Decoherence, Inaccuracy and Errors
in Quantum Information Processing
Proposta di Tesi
Supervisore: Dott. A. Leporati
Dottorato di Ricerca in Informatica Ciclo XXI
Sara Felloni
sara.felloni@disco.unimib.it
DISCo
http://www.disco.unimib.it
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 1/22
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Quantum Information Processing and
Quantum Errors
Codes
Research, Roadmaps and
Projects
Partial Results and Future
Research
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 2/22
Quantum Computing
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
Quantum computing is the study of the information processing
tasks that can be accomplished using quantum mechanical
systems:
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
▲
the elementary physical carrier of information is a qubit (a
quantum system with a 2-dimensional state space);
▲
the state of a n-qubit register lives in a 2n -dimensional
Hilbert space (tensor product of n 2-dimensional Hilbert
spaces);
▲
by imitating classical computations, quantum computations
comprise three steps in sequence:
● preparation of the initial state of a quantum register;
● computation, by means of a quantum gate array
(sequence of unitary transformations of the register state);
● output of a result (by probabilistic measurement of all or
part of the register).
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Codes
Research, Roadmaps and
Projects
Partial Results and Future
Research
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 3/22
Realistic Quantum Computation
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
How to reliably transmit quantum information through a noisy
quantum channel is a central problem of quantum information
theory:
Quantum Computing
➲
➲ Quantum Error Correction
▲
➲ Quantum Error Correcting
Codes
Research, Roadmaps and
Projects
Partial Results and Future
▲
qubits unavoidably interact with the external world
(decoherence and introduction of noise);
under realistic conditions, the quantum operations
themselves are unavoidably affected by errors.
Research
Studies of the impact of noise on the stability of quantum
computation and communication are of primary importance for
the practical implementation of quantum information
protocols.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 4/22
Practical Difficulties of Quantum
Computing
Scaling
Quantum Information
Processing and Quantum
Errors
▲
➲ Quantum Computing
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Codes
▲
▲
Research, Roadmaps and
Projects
Several and very different experimental designs for quantum
computers have been proposed (cold ions and atoms, ion
traps, linear optics, solid states, nuclear magnetic
resonance).
Small working prototypes can successfully be built, but
the issue of scaling these up to computers large enough to
yield useful computations is still under (physical) research.
Partial Results and Future
Research
Decoherence
▲
▲
Sara Felloni, August 19, 2007
Quantum computation involves manipulating the quantum
states of objects that are in coherent quantum
superpositions.
These superpositions decay easily, as the (classical)
environment interact with the quantum system.
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 5/22
Inaccuracy
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Codes
Quantum computers are fundamentally analog-type devices:
▲ the state of a quantum superposition depends on
continuous parameters;
▲ all quantum gates are potentially analog in that there will be
some amount of inaccuracy in any physical implementation.
Research, Roadmaps and
Fault Tolerance
Projects
Partial Results and Future
Research
▲
▲
Sara Felloni, August 19, 2007
For a quantum computation to successfully yield the correct
result, this inaccuracy must be less than the amount of
noise which the computation can tolerate.
The tolerance of the computation to these inaccuracies must
be large enough to allow physical (imperfect)
implementation of quantum gates.
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 6/22
Quantum Error Correction
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
➲ Realistic Quantum
Computation
➲ Practical Difficulties of
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Codes
Close relation with the theory of quantum error correction:
▲ decoherence can be expressed in terms of inaccuracies in
the overall state of the quantum system and an auxiliary
quantum system interacting with it representing the
“environment”;
▲
decoherence-reduction methods can be used to correct
inaccuracy, and vice versa;
▲
the use of quantum error correcting codes can reduce
both decoherence and inaccuracy
● during transmission and storage of quantum data and
● while performing computation on quantum data.
Research, Roadmaps and
Projects
Partial Results and Future
Research
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 7/22
Quantum Error Correcting Codes
Quantum Information
Processing and Quantum
Errors
➲ Quantum Computing
➲ Realistic Quantum
Quantum error-correcting codes map qubits into blocks of
qubits so that a small number of errors in the qubits of any
block has little or no effect on the encoded qubits.
