Quantum computation over the butterfly network

Modern quantum computing architectures are built to provide low-error environments within which to perform the unitary operations that form the bases of quantum computations. These low-error environments necessary for successful quantum computation require use of complex structures that make large-scale implementation difficult, leading to the investigation of coupled magnon-photon systems [1-6] that can achieve strong coupling rates (> 100 MHz) as a potential platform for hybrid quantum computing architectures. Such coupled magnon-photon systems benefit from the magnetic field tunability of the resonant magnon frequency, where dynamic tuning of resonant magnon frequencies with a pulsed magnetic field at rates comparable to or less than the order of the magnon-photon energy exchange time can realize unique operations not possible with static tuning [7-8]. Previously, we analyzed this problem numerically and found that a parabolically shaped pulsed magnetic field profile can realize the coherent exchange of quantum data and the generation of entanglement by changing the curvature λ and maximum frequency Δ of the parabolic profile [8]. That is, in the same physical structure, depending on the shape of the pulsed magnetic field, different unitary magnon-mediated quantum gates can be realized [8].Here, we consider an analytical approach to study this system. Namely, we consider a hybrid magnon-photon system, consisting of two photonic resonators having degenerate frequencies ωp and one magnonic resonator with a dynamically-tuned resonance frequency ωm(t). The photonic resonators are not directly coupled with each other but are both coupled with the magnonic mode with the coupling rate κ.We consider a simple analytically-solvable interaction protocol in this hybrid system, which can be experimentally realized with rectangular magnetic field pulses of sufficiently large magnitude. We assume that initially (t<0) the frequency difference │ωm–ωp│>>κ and interaction between the subsystems can be ignored. At t=0 the magnonic mode frequency ωm(t) is brought within the efficient interaction interval Δ=ωm–ωp∼κ and is held constant for the duration T. After t=T, the magnonic frequency is once again moved out of the photonic resonance region and any interaction between the subsystems stops. Depending on the control pulse parameters Δ and T, different behaviors of the hybrid system can be realized.Of particular interest is the situation, in which the magnonic and photonic subsystems stay completely decoupled after the action of the control pulse. Such a case corresponds to a unitary operation on the photonic modes, realized through the indirect magnon-mediated coupling, and is of great importance for practical applications of magnonic resonators in quantum computing systems.Using our model, we derived a simple analytical condition of realizing unitary magnon-mediated operations, namely, for each detuning Δ there exist an infinite series of pulse durationsTn(Δ)=2nπ/(8κ2+Δ2)½providing unitary conditions (here, n is a positive integer). The resulting effect of the magnon interaction on the photonic subsystem, in this case, can be described by 2×2 unitary matrix Un=Uei(nπ/2–Tn(Δ)Δ/4) relating photonic states before and after the pulse, which has components:U11=ei2πcos(θn), U12=eiπ/2sin(θn), U21=–e–iπ/2sin(θn), U22=e–i2πcos(θn)Here θn=nπ/2–Tn(Δ)Δ/4 has the meaning of inter-mode energy exchange angle. For example, for θn=π/2+mπ (m is an integer number) a complete exchange of two photonic states is achieved, while θn=π/4+mπ describes 50% mixing behavior that allows one to entangle the two photonic modes.The obtained results allow one to select pulse parameters realizing the desired unitary operation and, also, to analyze the limitations on the performance of the proposed magnon-mediated gates. Thus, it is clear that the simple rectangular pulse cannot realize all possible unitary gates since the matrix Un depends only on one continuous parameter (θn), while the general 2×2 unitary matrix has 4 independent parameters. On the other hand, this pulse profile is sufficient to achieve an arbitrary exchange angle θn. The shortest pulse duration for which a complete exchange (θn=π/2) is possible is Tmin=π/(κ√2)and is realized at exact resonance Δ=0.Our results, also, show a possibility to realize magnon-mediated unitary gates working in off-resonance conditions │Δ│≥κ. For example, θn=π/2 gate can be realized with Δ=18(2/19)½κ and T10=(19/2)½π/κ. While such off-resonance operations require longer interaction times, their advantage in comparison to resonance-type operations is that the magnonic and photonic subsystems remain only weakly coupled throughout the whole process. This may lead to weaker decoherence of the quantum photonic subsystem caused by spin-lattice relaxation processes in the magnonic resonator. In general, n-th unitary branch Tn(Δ) has n different detuning values Δ, for which a given exchange angle θn (up to multiplies of π) is achieved, which leaves a lot of space for optimization of practical magnon-mediated unitary gates.In summary, we proposed a simple model of magnon-mediated interaction in dynamically-tuned hybrid magnon-photon quantum systems. We derived analytically the conditions under which such interaction results in a unitary operation on the photonic subsystem and obtained expressions for the parameters of the control pulse necessary for achieving the desired operation. Our results can be used for estimation and optimization of performance parameters of practical hybrid magnon-photon systems. **

