Scalable quantum computation with cavity QED systems

The Controlled-NOT (CNOT) quantum gate associated with single qubit gates is universal for building all quantum circuits. Although CNOTs based on trapped ions and superconducting qubits have become experimentally feasible, the deterministically functioning quantum CNOT gate with photonic qubits will lead to practical realization of quantum computers in a deterministic manner, and will therefore provide an exponential increase in processing power with lowest decoherence time. Based on a previously proposed CNOT model utilizing a quantum cloner (with a fidelity of 78 %), we present in this work our design of the Compact CNOT gate providing a fidelity of 90.31 %, and give physical realization constraints for building CNOT based decomposition of the Toffoli gate and more generalized photonic CnNOT gates. The photonic qubits are encoded by circularly polarized single photons and the design of the gate is based on copying the polarization degree of freedom by quantum cloners. Our model is based on spin of electron in a quantum dot embedded in a double sided optical microcavity which behaves like a beam splitter. Circularly polarizing beam splitters, beam splitters, delay lines, photonic circulators, single qubit operations and other photonic devices are used for the design and decomposition. We show that the fidelity of the decomposed CnNOT gates is highly dependent on the errors of the non-perfect photonic devices used and more specifically to the theoretical optimal limit of 5/6 of the quantum cloners required for each CNOT in the decomposition.

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

Scalable quantum computation and communication require error control to protect quantum information against unavoidable noise. Quantum error correction protects information stored in two-level quantum systems (qubits) by rectifying errors with operations conditioned on the measurement outcomes. Error-correction protocols have been implemented in nuclear magnetic resonance experiments, but the inherent limitations of this technique prevent its application to quantum information processing. Here we experimentally demonstrate quantum error correction using three beryllium atomic-ion qubits confined to a linear, multi-zone trap. An encoded one-qubit state is protected against spin-flip errors by means of a three-qubit quantum error-correcting code. A primary ion qubit is prepared in an initial state, which is then encoded into an entangled state of three physical qubits (the primary and two ancilla qubits). Errors are induced simultaneously in all qubits at various rates. The encoded state is decoded back to the primary ion one-qubit state, making error information available on the ancilla ions, which are separated from the primary ion and measured. Finally, the primary qubit state is corrected on the basis of the ancillae measurement outcome. We verify error correction by comparing the corrected final state to the uncorrected state and to the initial state. In principle, the approach enables a quantum state to be maintained by means of repeated error correction, an important step towards scalable fault-tolerant quantum computation using trapped ions.

1D topological systems for next-generation electronics

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.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

Using a multi-layered printed circuit board, we propose a 3D architecture suitable for packaging superconducting chips, especially chips that contain two-dimensional qubit arrays. In our proposed architecture, the center strips of the buried coplanar waveguides protrude from the surface of a dielectric layer as contacts. Since the contacts extend beyond the surface of the dielectric layer, chips can simply be flip-chip packaged with on-chip receptacles clinging to the contacts. Using this scheme, we packaged a multi-qubit chip and performed single-qubit and two-qubit quantum gate operations. The results indicate that this 3D architecture provides a promising scheme for scalable quantum computing.

Improving gate performance is vital for scalable quantum computing. Universal quantum computing also requires gate fidelity to reach a high level. For a superconducting quantum processor, which operates in the microwave band, the single-qubit gates are usually realized with microwave driving. The crosstalk between microwave pulses is a non-negligible error source. In this article, we propose an error mitigation scheme to address this crosstalk issue for single-qubit gates. There are three steps in our method. First, by controlling the detuning between qubits, the microwave induced classical crosstalk error can be constrained within the computational subspace. Second, by applying the general decomposition procedure, the arbitrary single-qubit gate can be decomposed as a sequence of X and virtual Z gates. Finally, by optimizing the parameters in virtual Z gates, the error constrained in the computational space can be corrected. Using our method, no additional compensation signals are needed, arbitrary single-qubit gate time will not be prolonged, and the circuit depth containing simultaneous single-qubit gates will also not increase. The simulation results show that, in a specific regime of qubit–qubit detuning, the infidelities of simultaneous single-qubit gates can be as low as that without microwave crosstalk.

A way of designing a scalable optical quantum computer based on the photon echo effect is proposed. Individual rare earth ions , regularly located in the lattice of the orthosilicate crystal, are suggested to be used as optical qubits. Operations with qubits are performed using coherent and incoherent laser pulses. The operation protocol includes both the method of measurement-based quantum computations and the technique of optical computations. Modern hybrid photon echo protocols, which provide a sufficient quantum efficiency when reading recorded states, are considered as most promising for quantum computations and communications.

