Quantum Computing with Photons: Introduction to the Circuit Model, the One-Way Quantum Computer, and the Fundamental Principles of Photonic Experiments
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Quantum Noise in Quantum Optics: the Stochastic Schr\" Odinger
Quantum Noise in Quantum Optics: the Stochastic Schr¨odinger Equation Peter Zoller † and C. W. Gardiner ∗ † Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria ∗ Department of Physics, Victoria University Wellington, New Zealand February 1, 2008 arXiv:quant-ph/9702030v1 12 Feb 1997 Abstract Lecture Notes for the Les Houches Summer School LXIII on Quantum Fluc- tuations in July 1995: “Quantum Noise in Quantum Optics: the Stochastic Schroedinger Equation” to appear in Elsevier Science Publishers B.V. 1997, edited by E. Giacobino and S. Reynaud. 0.1 Introduction Theoretical quantum optics studies “open systems,” i.e. systems coupled to an “environment” [1, 2, 3, 4]. In quantum optics this environment corresponds to the infinitely many modes of the electromagnetic field. The starting point of a description of quantum noise is a modeling in terms of quantum Markov processes [1]. From a physical point of view, a quantum Markovian description is an approximation where the environment is modeled as a heatbath with a short correlation time and weakly coupled to the system. In the system dynamics the coupling to a bath will introduce damping and fluctuations. The radiative modes of the heatbath also serve as input channels through which the system is driven, and as output channels which allow the continuous observation of the radiated fields. Examples of quantum optical systems are resonance fluorescence, where a radiatively damped atom is driven by laser light (input) and the properties of the emitted fluorescence light (output)are measured, and an optical cavity mode coupled to the outside radiation modes by a partially transmitting mirror. -
Quantum Computation and Complexity Theory
Quantum computation and complexity theory Course given at the Institut fÈurInformationssysteme, Abteilung fÈurDatenbanken und Expertensysteme, University of Technology Vienna, Wintersemester 1994/95 K. Svozil Institut fÈur Theoretische Physik University of Technology Vienna Wiedner Hauptstraûe 8-10/136 A-1040 Vienna, Austria e-mail: [email protected] December 5, 1994 qct.tex Abstract The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applicationsto quantum computing. Standardinterferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed. hep-th/9412047 06 Dec 94 1 Contents 1 The Quantum of action 3 2 Quantum mechanics for the computer scientist 7 2.1 Hilbert space quantum mechanics ..................... 7 2.2 From single to multiple quanta Ð ªsecondº ®eld quantization ...... 15 2.3 Quantum interference ............................ 17 2.4 Hilbert lattices and quantum logic ..................... 22 2.5 Partial algebras ............................... 24 3 Quantum information theory 25 3.1 Information is physical ........................... 25 3.2 Copying and cloning of qbits ........................ 25 3.3 Context dependence of qbits ........................ 26 3.4 Classical versus quantum tautologies .................... 27 4 Elements of quantum computatability and complexity theory 28 4.1 Universal quantum computers ....................... 30 4.2 Universal quantum networks ........................ 31 4.3 Quantum recursion theory ......................... 35 4.4 Factoring .................................. 36 4.5 Travelling salesman ............................. 36 4.6 Will the strong Church-Turing thesis survive? ............... 37 Appendix 39 A Hilbert space 39 B Fundamental constants of physics and their relations 42 B.1 Fundamental constants of physics ..................... 42 B.2 Conversion tables .............................. 43 B.3 Electromagnetic radiation and other wave phenomena ......... -
Quantum Inductive Learning and Quantum Logic Synthesis
Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 2009 Quantum Inductive Learning and Quantum Logic Synthesis Martin Lukac Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Lukac, Martin, "Quantum Inductive Learning and Quantum Logic Synthesis" (2009). Dissertations and Theses. Paper 2319. https://doi.org/10.15760/etd.2316 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. For more information, please contact [email protected]. QUANTUM INDUCTIVE LEARNING AND QUANTUM LOGIC SYNTHESIS by MARTIN LUKAC A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in ELECTRICAL AND COMPUTER ENGINEERING. Portland State University 2009 DISSERTATION APPROVAL The abstract and dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering were presented January 9, 2009, and accepted by the dissertation committee and the doctoral program. COMMITTEE APPROVALS: Irek Perkowski, Chair GarrisoH-Xireenwood -George ^Lendaris 5artM ?teven Bleiler Representative of the Office of Graduate Studies DOCTORAL PROGRAM APPROVAL: Malgorza /ska-Jeske7~Director Electrical Computer Engineering Ph.D. Program ABSTRACT An abstract of the dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering presented January 9, 2009. Title: Quantum Inductive Learning and Quantum Logic Synhesis Since Quantum Computer is almost realizable on large scale and Quantum Technology is one of the main solutions to the Moore Limit, Quantum Logic Synthesis (QLS) has become a required theory and tool for designing Quantum Logic Circuits. -