Quantum teleportation is one of the most basic quantum protocols, which transfers an unknown quantum state from one location to another through local operation and classical communication by using shared quantum entanglement without physical transfer of the information carrier. And it has been widely used in various quantum information protocols such as entanglement swapping, quantum repeaters, quantum gate teleportation, quantum computation based on measurement, and quantum teleportation networks, which have important application value in quantum computation and quantum information. Quantum teleportation is a naturally bipartite process, in which an unknown quantum state can only be transmitted from one node to another. With the further development of quantum information research, it is necessary to transfer quantum states or quantum information among more and more nodes. Multipartite quantum protocols are expected to form fundamental components for larger-scale quantum communication and computation. A bipartite quantum teleportation should be extended to a multipartite protocol known as a quantum teleportation network. In this paper, a multifunctional quantum teleportation network is proposed theoretically. We first propose a special method of constructing four-partite quantum resources in continuous variables (CVs), and based on this, construct two different types of CV quantum teleportation networks. One type of network contains just one quantum teleportation process consisting of a sender, a receiver and two controllers. In this type of network, the unknown quantum state can be recovered at any other node according to the requirement after the measurement in the input node, which enriches the transfer direction and transfer mode of the unknown quantum state. And meanwhile, the two controllers can control the transfer of a quantum state from the sender to the receiver by restricting the sender and receiver’s access to their information, which makes the quantum teleportation network controllable. The other type of network has two quantum teleportation processes, each containing only a sender, a receiver and no controllers, which increases the number of quantum states that can be transmitted. Then we analyze the dependence of the fidelity of each quantum teleportation network on different physical parameters, and compare the characteristics, advantages and disadvantages among different types of quantum teleportation networks. The scheme for constructing a multifunctional quantum teleportation network in this paper shows some advantages, such as the greater number of quantum nodes, diversity of types, simple operation procedure. And all these advantages provide a broader application prospect for establishing larger and more complex quantum information networks in the future and quicken the pace of the application of quantum information.

Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition

The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations (U1,U2) into a controlled unitary operation that coherently applies them in different orders as |0⟩⟨0|⊗U1U2+|1⟩⟨1|⊗U2U1. This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.

It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$, and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(r\log r)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively $1$ ebit and some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits.

Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state holden by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this article, we show that a perfect discrimination between two unitary operations acting on a two-qudits can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on two-qudits by LOCC.

We show how to implement an arbitrary two-qubit unitary operation using any of several quantum gate libraries with small a priori upper bounds on gate counts. In analogy to library-less logic synthesis, we consider circuits and gates in terms of the underlying model of quantum computation, and do not assume any particular technology. As increasing the number of qubits can be prohibitively expensive, we assume throughout that no extra qubits are available for temporary storage. Using quantum circuit identities, we improve an earlier lower bound of 17 elementary gates by Bullock and Markov to 18, and their upper bound of 23 elementary gates to 18. We also improve upon the generic circuit with six CNOT gates by Zhang et al. (our circuit uses three), and that by Vidal and Dawson with 11 basic gates (we use 10). We study the performance of our synthesis procedures on two-qubit operators that are useful in quantum algorithms and communication protocols. With additional work, we find small circuits and improve upon previously known circuits in some cases.

