[Quantum Mathematics]

By

ABSTRACT

The diagnostic problem of fault localization for quantum circuits is considered for the first time. In this study we introduce an algorithm used to identify and localize errors in quantum circuits; we apply our algorithm to example circuits using known errors that occur in quantum circuits [ 3, 4, 5], and we define the quantum fault model. An overview is presented contrasting classical test theory with the new quantum test theory presented here. Two new types of faults having no parallel in other types of circuit technology are presented, named and formalized in this work. We introduce and define the quantum fault table, and categorize its entries; a method is presented illustrating how to construct a class of statistical decision diagrams from entries in a quantum fault table. This study has been motivated for the following reasons: (1) various quantum error-correcting codes have been developed, but no research has been done on the problem of formulating a general approach for minimizing the number of tests needed to identify errors in the quantum circuit itself, (2) such research may lead to the development of improved error-correcting codes, (3) physicists currently resort to exhaustively testing quantum circuits, where all possible inputs and outputs are compared; based on the current trends in quantum computing technology, quantum circuits will soon be larger than what is feasible to test exhaustively, forcing a need for the diagnostic testing method introduced here, and finally (4) Quantum test set generation has been mentioned in the literature several times, yet at the time of this writing no in depth study has been done.

TABLE OF CONTENTS

ABSTRACT2

CHAPTER 1: INTRODUCTION4

CHAPTER 2: THE PROBLEM OF TESTING A QUANTUM CIRCUIT7

Selecting a Quantum Fault Model8

The existence of complete test sets12

CHAPTER 3: GENERATION OF THE QUANTUM FAULT TABLE14

The Quantum Covering Problem17

CHAPTER 4: LOCALIZATION OF FAULTS21

Fault Trees22

Testing CAD Tool and Simulations - Q-Fault24

CHAPTER 5: CONCLUSION25

REFERENCES26

CHAPTER 1: INTRODUCTION

In classical circuit design an error in a circuit is typically referred to as a fault - in this paper we use the word error and fault interchangeably when referring to both classical and quantum circuits (Penrose, 1997). In classical circuits we represent the occurrence of faults digitally as the inversion of one or more bits of information at one or more locations in the circuit. In analog circuits a fault is manifested by lost signal integrity at one or more stages in the circuit. The detection of faults broudly falls in two distinct types of tests: (1) Parametric tests, as those typically adapted by an analog test engineer, taking into account parametric measures. (2) Logical test, (known also as functional test in which the functional output of a system is compared with the expected output value for a given input test (Roman & Carl, 2007).

A quantum computer can suffer from three general types of errors; measurement, preperation, or evolution. In this work, we concern ourselves with Logical testing as applied to quantum circuits, where we inspect the logical data processed by the quantum circuit and compare this data with ...