For a disentangled state left the visibility of the resulting Ramsey interference pattern is rather high, whereas for an entangled state middle the visibility almost vanishes. A further application of the quantum gates can however restore the interference pattern again right. Download Image 75 KB. Initially an atom in the yellow state is split into a green and red state, which are moved in opposite directions. There they interact wirth neighbouring atoms via controlled cold collisions.
Download Image 1 MB. Kristina Schuldt. Ildiko Kecskesi. Collisional Quantum Gate Arrays.
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Schematic controlled collision sequence. Entanglement oscillations made visible through a matter wave interference pattern I. A further application of the quantum gates can however restore the interference pattern again right Download Image 75 KB. E-mails: reiser inf. The simulation of quantum algorithms in classical computers demands high processingand storing capabilities. However, optimizations to reduce temporal and spatial complexities are promising and capable of improving the overall performance of simulators.
The main contribution of this work consists in designing optimizations to describe quantum transformations using Quantum Processes and Partial Quantum Processes, as conceived in the qGM theoretical model. These processes, when computed on the VPE-qGM execution environment, reduce the execution time of the simulation.
Two-Qubit Quantum Gate Successfully Built for the First Time
The performance evaluation of this proposal was carried out by benchmarks that include sequential simulation of quantum algorithms up to 24 qubits and instances of Grover's Algorithm. The results show improvements in the simulation of general, controled transformations since their execution time was significantly low, even for systems with several qubits.
Furthermore, a solution based on GPU computing for dealing with transformations that still have a high simulation cost in the VPE-qGM is also discussed. Quantum Computing QC predicts the development of quantum algorithms that, in various scenarios, are much faster than their classical versions [1,2].
However, such algorithms can only be efficiently executed on quantum computers, which are currently unavailable for general purpose use. In this context, quantum simulation softwares, such as [3,4,5,6] and , were proposed so researchers can anticipate the behaviors of the algorithms when executed on quantum hardware.
Despite all the work already done, several approaches for simulation can still be explored. The VPE-qGM Visual Programming Environment for the Quantum Geometric Machine Model  is a quantum simulator under development including both characterizations, visual modeling, and distributed simulation of quantum algorithms, showing the application and evolution of quantum computing through integrated graphical interfaces.
The current focus of this project is related to the exponential growth in the matrices associated to multi-qubit transformations, where the efforts are towards the reduction of the temporal complexities associated to the execution of a muti-qubit quantum transformation. According to the specifications of the qGM model, these new concepts can be explored for modeling quantum transformations and reducing the computations in a simulation.
They are the mathematical structure underling the modeling of quantum parallelism in massive parallel architectures, such as GPUs Graphic Processing Units , as described in . This article is structured as follows: Section comprehends the conceptual background related to this work. Discussions concerning the results and main contributions of this work are presented in Section 5. Some concepts of QC are necessary to understand the contribution proposed in this work.
Thus, an introduction of quantum computing and the qGM Quantum Geometric Machine model  are presented in the following subsections. In QC , the qubit is the basic information unit, being the simplest quantum system, defined by a unitary and bi-dimensional state vector.
The state space of a quantum system with multiple qubits is obtained by the tensor product of the space states of its subsystems. Transition states in a N -dimensional quantum system is performed by unitary quantum transformations, defined by square matrices of order N 2 N components since N is the number of qubits in the system.
The matrix notation of Hadamard and its application over a one-qubit system are, respectively, given as. Quantum transformations simultaneously applied to different qubits are obtained by the application of the tensor product between the corresponding matrices, as in the following example:. Besides the multi-dimensional transformations obtained by the tensor product, controlledtransformations can also be used in quantum systems. Figure 1 a shows the matrix notation of the CNOT transformation and its application to a generic two-qubit quantum state.
The corresponding representation quantum gate in the quantum circuit model is presented in Figure 1 b. By the composition and synchronization of quantum transformations, computations exploring the potentialities of quantum parallelism are created. However, the exponential increase of memory usually arises in such computations. As a consequence, there is a loss of performance in the simulation of multidimensional quantum systems. Therefore, optimizations for efficient representation of multi-qubit quantum transformations are necessary.