Arxiv:Quant-Ph/0002039V2 1 Aug 2000
Methodology for quantum logic gate construction 1,2 3,2 2 Xinlan Zhou ∗, Debbie W. Leung †, and Isaac L. Chuang ‡ 1 Department of Applied Physics, Stanford University, Stanford, California 94305-4090 2 IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120 3 Quantum Entanglement Project, ICORP, JST Edward Ginzton Laboratory, Stanford University, Stanford, California 94305-4085 (October 29, 2018) We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation (Gottesman and Chuang, Nature 402, 390, 1999) and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the π/8 and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed. I. INTRODUCTION Practical realization of quantum information processing requires specific types of quantum operations that may be difficult to construct. In particular, to perform quantum computation robustly in the presence of noise, one needs fault-tolerant implementation of quantum gates acting on states that are block-encoded using quantum error correcting codes [1–4]. Fault-tolerant quantum gates must prevent propagation of single qubit errors to multiple qubits within any code block so that small correctable errors will not grow to exceed the correction capability of the code. This requirement greatly restricts the types of unitary operations that can be performed on the encoded qubits. Certain fault-tolerant operations can be implemented easily by performing direct transversal operations on the encoded qubits, in which each qubit in a block interacts only with one corresponding qubit, either in another block or in a specialized ancilla. -
Fault-Tolerant Interface Between Quantum Memories and Quantum Processors
ARTICLE DOI: 10.1038/s41467-017-01418-2 OPEN Fault-tolerant interface between quantum memories and quantum processors Hendrik Poulsen Nautrup 1, Nicolai Friis 1,2 & Hans J. Briegel1 Topological error correction codes are promising candidates to protect quantum computa- tions from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation 1234567890 needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. Here we propose a practical solution, subsystem lattice surgery, which requires only two-body nearest-neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions and beyond. In particular, it can be employed to create a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors. 1 Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria. 2 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to H.P.N. (email: [email protected]) NATURE COMMUNICATIONS | 8: 1321 | DOI: 10.1038/s41467-017-01418-2 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01418-2 oise and decoherence can be considered as the major encoding k = n − s qubits. We denote the normalizer of S by Nobstacles for large-scale quantum information processing. -
Realization of the Quantum Toffoli Gate
Realization of the quantum Toffoli gate Realization of the quantum Toffoli gate Based on: Monz, T; Kim, K; Haensel, W; et al Realization of the quantum Toffoli gate with trapped ions Phys. Rev. Lett. 102, 040501 (2009) Lanyon, BP; Barbieri, M; Almeida, MP; et al. Simplifying quantum logic using higher dimensional Hilbert spaces Nat. Phys. 5, 134 (2009) Realization of the quantum Toffoli gate Outline 1. Motivation 2. Principles of the quantum Toffoli gate 3. Implementation with trapped ions 4. Implementation with photons 5. Comparison and conclusion 6. Summary 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 1 Realization of the quantum Toffoli gate 1. Motivation • Universal quantum logic gate sets are needed to implement algorithms • Implementation of algorithms is difficult due to the finite fidelity and large amount of gates • Use of other degrees of freedom to store information • Reduction of complexity and runtime 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 2 Realization of the quantum Toffoli gate 2. Principles of the Toffoli gate • Three-qubit gate (C1, C2, T) • Logic flip of T depending on (C1 AND C2) Truth table Matrix form Input Output C1 C2 T C1 C2 T 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 3 Realization of the quantum Toffoli gate 2. Principles of the Toffoli gate • Qubit Implementation • Qutrit Implementation Qutrit states: and 6. -
