Anderson Braga de Avila

Quantum computing simulation through reduction and decomposition optimizations with a case study of Shor's algorithm

Because of the expansion of transformations and read/write memory states by tensor products in multidimensional quantum applications, the exponential increase in temporal and spatial complexities constitutes one of the main challenges for quantum computing simulations. Simulation of these systems is very relevant to develop and test new quantum algorithms. In order to overcome the increase in simulation complexity, this work presents reduction and decomposition optimizations for the Distributed Geometric Machine environment. By exploring properties as the sparsity of the Identity operator and partiality of dense unitary transformations, better storage and distribution of quantum information are achieved. The main improvements are reached by decreasing replication and void elements inherited from quantum operators. In the evaluation of this proposal, Hadamard transformations from 21 to 28 qubits and Quantum Fourier Transforms from 26 to 28 qubits were simulated over CPU, sequentially and in parallel, and in graphics processing unit, showing reduced temporal complexity and, consequently, shorter simulation time. Moreover, evaluations of the Shors algorithm considering 2n + 3 qubits in the order‐finding quantum algorithm were simulated up to 25 qubits. When comparing our implementations running on the same hardware with language‐integrated quantum operation, academic release version, our new simulator was faster and allowed for the simulation of more qubits. Copyright

Fuzzy Xor Classes from Quantum Computing

By making use of quantum parallelism, quantum processes provide parallel modelling for fuzzy connectives and the corresponding computations of quantum states can be simultaneously performed, based on the superposition of membership degrees of an element with respect to the different fuzzy sets. Such description and modelling is mainly focussed on representable fuzzy Xor connectives and their dual constructions. So, via quantum computing not only the interpretation based on traditional quantum circuit is considered, but also the notion of quantum process in the qGM model is applied, proving an evaluation of a corresponding simulation by considering graphical interfaces of the VPE-qGM programming environment. The quantum interpretations come from measurement operations performed on the corresponding quantum states.

Optimizing Quantum Simulation for Heterogeneous Computing: a Hadamard Transformation Study

The D-GM execution environment improves distributed simulation of quantum algorithms in heterogeneous computing environments comprising both multi-core CPUs and GPUs. The main contribution of this work consists in the optimization of the environment VirD-GM, conceived in three steps: (i) the theoretical studies and implementation of the abstractions of the Mixed Partial Process defined in the qGM model, focusing on the reduction of the memory consumption regarding multidimensional QTs; (ii) the distributed/parallel implementation of such abstractions allowing its execution on clusters of GPUs; (iii) and optimizations that predict multiplications by zero-value of the quantum states/transformations, implying reduction in the number of computations. The results obtained in this work embrace the distribute/parallel simulation of Hadamard gates up to 21 qubits, showing scalability with the increase in the number of computing nodes.

Optimizing D-GM quantum computing by exploring parallel and distributed quantum simulations under GPUs arquitecture

The exponential increase in the temporal and spatial complexities is one of the main challenges in the widespread use of quantum algorithm simulation, especially in dense quantum transformations (QTs) such as the Hadamard transformation (H), which has found wide applications in computer and communication science and also comprising the simplest quantum universal set of QTs. The main reason for these costs is the expansion of QTs by using tensor product in multi-dimension quantum applications. In this work, new optimizations for the execution of reduction and decomposition based on the Identity operator are introduced in the Distributed Geometric Machine framework (D-GM). Instead of executing the quantum transformation in a single step, they are divided in sub-quantum transformations and only the values different from Identity transformations are stored. Mixed Partial Processes provide control over the increase in the size of read/write memory states in the calculation of a QT, thus contributing to increase the scalability of applications regarding hardware-GPUs memory limit. In the evaluation of this D-GM extension, Hadamard Transformations were simulated up to 28 qubits applications over a single GPU. Our new simulator is 10, 829χ faster and allows for the simulation of more qubits when compared to our previous implementation running on the same GPU.

Scalable quantum simulation by reductions and decompositions through the Id-operator

One of the main obstacles for the adoption of quantum algorithm simulation is the exponential increase in temporal and spatial complexities, due to the expansion of transformations and read/write memory states by using tensor product in multi-dimension applications. Reduction and decomposition optimizations via the Id-operator provide a smart and appropriate storage and distribution of quantum information. Reductions are achieved by avoiding replication and sparsity inherited from such operators. By using decompositions, applications may be divided into sub-steps to store only distinct values from Id-operators, instead of executing quantum transformations in a single step. Additional optimizations based on mixed partial processes provide control over increase in read/write memory states in quantum transformations, also contributing to increase the scalability regarding hardware-GPUs memory limit. Hadamard and Discret Quantum Fourier Transforms were simulated up to 28 qubits applications over a single GPU with drastic temporal complexity reduction and simulation time.