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.

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit much more faults than classical protocol designers since human intuition is much better adapted to the classical world than the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. One of the most serious issues in quantum process algebra is to discover a quantum generalization of the notion of bisimulation, which lies in a central position in process algebra, preserved by parallel composition in the presence of quantum entanglement, which has no counterpart in classical computation. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communications are involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communications) scheme, where classical communications must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communications is crucial for process algebra approach to verification of quantum cryptographic protocols. In this paper we introduce a novel notion of bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communications are present. We also establish some basic algebraic laws for this bisimulation. In particular, we prove uniqueness of the solutions to recursive equations of quantum processes, which provides a powerful proof technique for verifying complex quantum protocols.

Entanglement concentration can be achieved in a deterministic fashion [Morikoshi, Phys. Rev. Lett. 84, 3189 (2000); Phys. Rev. A 64, 022316 (2001)]. We present a protocol for deterministic extraction of an Einstein-Podolsky-Rosen pair from two partially entangled pairs via local operations and classical communication. This protocol consists of a generalized measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding unitary operation or a choice between the two pairs. Based on the theory of majorization, we explicitly construct the required POVM, which can be realized by performing a unitary operation in the extended space and a conventional Von Neuman orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates, and thus provide a physical means to realize the deterministic entanglement concentration. Our method for constructing quantum circuits differs from the usual method based on decomposition of unitary matrices, and is convenient for a large class of quantum processes involving generalized measurements.

Quantum machine learning recently gained prominence due to the computational ability of quantum computers in solving machine learning problems that are intractable on a classical computer. However, achieving a quantum advantage on present-day quantum computers remains an open challenge. In this work, we primarily focus on solving machine learning problems using a hybrid model based on both quantum and classical computers together for the classification task. We propose the quantum multi-class classifier (QMCC) as a variational circuit with a hybrid classical-quantum approach using quantum mechanical properties such as superposition and entanglement. A unitary operation on a single qubit for the state preparation is designed and also demonstrated using a real quantum computer on the IBMQX platform. The entire variational circuit for the classification task is implemented on a quantum simulator. We performed our quantum simulations on three benchmark datasets: Iris dataset, Banknote Authentication (BNA) dataset, and Wireless Indoor Localization (WIL) dataset for machine learning algorithms. Our simulation results show that the proposed QMCC model classified Iris dataset with an accuracy of 92.10%, BNA dataset with an accuracy of 89.50%, and WIL dataset with an accuracy of 91.73%. The proposed model can also be extended to multiple class classifiers.

The high-dimensional quantum system greatly improve the quantum channel capacity and information storage space, and achieve high-dimensional quantum information transmission, which enhance the speed of quantum computing and quantum information processing. In this paper, a high-dimensional quantum teleportation protocol without information loss is proposed. We consider pre-sharing a high-dimensional non-maximum entangled state as a quantum channel between sender and receiver. By adding auxiliary particle and performing high-dimensional local operations, it is possible to achieve high-dimensional quantum teleportation without information loss. Simultaneously, we apply the protocol to butterfly network, and propose a novel high-dimensional quantum network coding based on prediction mechanism. In our scheme, we use Z-{|0⟩, |1⟩} basis to predict the transmission of high dimensional states over the butterfly network. When the prediction is successful, the deterministic transmission of high-dimensional quantum states can be realized over the butterfly network. Our scheme greatly saves the usage of quantum and classical channels, which improves the utilization efficiency of both channels.