Computation
➲ Practical Difficulties of
Quantum Computing
➲
➲ Quantum Error Correction
➲ Quantum Error Correcting
Codes
Research, Roadmaps and
Projects
Partial Results and Future
Research
To prevent errors during quantum computation, it is necessary
to compute with encoded qubits without decoding them.
▲ Decoding or correcting errors in quantum codes is in itself a
quantum computation.
▲ Decoding the qubits in order to compute exposes them to
potential errors.
▲ Correcting errors using noisy quantum gates may introduce
worse errors.
Approximate quantum error correcting codes show strong
connections with secret sharing schemes.
Secret sharing seems to be a better classical analogue to
quantum error correction than classical error correction.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 8/22
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
➲ US Research
➲ European Research
Research, Roadmaps and Projects
➲ Quantum Codes for
Classical Simulation
Partial Results and Future
Research
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 9/22
US Research
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
➲ US Research
➲ European Research
➲ Quantum Codes for
Classical Simulation
Partial Results and Future
Research
Sara Felloni, August 19, 2007
Quantum Information Science and Technology
Roadmapping Project
Los Alamos National Laboratory and University of California for
the National Nuclear Security Administration,
of the US Department of Energy
http://qist.lanl.gov/
▲
A Quantum Information Science and Technology Roadmap,
Part 1: Quantum computation, report of the quantum
information science and technology experts panel,
version 2.0, April 2, 2004;
▲
A Quantum Information Science and Technology Roadmap,
Part 2: Quantum cryptography, report of the quantum
cryptography and technology experts panel,
version 1.0, July 19, 2004.
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 10/22
European Research
▲
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
➲ US Research
➲ European Research
▲
➲ Quantum Codes for
Classical Simulation
Partial Results and Future
Research
▲
Sara Felloni, August 19, 2007
P. Zoller et al., Quantum information processing and
communication. Strategic report on current status, visions
and goals for research in Europe,
Eur. Phys. J. D 36, 203-228, 2005.
R.F. Werner et al., Some open problems in quantum
information theory, Institut fur Mathematische Physik, TU
Braunschweig, Germany;
online preprint quant-ph/0504166, 2005;
http://www.imaph.tu-bs.de/qi/problems/problems.html
EDIQIP Project: Effects of Decoherence and Imperfections
for Quantum Information Processing,
contract IST-2001-38869, 2003-2006, Universitè Paul
Sabatier, Toulouse; Technische Universitat Darmstadt;
Istituto Nazionale per la Fisica della Materia, Como; Royal
Holloway, University of London;
periodic progress report:
http://www.quantware.ups-tlse.fr/EDIQIP/report05.pdf
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 11/22
Quantum Codes for Classical Simulation
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
➲ US Research
Quantware Library
http://www.quantware.ups-tlse.fr/EDIQIP/index.html
opened at 19/12/2005.
➲ European Research
➲ Quantum Codes for
Classical Simulation
Partial Results and Future
Research
This library is established in the framework of the EC IST-FET
projects EDIQIP (Effects of Decoherence and Imperfections for
Quantum Information Processing) and EuroSQIP (European
Superconducting Quantum Information Processor ).
Several quantum computations and processes can be
simulated with the given codes on a classical computer, for a
moderate number of qubits (∼ 20), in presence of realistic
internal imperfections and external decoherence.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 12/22
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
Partial Results and Future Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 13/22
Noise Models in Quantum Computing
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
Noise models that can be currently found in literature do not
seem to be the most general ones.
In particular, error channels corresponding to physical effects
of amplitude damping or thermal excitations do not seem to be
considered.
After having studied and compared all single-qubit quantum
noise channels, showing their different impact on quantum
protocols of useful application, the following step is to further
generalize the model including two-qubit errors.
➲ Main References
An error model constituted by single-qubit and two-qubit
quantum noise channels would allow to reach a satisfiable
degree of generalization: every quantum computation can be
expressed in terms of single-qubit and two-qubit gates.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 14/22
Aims
Quantum Information
Processing and Quantum
Errors
The main aims of the PhD Thesis are:
▲
to develop circuit models and mathematical descriptions
of quantum noise channels;
▲
to perform a systematic numerical study of the effects of
different noise channels on known quantum protocols;
▲
to show the very different impact of the various noise
channels on the algorithms considered, in terms of fidelity
behavior and of dependence of the maximum noise strength
tolerable for the protocols on the noise channels.