Interpretations on Quantum Fuzzy Computing

Quantum processes provide a parallel model for fuzzy connectives. Calculations of quantum states may be simultaneously performed by the superposition of membership and non-membership degrees of each element regarding the intuitionistic fuzzy sets. This work aims to interpret Atanassovs intuitionistic fuzzy logic through quantum computing, where not only intuitionistic fuzzy sets, but also their basic operations and corresponding connectives (negation, conjuntion, disjuntion, difference, codifference, implication, and coimplication), are interpreted based on the traditional quantum circuit model.

Representing Intuistionistic Fuzzy Bi-implications Using Quantum Computing

Computer systems based on intuitionistic fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system, even with rules that are generated with imprecision. The main contribution of this paper is to show that quantum computing can be used to extend the class of intuitionistic fuzzy sets with respect to representing intuitionistic fuzzy bi-implications. This paper describes a multi-dimensional quantum register using aggregations operators such as t-(co)norms and implications based on quantum gates allowing the modeling and interpretation of intuitionistic fuzzy bi-implications.

Efficient In-Situ Quantum Computing Simulation of Shor's and Grover's Algorithms

Exponential increase and global access to read/write memory states in quantum computing simulation limit both the number of qubits and quantum transformations that can be currently simulated. Although quantum computing simulation is parallel by nature, spatial and temporal complexity are major performance hazards, making this an important application for HPC. A new methodology employing reduction and decomposition optimizations has shown great results, but its GPU implementation could be further improved. In this work, we intend to do a new implementation for in-situ GPU simulation that better explores its resources without requiring further HPC hardware. Shors and Grovers algorithms are simulated and compared to the previous version and to LIQUi|s simulator, showing better results with relative speedups up to 15.5x and 765.76x respectively.

Reduction and Decomposition Optimizations in Quantum Computing Simulation Applied to the Shor’s Algorithm

Due to the expansion of transformations and read/write memory states by tensor products in multidimensional quantum applications, the exponential increase in temporal and spatial complexities constitutes one of the main challenges for quantum computing simulations. Simulation of these systems is important in order to develop and test new quantum algorithms. This work presents reduction and decomposition optimizations for the Distributed Geometric Machine environment. By exploring properties as the sparsity of the Identity operator and partiality of dense unitary transformations, better storage and distribution of quantum information are achieved. The main improvements are implemented by decreasing replication and void elements inherited from quantum operators. In the evaluation of this proposal, Shor’s algorithm considering 2n+3 qubits in the order-finding quantum algorithm was simulated up to 25 qubits over CPU, sequentially and in parallel, and over GPU. Results confirm that temporal complexity is reduced. When comparing our implementations running on the same hardware with LIQUi|i, academic release version, our new simulator was faster and allowed for the simulation of more qubits.