The qGM model follows the concepts of the domain theory closely related to the Girard's coherent spaces . The processes and states are labeled by points in a geometric space, which characterizes the computational basis as an n-dimensional subspace of the Hilbert space. In the qGM model, an elementary process EP may read data from many memory positions state space , but can only write in one.
These operations correspond to the computation defined by each component vector of the matrix H and generate the new amplitudes of the state vector. The QP for the H transformation is obtained by the synchronization of two EPs associated to each one of these operations. During the simulation, both EPs are simultaneously executed, modifying the data in the memory positions. Such execution is performed accordingly to the behavior of the transformation, simulating the evolution of the quantum system. The interpretation of QPPs is obtained from the partial application of a quantum gate due to the existence of uncertainties related to some sets of vectors.
By this statement, it is possible to define partial states as in the following matrix-notation:. Although it is not the focus of this work, the qGM model provides interpretation for other quantum transformations, such as projections for measure operations.
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The VPE-qGM environment is being developed aiming at the support for modeling and distributed simulation of algorithms from QC , considering abstraction of the qGM model. The main extensions consider the representation of controlled and non-controlled transformations, and related possible synchronization. The specifications of these and other new features are described in the following subsections. A component QP is able to model a quantum gate. Figure 2 shows a QP associated to a three-dimensional quantum system, including its representation using EPs and the structure of such component in the qGM-Analyzer.
In Figure 2 , ML stores the matrices associated with quantum transformations. The tuples of each line are obtained by changing the values of the parameters x 1 and x 2. The first tuple corresponds to the value obtained by the scalar product between the corresponding functions. The second indicates the column in which the value will be stored.
The matrix-order in ML is defined from the number of functions n grouped together. It is interesting that the order of each matrix in ML can be arbitrarily determined. Although, there is an exponential growth in memory consumption. Hence, a balance between the order and the number of matrices in ML ML interferes directly in the performance of an application. Besides the ML , it is necessary to create a list see in 3. In such list, q indicates the total number of qubits in the quantum application.
Based on the concept of partial processes defined as partial objects in the qGM model, it is possible to split the QP described in Figure 2 in two QPPs. The QPPs contribute with the possibility of establishing partial interpretations of a quantum transformation. Complementary QPPs that interpret distinct line sets can be synchronized and executed independently in different processing nodes of a multiprocessor system. The bigger the number of QPPs synchronized, the smaller is the computation executed by each one, resulting in a low-cost of execution. For non-controlled quantum gates, it is possible to model all the evolution of the global state of a quantum system with a single QP.
However, this possibility can not be applied to controlled quantum gates. As these states are not modified, the execution of the QPP 2 is not mandatory. In a synchronization mixing controlled and non-controlled gates different from Id , all theamplitudes are modified. The configuration illustrated in the Figure 5 b is modeled by the expressions in 3. However, it is not possible to discard the execution of the QPP 2 , once it modifies the amplitudes of some states. Those changes are due to the H transformation, which is always applied to the last qubit, despite the control state of the CNOT transformation.
After building the QPPs , a recursive operator is applied to the matrices in ML for computing the amplitudes of the new global state of the quantum system. This operator dynamically generates all values associated to the resulting matrix obtained by the tensor product of the transformations, defining the quantum application. Besides, a value indexing the amplitude is also generated. The algorithmic description of this procedure with some optimizations is shown in Figure 6. The execution time of this algorithm grows exponentially when new qubits are added.
When analyzing the use of QPs and QPPs exclusively for the representation of quantum gates, there is a high-cost related to temporal complexity, specially when Hadamard gates are applied. Such cost reflects directly in the execution time. For validation and performance analysis of the simulation with QPs and QPPs , the following three study-cases were considered:.
C1: Reversible circuit benchmarks from ;. C2: Hadamard gates up to 14 qubits;.