without Magic explains the requisite quantum physics in some depth, and then explains the devices themselves. It is a book for readers who, having already encountered quantum algorithms, may ask, Yes, I can see how the algebra does the trick, but how can we actually do it? By explaining the details in the context of the topics covered, this book strips the subject of the magic with which it is so often cloaked. Quantum Computing without Magic covers the essential probability calculus; the qubit, its physics, manipulation and measurement, and how it can be implemented using superconducting electronics; quaternions and density operator formalism; unitary formalism and its application to Berry phase manipulation; the biqubit, the mysteries of entanglement, nonlocality, separability, biqubit classification, and the Schroedinger's Cat paradox; the controlled-NOT gate, its applications and implementations; and classical analogs of quantum devices and quantum processes. Quantum Computing without Magic can be used as a complementary text for physics and electronic engineering undergraduates studying quantum computing and basic quantum mechanics, or as an introduction and guide for electronic engineers, mathematicians, computer scientists, or scholars in these fields who are interested in quantum computing and how it might fit into their research programs.

In the emerging field of quantum information technology the two basic subfields are quantum communication and quantum computation. Photonic qubits are considered as most promising information carriers for this new technology due to the immense advantage of suffering negligible decoherence. Additionally, the very small photon-photon interactions can be replaced by inducing effective nonlinearities via measurements which allow for the implementation of crucial two-qubit gate operations. Although the spontaneous parametric down-conversion gives access to the generation of highly entangled few-photon states, such as four-qubit cluster states which allow to demonstrate the new concept of the one-way quantum computer, its applicability is highly limited due to the poor scaling of the simultaneous emission of more than one-entangled photon pair. Therefore of particular interest is the reversible mapping of qubits from photon states to atomic states. This might allow the implementation of photonic quantum repeaters for long-distance quantum communication or the generation of arbitrary multi-photon states as required for linear-optics quantum computing. Thus for the realization of such a quantum network several approaches for achieving the required quantum control between matter and photons have been studied during the past few years. Recent experiments demonstrating the generation of narrow-bandwidth single photons using a room-temperature ensemble of ⁸⁷Rb atoms and electromagnetically induced transparency should emphasize the progress towards such a quantum network.

This chapter presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

Quantum information science is one of the frontier areas of science and technology. It is also an interdisciplinary area of research where scientists from physics, mathematics, and computer science can contribute alike. One of the main goals is how well one can process information using laws of quantum theory. Quantum information theory aims to process information that is not amenable with classical devices. At the same time this also provides new insights into the nature of the quantum world. Quantum information processing includes quantum computation, quantum communication, quantum cryptography and various information processing tasks. Not only does this provide a fundamental arena to investigate quantum phenomena but also gives new technological benefits like quantum teleportation, remote state preparation and secure communications. In India there is a growing interest in this emerging field. In view of this we had organized an International Program on Quantum Information (IPQI-2010) at the Institute of Physics, Bhubaneswar during January 4–January 30, 2010. In this programme, there were experts from all over the world and topics of focus were on various issues related to quantum information science and general quantum theory, characterization and quantification of entanglement, quantum channels and quantum operations, role of entanglement in quantum algorithms and quantum computation, developing new quantum algorithms, understanding of capacities of quantum channels, and new communication tasks in the multiparty case.

Among environmental effects, dissipation allows the stabilization of quantum resources, a fact that may accomplish key advantages over unitary quantum state manipulation. In this paper, we study an open quantum system composed ofNtwo-mode Bose–Einstein condensates (BECs), namely, BEC qubits, interacting with their environments. Usually, the coupling of a system with its environment results in a dissipation process that washes out the quantum effects, but the environment may also play exactly the opposite role. Here, we show that the coupling of a BEC qubit with its local environments can be exploited to establish dissipative state engineering and dissipative quantum computation. Also, we discuss how to generate entanglement and transfer it using such macroscopic BEC qubits. Generally, the couplings can be manipulated such that the system is driven to a desired steady state. We demonstrate that the steady state of a BEC qubit just depends on the tunneling strength between its two modes and the features of the local environments. By preparing the proper physical conditions and adjusting the parameters involved in the model, it can be shown that the steady state of the system may coincide with its initial state. Also, the interaction between the BEC qubits and local environments can be engineered to provide physically controlled single- and two-qubit operations, which are necessary for performing dissipative universal quantum computation. With the prosperous development of quantum computing, such a system can be employed to implement quantum computers based on non-unitary operations.

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics.

Quantum computing is an extremely promising research combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in “quantum computer science” is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present paper is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon’s algorithm, the Deutsch-Josza algorithm, Grover’s algorithm, Shor’s algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.KeywordsDeutsch-Josza algorithmGrover’s algorithmQuantum computingQuantum error-correctingShor’s algorithmSimon’s algorithmSimulation of quantum algorithms

Estimating the eigenstate properties of quantum systems is a long-standing, challenging problem for both classical and quantum computing. Existing universal quantum algorithms typically rely on ideal and efficient query models (e.g., time evolution operator or block encoding of the Hamiltonian), which, however, become suboptimal for actual implementation at the quantum circuit level. Here, we present a full-stack design of quantum algorithms for estimating the eigenenergy and eigenstate properties, which can achieve high precision and good scaling with system size. The gate complexity per circuit for estimating generic Hamiltonians’ eigenstate properties is , which has a logarithmic dependence on the inverse precision ε. For lattice Hamiltonians, the circuit depth of our design achieves near-optimal system-size scaling, even with local qubit connectivity. Our full-stack algorithm has low overhead in circuit compilation, which thus results in a small actual gate count (cnot and non-Clifford gates) for lattice and molecular problems compared to advanced eigenstate algorithms. The algorithm is implemented on IBM quantum devices using up to 2000 two-qubit gates and 20,000 single-qubit gates and achieves high-precision eigenenergy estimation for Heisenberg-type Hamiltonians, demonstrating its noise robustness.