Simulação quântica em arquiteturas multicore via biblioteca qgmc-analyzer

Atualmente situado sob o contexto do ambiente de simulacao quântica VPE-qGM (Visual Programming Environment for the Quantum Geometric Machine Model), este trabalho tem o objetivo de estabelecer o suporte a aceleracao da biblioteca de execucao do ambiente atraves de processadores multicore, beneficiando-se dos recursos providos pela biblioteca OpenMP [3]. Consolida-se assim a biblioteca qGMC-Analyzer, com a implementacao, desenvolvimento e validacao de algoritmo para simulacao quântica em arquiteturas multicore, cuja modelagem foi introduzida em [5]. O ambiente VPE-qGM, fundamentado no modelo de processos qGM (Quantum Geometric Machine Model) [4], e constituido de construtores para modelagem e simulacao grafica de aplicacoes quânticas. De acordo com o modelo qGM, a nocao de portas quânticas pode ser substituida pelo conceito de sincronizacao de processos elementares (PEs). No ambiente VPE-qGM, o PE e um elemento estruturado por tres atributos: (i) Acao: Corresponde as transformacoes aplicadas a diferentes qubits em um mesmo instante de tempo; (ii) Parâmetros: Contem dados auxiliares associados a definicao das transformacoes quânticas; (iii) Posicao: Posicao de escrita em um espaco de memoria global e compartilhada, na qual e armazenado o resultado calculado pelo PE. Neste contexto, uma transformacao quântica, aplicada a N qubits, pode ser modelada pela sincronizacao de 2N PEs, cujas parametrizacoes satisfazem as condicoes equivalentes a definicao dos vetores componentes da matriz (transformacao unitaria ou de medida) associada. Assim, durante a simulacao, ocorre a execucao (sequencial ou sincrona) dos PEs, os quais tem suas correspondentes computacoes efetuadas pela biblioteca qGM-Analyzer, manipulando os dados presentes nas posicoes de memoria e simulando o comportamento de um sistema quântico. A biblioteca de execucao dos PEs, denominada qGM-Analyzer, implementa otimizacoes que controlam o aumento exponencial dos vetores componentes das matrizes de definicao do operador de multiplos qubits [1]. Entretanto, o tempo total de simulacao obtido permanece elevado, devido a quantidade de operacoes necessarias para simular uma transformacao quântica. A implementacao da biblioteca qGM-Analyzer em C segue as otimizacoes introduzidas em [1], apenas alterando as estruturas de dados utilizados e fazendo uso de recursos nativos oferecidos pela linguagem visando a otimizacao da execucao. Para a implementacao do paralelismo foi utilizada a biblioteca OpenMP [3]. Na implementacao paralela, cada thread possui uma copia privada da pilha, e as memorias de escrita e leitura sao compartilhadas entre todos os threads, pois cada valor calculado e escrito em uma posicao diferente da memoria. A divisao dos threads e feita de forma a manter juntos os valores das matrizes necessarios para o calculo de uma posicao. As iteracoes sao divididas levando em conta o numero de colunas da primeira matriz envolvida na transformacao. Nos casos em que a transformacao possui apenas uma matriz, os threads sao divididos de forma diferente, para que o trabalho seja distribuido de forma Bolsista de Iniciacao Cientifica PIBIC/CNPq †Bolsista de PROBIC/FAPERGS balanceada. A definicao do numero de threads utilizadas pela biblioteca e definida por uma variavel de ambiente (OMP NUM THREADS), gerando uma implementacao mais flexivel. Ou seja, dependendo da transformacao quântica, tem-se um controle da granulosidade visando melhor desempenho da biblioteca. Para validacao e analise de desempenho da implementacao, foram desenvolvidos estudos de caso de transformacoes quânticas Hadamard e Pauli X [2], contemplando sistemas entre 13 e 20 qubits. Para cada estudo de caso foram realizadas 15 simulacoes. A maquina utilizada nas simulacoes possui as seguintes caracteristicas principais: processador Intel Core i7-3770 @ 3.4 GHz com Hyper-Threading abilitado e Turbo Boost desabilitado, 8GB RAM e sistema operacional Ubuntu 12.04 64 bits. A principal comparacao de desempenho se da com a execucao da biblioteca em diferentes numeros de threads. Os testes foram realizados com transformacoes Hadamards, representada por uma matriz densa de ordem 2n, onde n e o numero de qubits e transformacoes Pauli X, uma matriz esparsa de ordem 2n. Estas execucoes compreenderam duas etapas: (i) Geracao dos valores nao nulos associados ao correspondente vetor componente da matriz de definicao da transformacao quântica modelada; (ii) Multiplicacao desses valores pelas amplitudes obtidas da estrutura de memoria que modela o espaco de estados do sistema quântico. Podemos observar o speedup das simulacoes em relacao a um thread na Figura 1: Speedup das transformacoes Hadamard e Pauli X em relacao a um thread. Figura 1. Nas simulacoes das Hadamards, observa-se que para as configuracoes de 2 e 4 threads, o speedup aproxima-se do ideal conforme aumenta-se o numero de qubits. Para 8 threads o desempenho nao segue o mesmo padrao, justificado pelo uso do hyper-threading. Nas transformacoes controladas Pauli X o speedup diminui conforme aumenta-se o numero de threads. Justifica-se este fato pelo baixo numero de operacoes envolvidas. Sendo a Pauli X uma matriz esparsa, seus valores zerados nao sao computados pelo algoritmo otimizado, tornando o overhead da criacao e troca de contexto dos threads o principal custo da transformacao. As otimizacoes realizadas e a implementacao paralela representaram um ganho significativo no tempo de execucao das transformacoes quânticas, permitindo a simulacao de transformacoes de 18 qubits com elevado numero de operacoes, como no caso das Hadamards.