Decoherence causes the decay of the quantum information that is stored in highly correlated states during quantum computation. It is thus a limiting factor for all implementations of a quantum computer. Because a scalable quantum computer with hundreds or thousands of qubits is not available yet, experimental data about decoherence rates was restricted to small quantum registers. With solid state nuclear magnetic resonance we create highly correlated multiqubit states that serve as a model quantum register and measure their decay. By measuring the decay as a function of the system size, we determined the scaling of the decoherence rate with the number of qubits. Using the same system, we also used decoupling techniques to reduce the coupling between system and environment and thereby the decoherence rate by more than an order of magnitude, independent of the system size. For the free decay as well as for the decoupled case, we found a relatively weak scaling with system size, which could be fitted to a power law $\ensuremath{\propto}{\overline{K}}^{p}$ with an exponent $p\ensuremath{\approx}1∕2$. This raises the prospect for large-scale quantum computation.

We show how the quantum Zeno effect can be exploited to control quantum many-body dynamics for quantum information and computation purposes. In particular, we consider a one dimensional array of three level systems interacting via a nearest-neighbour interaction. By encoding the qubit on two levels and using simple projective frequent measurements yielding the quantum Zeno effect, we demonstrate how to implement a well defined quantum register, quantum state transfer on demand, universal two-qubit gates and two-qubit parity measurements. Thus, we argue that the main ingredients for universal quantum computation can be achieved in a spin chain with an {\em always-on} and {\em constant} many-body Hamiltonian. We also show some possible modifications of the initially assumed dynamics in order to create maximally entangled qubit pairs and single qubit gates.

Fault-tolerant cluster states form the basis for scalable measurement-based quantum computation. Recently, new stabilizer codes for scalable circuit-based quantum computation have been introduced that have very high thresholds under biased noise where the qubit predominantly suffers from one type of error, e.g. dephasing. However, extending these advances in stabilizer codes to generate high-threshold cluster states for biased noise has been a challenge, as the standard method for foliating stabilizer codes to generate fault-tolerant cluster states does not preserve the noise bias. In this work, we overcome this barrier by introducing a generalization of the cluster state that allows us to foliate stabilizer codes in a bias-preserving way. As an example of our approach, we construct a foliated version of the XZZX code which we call the XZZX cluster state. We demonstrate that under a circuit-level-noise model, our XZZX cluster state has a threshold more than double the usual cluster state when dephasing errors are more likely than errors that cause bit flips by a factor of order ~100 or more.

In 2001, all-optical quantum computing became feasible with the discovery that scalable quantum computing is possible using only single-photon sources, linear optical elements, and single-photon detectors. Although it was in principle scalable, the massive resource overhead made the scheme practically daunting. However, several simplifications were followed by proof-of-principle demonstrations, and recent approaches based on cluster states or error encoding have dramatically reduced this worrying resource overhead, making an all-optical architecture a serious contender for the ultimate goal of a large-scale quantum computer. Key challenges will be the realization of high-efficiency sources of indistinguishable single photons, low-loss, scalable optical circuits, high-efficiency single-photon detectors, and low-loss interfacing of these components.

A scalable quantum computer could be built by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. A solution is for cells to repeatedly communicate with each other and so purify any imperfections; however prior studies suggest that the cells themselves must then have prohibitively low internal error rates. Here we describe a method by which even error-prone cells can perform purification: groups of cells generate shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (≥10% error rate) we find that our protocol can succeed provided that intra-cell error rates for initialisation, state manipulation and measurement are below 0.82%. This level of fidelity is already achievable in several laboratory systems.

Scaling quantum computers, i.e., quantum processing units (QPUs) to enable the execution of large quantum circuits is a major challenge, especially for applications that should provide a quantum advantage over classical algorithms. One approach to scale QPUs is to connect multiple machines through quantum and classical channels to form clusters or even quantum networks. Using this paradigm, several smaller QPUs can collectively execute circuits that each would not be able on its own. However, communication between QPUs is costly as it requires generating and maintaining entanglement, and hence it should be used wisely. In this paper, we evaluate the architectures, and in particular the entanglement patterns, of variational quantum circuits in a distributed quantum computing (DQC) setting. That is, using Qiskit, we simulate the execution of an eight qubit circuit using four QPUs each with two computational and two communication qubits where non-local CX-gates are performed using the remote-CX protocol. We compare the performance of various circuits on a binary classification task where training is executed under ideal and testing under noisy conditions. The study provides initial results on suitable VQC architectures for the DQC paradigm, and indicates that a standard VQC baseline is not always the best choice, and alternative architectures that use entanglement between QPUs sparingly deliver better results under noise.