Machine Learning for Designing Fast Quantum Gates
University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2016-01-26 Machine Learning for Designing Fast Quantum Gates Zahedinejad, Ehsan Zahedinejad, E. (2016). Machine Learning for Designing Fast Quantum Gates (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26805 http://hdl.handle.net/11023/2780 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Machine Learning for Designing Fast Quantum Gates by Ehsan Zahedinejad A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA January, 2016 c Ehsan Zahedinejad 2016 Abstract Fault-tolerant quantum computing requires encoding the quantum information into logical qubits and performing the quantum information processing in a code-space. Quantum error correction codes, then, can be employed to diagnose and remove the possible errors in the quantum information, thereby avoiding the loss of information. Although a series of single- and two-qubit gates can be employed to construct a quan- tum error correcting circuit, however this decomposition approach is not practically desirable because it leads to circuits with long operation times. An alternative ap- proach to designing a fast quantum circuit is to design quantum gates that act on a multi-qubit gate. -
Quantum Memories for Continuous Variable States of Light in Atomic Ensembles
Quantum Memories for Continuous Variable States of Light in Atomic Ensembles Gabriel H´etet A thesis submitted for the degree of Doctor of Philosophy at The Australian National University October, 2008 ii Declaration This thesis is an account of research undertaken between March 2005 and June 2008 at the Department of Physics, Faculty of Science, Australian National University (ANU), Can- berra, Australia. This research was supervised by Prof. Ping Koy Lam (ANU), Prof. Hans- Albert Bachor (ANU), Dr. Ben Buchler (ANU) and Dr. Joseph Hope (ANU). Unless otherwise indicated, the work present herein is my own. None of the work presented here has ever been submitted for any degree at this or any other institution of learning. Gabriel H´etet October 2008 iii iv Acknowledgments I would naturally like to start by thanking my principal supervisors. Ping Koy Lam is the person I owe the most. I really appreciated his approachability, tolerance, intuition, knowledge, enthusiasm and many other qualities that made these three years at the ANU a great time. Hans Bachor, for helping making this PhD in Canberra possible and giving me the opportunity to work in this dynamic research environment. I always enjoyed his friendliness and general views and interpretations of physics. Thanks also to both of you for the opportunity you gave me to travel and present my work to international conferences. This thesis has been a rich experience thanks to my other two supervisors : Joseph Hope and Ben Buchler. I would like to thank Joe for invaluable help on the theory and computer side and Ben for the efficient moments he spent improving the set-up and other innumerable tips that polished up the work presented here, including proof reading this thesis. -
What Can Quantum Optics Say About Computational Complexity Theory?
week ending PRL 114, 060501 (2015) PHYSICAL REVIEW LETTERS 13 FEBRUARY 2015 What Can Quantum Optics Say about Computational Complexity Theory? Saleh Rahimi-Keshari, Austin P. Lund, and Timothy C. Ralph Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia (Received 26 August 2014; revised manuscript received 13 November 2014; published 11 February 2015) Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational complexity theory point of view. We derive a general formula for calculating the output probabilities, and by considering input thermal states, we show that the output probabilities are proportional to permanents of positive-semidefinite Hermitian matrices. It is believed that approximating permanents of complex matrices in general is a #P-hard problem. However, we show that these permanents can be approximated with an algorithm in the BPPNP complexity class, as there exists an efficient classical algorithm for sampling from the output probability distribution. We further consider input squeezed- vacuum states and discuss the complexity of sampling from the probability distribution at the output. DOI: 10.1103/PhysRevLett.114.060501 PACS numbers: 03.67.Ac, 42.50.Ex, 89.70.Eg Introduction.—Boson sampling is an intermediate model general formula for the probabilities of detecting single of quantum computation that seeks to generate random photons at the output of the network. Using this formula we samples from a probability distribution of photon (or, in show that probabilities of single-photon counting for input general, boson) counting events at the output of an M-mode thermal states are proportional to permanents of positive- linear-optical network consisting of passive optical ele- semidefinite Hermitian matrices. -
