Towards a NMR implementation of a quantum lattice gas algorithm

Recent theoretical results suggest that an array of quantum information processors communicating via classical channels can be used to solve fluid dynamics problems. Quantum lattice-gas algorithms (QLGA) running on such architectures have been shown to solve the diffusion equation and the nonlinear Burgers equations. In this report, we describe progress towards an ensemble nuclear magnetic resonance (NMR) implementation of a QLGA that solves the diffusion equation. The methods rely on NMR techniques to encode an initial mass density into an ensemble of two-qubit quantum information processors. Using standard pulse techniques, the mass density can then manipulated and evolved through the steps of the algorithm. We provide the experimental results of our first attempt to realize the NMR implementation. The results qualitatively follow the ideal simulation, but the observed implementation errors highlight the need for improved control.     

Implementing a Quantum Algorithm with Exchange-Coupled Quantum Dots: A Feasibility Study

We present Monte Carlo wavefunction simulations for quantum computations employing an exchange-coupled array of quantum dots. Employing a combination of experimentally and theoretically available parameters, we find that gate fidelities greater than 98% may be obtained with current experimental and technological capabilities. Application to an encoded 3 qubit (nine physical qubits) Deutsch-Josza computation indicates that the algorithmic fidelity is more a question of the total time to implement the gates than of the physical complexity of those gates. PACS: 81.07.Ta, 02.70.Ss, 03.67.Lx, 03.65.Yz     

Qubitless Quantum Logic

We discuss the implementation of quantum logic in a system of strongly interacting particles. The implementation is qubitless since logical qubits do not correspond to any well-defined physical substates. As an illustration, we present the results of simulations of the quantum controlled-NOT gate and Shor's algorithm for a chain of spin-1/2 particles with Heisenberg coupling. Our proposal extends quantum information processing theory to systems with permanent strong coupling between the two-state subsystems. PACS: 03.67.Lx; 03.65.Ta     

Controllability and Universal Three-qubit Quantum Computation with Trapped Electron States

We show how to control and perform universal three-qubit quantum computation with trapped electron quantum states. The three qubits are the electron spin, and the first two quantum states of the cyclotron and axial harmonic oscillators. We explicitly show how universal three-qubit gates can be performed. As an example of a quantum algorithm, we outline the implementation of the three-qubit Deutsch-Jozsa algorithm in this system.      

Multiple RF Coil Nuclear Magnetic Resonance Quantum Computing

Recent work has demonstrated the feasibility of using an array of quantum information processors connected via classical channels (type II quantum computer) to implement a quantum lattice-gas algorithm. This paper describes work towards constructing a new experimental set-up for a type II quantum computer. This set-up has new hardware and software specifications but does follow previously published approaches of operation encoding the initial mass density onto a twoqubit processor and using standard pulse techniques to step through the algorithm. New hardware for this system includes the ability to read both qubits at once, effectively reducing the processing time by twofold. Hardware changes also include the use of multiple coils controlled by a single spectrometer and a hardware switch. New software includes a top level control system for the spectrometer for quick experimental configuration as well as configurable modeling software to verify results. Results are presented here from a system with the final software implementations and the two channel spectrometer configuration run on a single prototype coil. Progress towards the final multi-coil implementation is described.     

Improving Gate-Level Simulation of Quantum Circuits

Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and the vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a novel data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, a useful subset of operator matrices and state vectors can be represented in a form that grows polynomially with the number of qubits. This subset contains, but is not limited to, any equal superposition of n qubits, any computational basis state, n-qubit Pauli matrices, and n-qubit Hadamard matrices. It does not, however, contain the discrete Fourier transform (employed in Shor's algorithm) and some oracles used in Grover's algorithm. We first introduce and motivate decision diagrams and QuIDDs. We then analyze the runtime and memory complexity of QuIDD operations. Finally, we empirically validate QuIDD-based simulation by means of a general-purpose quantum computing simulator QuIDDPro implemented in C++. We simulate various instances of Grover's algorithm with QuIDDPro, and the results demonstrate that QuIDDs asymptotically outperform all other known simulation techniques. Our simulations also show that well-known worst-case instances of classical searching can be circumvented in many specific cases by data compression techniques. PACS: 03.67.Lx, 03.65.Fd, 03.65.Vd, 07.05.Bx     

A parallel sorting algorithm for a novel model of computation

The computational complexity of a parallel algorithm depends critically on the model of computation. We describe a simple and elegant rule-based model of computation in which processors apply rules asynchronously to pairs of objects from a global object space. Application of a rule to a pair of objects results in the creation of a new object if the objects satisfy the guard of the rule. The model can be efficiently implemented as a novel MIMD array processor architecture, the Intersecting Broadcast Machine. For this model of computation, we describe an efficient parallel sorting algorithm based on mergesort. The computational complexity of the sorting algorithm isO(nlog² n), comparable to that for specialized sorting networks and an improvement on theO(n ^1.5) complexity of conventional mesh-connected array processors.     

