Abstract

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. * Earlier results of this work were reported at ASPDAC '03 [18]. New material includes significantly better experimental results and a description of a class of matrices and vectors which can be manipulated in polynomial time and memory using QuIDDPro.

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Preface Recent scientific advances, both experimental and algorithmic, have brought quantum information processing closer to reality. The ability to process information "quantumly" already allows ultra-precise metrology and more secure communication, and quantum algorithms allow computations such as factoring to be done significantly faster than we know how to do them on classical computers. With these advances, quantum simulation has become an increasingly important topic for both theoretical and engineering reasons. On the theoretical front, progress toward defining the class of circuits that can be simulated efficiently on a classical computer has and will continue to lead to a deeper understanding of the power of quantum computation. Even though efficient simulation of all quantum circuits may not be possible, the circuits that will most likely make up the majority of operations on a quantum computer can in fact be simulated efficiently. These are fault-tolerant error-correction circuits; they are composed of only Clifford-group gates, which, as this book demonstrates, can be simulated surprisingly efficiently on a classical computer. Department of Electrical EngineeringRecent results suggest that larger classes of circuits can also be simulated efficiently on classical computers. From a circuit perspective, efficient simulation can result from operating with a restricted set of gates, such as the Clifford-group gates, or from operating with an arbitrary gate set if the circuit has a small treewidth, as Markov and Shi have shown. From a physical perspective, efficient simulation can also result from limiting the amount of entanglement in the intermediate states of the computation. For example, Vidal has shown that one-dimensional many-body systems can be simulated efficiently and more recently, Bravyi and Terhal have identified a large class of quantum Hamiltonians for which the adiabatic evolution can be simulated efficiently on a classical computer. Ideally, we would like to characterize the class of quantum circuits that are computationally no more efficient than the equivalent classical versions. Such developments would lead to new theoretical results, sharpening our understanding of the differences between classical complexity classes and their quantum counterparts. It would also help us understand the true power of quantum computation. VI PrefaceOn the engineering front, efficient simulation provides the ability to validate and perform sanity checks on circuit components prior to their implementation. It is also useful for evaluating circuits, in particular for benchmarking the error rates of gates a...

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.

While thousands of experimental physicists and chemists are currently trying to build scalable quantum computers, it appears that simulation of quantum computation will be at least as critical as circuit simulation in classical VLSI design. However, since the work of Richard Feynman in the early 1980s little progress was made in practical quantum simulation. Most researchers focused on polynomial-time simulation of restricted types of quantum circuits that fall short of the full power of quantum computation [7].Simulating quantum computing devices and useful quantum algorithms on classical hardware now requires excessive computational resources, making many important simulation tasks infeasible. In this work we propose a new technique for gate-level simulation of quantum circuits which greatly reduces the difficulty and cost of such simulations. The proposed technique is implemented in a simulation tool called the Quantum Information Decision Diagram (QuIDD) and evaluated by simulating Grover's quantum search algorithm [8]. The back-end of our package, QuIDD Pro, is based on Binary Decision Diagrams, well-known for their ability to efficiently represent many seemingly intractable combinatorial structures. This reliance on a well-established area of research allows us to take advantage of existing software for BDD manipulation and achieve unparalleled empirical results for quantum simulation.