The paradigm of variational quantum classifiers (VQCs) encodes classical information as quantum states, followed by quantum processing and then measurements to generate classical predictions. VQCs are promising candidates for efficient utilizations of noisy intermediate scale quantum (NISQ) devices: classifiers involving M-dimensional datasets can be implemented with only lceil log _2 M rceil qubits by using an amplitude encoding. A general framework for designing and training VQCs, however, is lacking. An encouraging specific embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: a circuit with a predetermined circuit geometry and parametrized gates expressing a time-evolution unitary operator; training involves learning the gate parameters through a gradient-descent algorithm where the gradients themselves can be efficiently estimated by the quantum circuit. The representational power of QCL, however, depends strongly on the choice of the ansatz, as it limits the range of possible unitary operators that a VQC can search over. Equally importantly, the landscape of the optimization problem may have challenging properties such as barren plateaus and the associated gradient-descent algorithm may not find good local minima. Thus, it is critically important to estimate (i) the price of ansatz; that is, the gap between the performance of QCL and the performance of ansatz-independent VQCs, and (ii) the price of using quantum circuits as classical classifiers: that is, the performance gap between VQCs and equivalent classical classifiers. This paper develops a computational framework to address both these open problems. First, it shows that VQCs, including QCL, fit inside the well-known kernel method. Next it introduces a framework for efficiently designing ansatz-independent VQCs, which we call the unitary kernel method (UKM). The UKM framework enables one to estimate the first known computationally-determined bounds on both the price of ansatz and the price of any speedup advantages of VQCs: numerical results with datatsets of various dimensions, ranging from 4 to 256, show that the ansatz-induced gap can vary between 10 and 20%, while the VQC-induced gap (between VQC and kernel method) can vary between 10 and 16%. To further understand the role of ansatz in VQCs, we also propose a method of decomposing a given unitary operator into a quantum circuit, which we call the variational circuit realization (VCR): given any parameterized circuit block (as for example, used in QCL), it finds optimal parameters and the number of layers of the circuit block required to approximate any target unitary operator with a given precision.

Integrated optics is an engineering solution proposed for exquisite control of photonic quantum information. Here we use silicon photonics and the linear combination of quantum operators scheme to realise a fully programmable two-qubit quantum processor. The device is fabricated with readily available CMOS based processing and comprises four nonlinear photon-sources, four filters, eighty-two beam splitters and fifty-eight individually addressable phase shifters. To demonstrate performance, we programmed the device to implement ninety-eight various two-qubit unitary operations (with average quantum process fidelity of 93.2$\pm$4.5%), a two-qubit quantum approximate optimization algorithm and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear combination architecture with silicon photonics for future photonic quantum processors.

: We propose to use NMR as a testbed to develop general methods for solving computational problems on EQC's, to study the fundamental physics and computer science of these machines, and to learn how to make optimal use of the trade-offs that their unique capabilities permit us to make. Specifically, we intend to explore the critical issue of decoherence in a real quantum information processor, including its nature, its simulation, and methods of controlling it. The lessons thereby learned are expected to be broadly applicable throughout the field of quantum information processing, and particularly to proposed implementations based on solid-state NMR. Liquid-state NMR is thus an invaluable if not indispensable step in the field's efforts to bootstrap its way towards scalable quantum information processing. The results obtained during the two years covered by this report (July 1, 2001 - June 30, 2003) fall into four principal classes: 1) Development of methods for obtaining precise coherent control over nuclear spin systems with a well-characterized Hamiltonian and relaxation superoperator, and for quantifying the precision of such control. 2) Validation of these methods by implementing simple quantum algorithms, communications protocols, and other quantum phenomena that make essential use of entangling unitary operations and/or measurements. 3) Simulation of quantum systems using the unitary and nonunitary control operations that are available in liquid-state NMR spectroscopy. 4) Reviews, commentary and educational articles. We stress that although these studies utilized liquid-state NMR spectroscopy as a testbed for the development and validation of our techniques and simulations, the results will be directly applicable to a wide range of physical systems now being studied for quantum information processing purposes, once sufficient favorable ratios of gate operation to decoherence times have been obtained.