Quantum Error Correction
Quantum Error Correction Sri Rama Prasanna Pavani [email protected] ECEN 5026 – Quantum Optics – Prof. Alan Mickelson - 12/15/2006 Agenda Introduction Basic concepts Error Correction principles Quantum Error Correction QEC using linear optics Fault tolerance Conclusion Agenda Introduction Basic concepts Error Correction principles Quantum Error Correction QEC using linear optics Fault tolerance Conclusion Introduction to QEC Basic communication system Information has to be transferred through a noisy/lossy channel Sending raw data would result in information loss Sender encodes (typically by adding redundancies) and receiver decodes QEC secures quantum information from decoherence and quantum noise Agenda Introduction Basic concepts Error Correction principles Quantum Error Correction QEC using linear optics Fault tolerance Conclusion Two bit example Error model: Errors affect only the first bit of a physical two bit system Redundancy: States 0 and 1 are represented as 00 and 01 Decoding: Subsystems: Syndrome, Info. Repetition Code Representation: Error Probabilities: 2 bit flips: 0.25 *0.25 * 0.75 3 bit flips: 0.25 * 0.25 * 0.25 Majority decoding Total error probabilities: With repetition code: Error Model: 0.25^3 + 3 * 0.25^2 * 0.75 = 0.15625 Independent flip probability = 0.25 Without repetition code: 0.25 Analysis: 1 bit flip – No problem! Improvement! 2 (or) 3 bit flips – Ouch! Cyclic system States: 0, 1, 2, 3, 4, 5, 6 Operators: Error model: probability = where q = 0.5641 Decoding Subsystem probability -
Quantum Interference & Entanglement
QIE - Quantum Interference & Entanglement Physics 111B: Advanced Experimentation Laboratory University of California, Berkeley Contents 1 Quantum Interference & Entanglement Description (QIE)2 2 Quantum Interference & Entanglement Pictures2 3 Before the 1st Day of Lab2 4 Objectives 3 5 Introduction 3 5.1 Bell's Theorem and the CHSH Inequality ............................. 3 6 Experimental Setup 4 6.1 Overview ............................................... 4 6.2 Diode Laser.............................................. 4 6.3 BBOs ................................................. 6 6.4 Detection ............................................... 7 6.5 Coincidence Counting ........................................ 8 6.6 Proper Start-up Procedure ..................................... 9 7 Alignment 9 7.1 Violet Beam Path .......................................... 9 7.2 Infrared Beam Path ......................................... 10 7.3 Detection Arm Angle......................................... 12 7.4 Inserting the polarization analyzing elements ........................... 12 8 Producing a Bell State 13 8.1 Mid-Lab Questions.......................................... 13 8.2 Quantifying your Bell State..................................... 14 9 Violating Bell's Inequality 15 9.1 Using the LabVIEW Program.................................... 15 9.1.1 Count Rate Indicators.................................... 15 9.1.2 Status Indicators....................................... 16 9.1.3 General Settings ....................................... 16 9.1.4 -
Quantum Retrodiction: Foundations and Controversies
S S symmetry Article Quantum Retrodiction: Foundations and Controversies Stephen M. Barnett 1,* , John Jeffers 2 and David T. Pegg 3 1 School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK 2 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK; [email protected] 3 Centre for Quantum Dynamics, School of Science, Griffith University, Nathan 4111, Australia; d.pegg@griffith.edu.au * Correspondence: [email protected] Abstract: Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on cur- rent information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes’ theorem. Keywords: quantum foundations; bayesian inference; time reversal 1. Introduction Quantum theory is usually presented as a predictive theory, with statements made concerning the probabilities for measurement outcomes based upon earlier preparation events. In retrodictive quantum theory this order is reversed and we seek to use the Citation: Barnett, S.M.; Jeffers, J.; outcome of a measurement to make probabilistic statements concerning earlier events [1–6]. Pegg, D.T. Quantum Retrodiction: The theory was first presented within the context of time-reversal symmetry [1–3] but, more Foundations and Controversies. recently, has been developed into a practical tool for analysing experiments in quantum Symmetry 2021, 13, 586.