Quantum Computing Using Quadrupolar Spins in Solid State NMR

Nuclear magnetic resonance (NMR) is a successful method for experimental implementation of quantum information processing. Most of the successful NMR quantum processors are small molecules in liquid state. In this case each spin half particle represents a qubit. Another approach is the usage of higher spin particles as multi qubit systems. We present the first solid state virtual 2-Qubit system, represented by the spin-3/2 nucleus ²³Na in a NaNO₃ single crystal. For this system we show how to create the pseudo pure states and we derive a set of propagators and logic gates corresponding to the selective excitation of single quantum transitions. With this set, the preparation of an entangled state is experimentally verified by state tomography, adjusted to the spin-3/2 system. PACS: 0.367Lx; 76.60-k     

Experimental Concatenation of Quantum Error Correction with Decoupling

We experimentally explore the reduction of decoherence via concatenating quantum error correction (QEC) with decoupling in liquid-state NMR quantum information processing. Decoupling provides an efficient means of suppressing decoherence from noise sources with long correlation times, and then QEC can be used more profitably for the remaining noise sources. PACS: 03.67.Lx, 03.65.Bz     

CORDIC-Based MMSE-DFE Coefficient Computation

A modular parallel architecture for a MMSE-DFE coefficient computation processor is presented. The architecture is based on QR factorization of a channel-and-noise-dependent data matrix and is implemented using CORDIC processors within a systolic array architecture. Implementation issues including the number of CORDIC stages and the bit precision required in a fixed-point implementation are investigated through computer simulations. The proposed architecture accommodates fractionally spaced DFEs, co-channel interference, and multiple diversity paths.     

交互更新模式的量子遗传算法的几何约束求解

目的针对传统量子遗传算法无法充分利用种群中未成熟个体信息的不足,提出了基于交互更新模式的量子遗传算法(IUMQGA)并应用于几何约束求解中。方法几何约束问题的约束方程组可转化为优化模型,因此约束求解问题可以转化为优化问题。采用将遗传算法与量子理论相结合的量子遗传算法,使用双串量子染色体结构,使用交互更新策略将遗传算法中的交叉操作利用量子门变换来实现,根据不同情况采用不同的交互更新策略。这里的交互,指的是两个个体进行信息交换的过程,该过程用以产生新的个体。这不仅增加了个体间信息的交换而且充分利用了种群中未成熟个体的信息,提高了算法的收敛速度。结果通过非线性方程实例和几何约束实例测试并与其他方法比较表明,基于交互更新模式的量子遗传算法求解几何约束问题具有更好的求解精度和求解速率。双圆外公切线问题实例中,IUMQGA算法比QGA算法稳定;单圆填充问题和双圆外公切线问题实例中,通过实验求得各变量的最优值与其相应的精确值的误差在1E-2以下。结论采用交互更新模式的量子遗传算法可以很好地求解几何约束问题。     

Fast linear systolic matrix vector multiplication

The systolic concept in the parallel architecture design proposed by the H. T. Kung [1,2] obtains high throughput and speedups. The linear array for the matrix vector multiplication executes the algorithm in 2n ? 1 time steps using 2n ? 1 processors. Although the speedup obtained is very high, the efficiency is very poor (typical values of 25% efficiency for problem size greater than 10). H. T. Kung proposed an idea for a linear systolic array using two data streams flowing in opposite directions. However, the processors in the array perform operations in every second time moment.

Attempts to improve this design have been made by many researchers. Nonlinear and folding transformations techniques [3,4,5] only decrease the number of processors used to half the size, but do not affect the time.

We propose the use of a fast linear systolic computation procedure to obtain a solution that uses 3n/2 processors and executes the algorithm in 3n/2 time steps for the same cells, the same communication and the same regular data flow as the H. T. Kung linear array. Only the algorithm is restructured and more efficiently organized. Now the processors are utilized in every time step and no idle steps are required.     

