George F. Viamontes

Probabilistic transfer matrices in symbolic reliability analysis of logic circuits

We propose the probabilistic transfer matrix (PTM) framework to capture nondeterministic behavior in logic circuits. PTMs provide a concise description of both normal and faulty behavior, and are well-suited to reliability and error susceptibility calculations. A few simple composition rules based on connectivity can be used to recursively build larger PTMs (representing entire logic circuits) from smaller gate PTMs. PTMs for gates in series are combined using matrix multiplication, and PTMs for gates in parallel are combined using the tensor product operation. PTMs can accurately calculate joint output probabilities in the presence of reconvergent fanout and inseparable joint input distributions. To improve computational efficiency, we encode PTMs as algebraic decision diagrams (ADDs). We also develop equivalent ADD algorithms for newly defined matrix operations such as eliminate_variables and eliminate_redundant_variables, which aid in the numerical computation of circuit PTMs. We use PTMs to evaluate circuit reliability and derive polynomial approximations for circuit error probabilities in terms of gate error probabilities. PTMs can also analyze the effects of logic and electrical masking on error mitigation. We show that ignoring logic masking can overestimate errors by an order of magnitude. We incorporate electrical masking by computing error attenuation probabilities, based on analytical models, into an extended PTM framework for reliability computation. We further define a susceptibility measure to identify gates whose errors are not well masked. We show that hardening a few gates can significantly improve circuit reliability.

Improving Gate-Level Simulation of Quantum Circuits

AbstractSimulating 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 Shors algorithm) and some oracles used in Grovers 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 Grovers 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

Gate-level simulation of quantum circuits

Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a new data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, many of these matrices and vectors can be represented in a form that grows polynomially. Using QuIDDs, we implemented a general-purpose quantum computing simulator in C++ called QuIDDPro and tested it on Grovers algorithm. Our QuIDD technique asymptotically outperforms other known simulation techniques.

Checking equivalence of quantum circuits and states

Among the post-CMOS technologies currently under investigation, quantum computing (QC) holds a special place. QC offers not only extremely small size and low power, but also exponential speed-ups for important simulation and optimization problems. It also poses new CAD problems that are similar to, but more challenging, than the related problems in classical (non-quantum) CAD, such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complex-valued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalence-checking issues for simulation and synthesis. We identify several different equivalence-checking problems and present algorithms for practical benchmarks, including quantum communication and search circuits, which are shown to be very fast and robust for hundreds of qubits.

High-performance QuIDD-based simulation of quantum circuits

Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with the number of qubits. It has been shown experimentally that the QuIDD (Quantum Information Decision Diagram) datastructure greatly facilitates simulations using memory and runtime that are polynomial in the number of qubits. In this paper, we present a complexity analysis which formally describes this class of matrices and vectors. We also present an improved implementation of QuIDDs which can simulate Grovers algorithm for quantum search with the asymptotic runtime complexity of an ideal quantum computer up to negligible overhead.

Is quantum search practical

Gauging a quantum algorithms practical significance requires weighing it against the best conventional techniques applied to useful instances of the same problem. The authors show that several commonly suggested applications of Grovers quantum search algorithm fail to offer computational improvements over the best conventional algorithms.