Efficient parallel implementation of Bose Hubbard model: Exact numerical ground states and dynamics of gaseous Bose-Einstein condensates

We present a parallel implementation of the Bose Hubbard model, using imaginary time propagation to find the lowest quantum eigenstate and real time propagation for simulation of quantum dynamics. Scaling issues, performance of sparse matrix-vector multiplication, and a parallel algorithm for determining nonzero matrix elements are described. Implementation of imaginary time propagation yields an O(N) linear convergence on a single processor and slightly better than ideal performance on up to 160 processors for a particular problem size. The determination of the nonzero matrix elements is intractable using sequential non-optimized techniques for large problem sizes. Thus, we discuss a parallel algorithm that takes advantage of the intrinsic structural characteristics of the Fock-space matrix representation of the Bose Hubbard Hamiltonian and utilizes a parallel implementation of a Fock state look up table to make this task solvable within reasonable timeframes. Our parallel algorithm demonstrates near ideal scaling on thousand of processors. We include results for a matrix 22.6 million square, with 202 million nonzero elements, utilizing 2048 processors.     

多量子位Grover量子搜索算法的NMR仿真实现

核磁共振（NMR）技术目前是能有效实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。本文以核磁共振和多量子算符代数理论为基础,提出了实现多量子位Grover量子搜索算法的核磁共振脉冲序列设计方法,并在量子计算仿真程序上进行了3量子位的Grover量子搜索算法的实验验证。     

Natural Majorization of the Quantum Fourier Transformation in Phase-Estimation Algorithms

We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms. Our result relies on the fact that states which are mixed by Hadamard operators at any stage of the computation only differ by a phase. This property is a consequence of the structure of the initial state and of the QFT, based on controlled-phase operators and a single action of a Hadamard gate per qubit. The detail of our proof shows that Hadamard gates sort the probability distribution associated to the quantum state, whereas controlled-phase operators carry all the entanglement but are immaterial to majorization. We also prove that majorization in phase-estimation algorithms follows in a most natural way from unitary evolution, unlike its counterpart in Grover's algorithm. PACS: 03.67.-a, 03.67.Lx     

Progress in Quantum Algorithms

We discuss the progress (or lack of it) that has been made in discovering algorithms for computation on a quantum computer. Some possible reasons are given for the paucity of quantum algorithms so far discovered, and a short survey is given of the state of the field. PACS: 03.67.Lx     

Performance of SOR methods on modern vector and scalar processors

The building-cube method (BCM) is a new generation algorithm for CFD simulations. The basic idea of BCM is to simplify the algorithm in all stages of flow computation to achieve large-scale simulations. Calculation of a pressure field using the Successive Over Relaxation (SOR) method consumes most of the total execution time required for BCM. In this paper, effective implementations on modern vector and scalar processors are investigated. NEC SX-9 and Intel Nehalem-EX are the latest vector and scalar processors. Those processors have much higher peak performances than their previous-generation processors. However, their memory bandwidth improvement cannot catch up with the performance improvement of processors. This is the so-called memory wall problem. In our paper, we discuss optimization techniques for implementation of the SOR method based on architectural characteristics of these modern processors, and evaluate their effects on the sustained performances of these processors for BCM.     

Algorithms on ensemble quantum computers

In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this limitation on the model of computation, many algorithms cannot be processed directly on such computers, and must be modified, as the common strategy of delaying the measurements usually does not resolve this ensemble-measurement problem. Here we present several new strategies for resolving this problem. Based on these strategies we provide new versions of some of the most important quantum algorithms, versions that are suitable for implementing on ensemble quantum computers, e.g., on liquid NMR quantum computers. These algorithms are Shor’s factorization algorithm, Grover’s search algorithm (with several marked items), and an algorithm for quantum fault-tolerant computation. The first two algorithms are simply modified using a randomizing and a sorting strategies. For the last algorithm, we develop a classical-quantum hybrid strategy for removing measurements. We use it to present a novel quantum fault-tolerant scheme. More explicitly, we present schemes for fault-tolerant measurement-free implementation of Toffoli and s_z^1/4,\sigma_{z}^{1/4}, as these operations cannot be implemented “bitwise”, and their standard fault-tolerant implementations require measurement.     

A load-balancing algorithm for a parallel electromagnetic particle-in-cell code

Particle-in-cell simulations often suffer from load-imbalance on parallel machines due to the competing requirements of the field-solve and particle-push computations. We propose a new algorithm that balances the two computations independently. The grid for the field-solve computation is statically partitioned. The particles within a processor's sub-domain(s) are dynamically balanced by migrating spatially-compact groups of particles from heavily loaded processors to lightly loaded ones as needed. The algorithm has been implemented in the quicksilver electromagnetic particle-in-cell code. We provide details of the implementation and present performance results for quicksilver running models with up to a billion grid cells and particles on thousands of processors of a large distributed-memory parallel machine.