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
Studies like the present one aim to provide valuable
information for experimental implementations of the
quantum computations considered and, more generally, of
quantum computers.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 15/22
Methods
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
Each kind of imperfection will be described by means of
quantum circuits, composed by single-qubit and two-qubit
quantum gates.
The use of quantum circuits naturally provides:
▲ physical feasibleness, without requiring algebraic
conditions that may result hard to treat in several complex
cases;
▲
associated diagrams of states, for a clear visualization of
quantum information’s flow during computation;
▲
associated algorithms for quantum computation and for
classical simulation;
▲
a straightforward link to physical implementations.
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 16/22
Recent Research Results
▲
In (1), the stability under quantum noise effects of the
quantum privacy amplification protocol for the purification
of entanglement in quantum cryptography has been studied.
A systematic numerical study of the impact of all possible
single-qubit noise channels has been presented, finding that
both the qualitative behavior and the noise tolerance of the
iterative protocol strongly depend on the specific noise
channel.
▲
In (2), a graphic representation of quantum states has
been introduced, by means of a specific application, the
analysis of two models of quantum copying machines.
Elementary diagrams of states for single-qubit and two-qubit
systems, entanglement representation and quantum copying
machines have been presented, determining quantum
circuits of easier interpretation.
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 17/22
Current Studies
▲
The initial formulation of the graphic representation of
states in (2) will be extended in a more general formulation
of the method and will be followed by application to more
complex and meaningful quantum algorithms (Deutsch’s
Problem, single-qubit noise channels, quantum error
correcting codes, teleportation and dense coding,
probabilistic quantum cloning).
▲
An exhaustive study of a general transformation of a
two-qubit density matrix appears hardly feasible (240
parameters involved).
The first step towards a two-qubit error model is the study of
imperfect c-not gates, synthetized by means of known
c-phase gates schemes (c-not gates and single-qubit gates
reproduce any quantum computation).
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 18/22
PhD Thesis Preliminary Overview
▲
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
▲
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
▲
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
▲
An introduction to Quantum Computing and Information
Processing.
The diagrams of states:
● a graphic representation of states as an additional
description of quantum computations, for analysis and
synthesis of quantum circuits;
● formulation of the method;
● application to known quantum circuits and further studies.
Decoherence, inaccuracy, approximations and quantum
noise: an introduction to Quantum Error Theory.
A general model for single-qubit quantum noise channels:
characterization by means of quantum circuits, diagrams of
states and mathematical description.
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 19/22
▲
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
▲
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
Sara Felloni, August 19, 2007
▲
Impact of single-qubit noise channels on known quantum
algorithms, in terms of fidelity behavior and maximum noise
strength tolerable:
● entanglement purification in EPR cryptographic protocols;
● quantum simulations of not too complex physical systems;
● quantum error correcting codes.
A model for two-qubit quantum noise channels:
● universal set of quantum gates;
● imperfect implementation of the two-qubit c-not gate;
● further studies on two-qubit quantum noise channels.
Application of the overall error model to further known
quantum algorithms.
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 20/22
Publication List
Quantum Information
Processing and Quantum
Errors
Research, Roadmaps and
Projects
Partial Results and Future
Research
➲ Noise Models in Quantum
Computing
➲ Aims
➲ Methods
➲ Recent Research Results
➲ Current Studies
➲ PhD Thesis Preliminary
Overview
➲
➲ Publication List
➲ Main References
References
[1] G. Benenti, S. Felloni, G. Strini, Effects of single-qubit quantum noise on
entanglement purification, Eur. Phys. J. D 38, 389, 2006.
[2] S. Felloni, G. Strini, A graphic representation of states for quantum
copying machines, EJTP 3, No. 11, 159-187, 2006.
[3] A. Leporati, S. Felloni, Three “quantum” algorithms to solve 3-SAT, Vol. II
of Proc. of the Fourth Brainstorming Week on Membrane Computing
(BWMC 2006), Sevilla, Spain, January 30 - February 3 2006, 137-160.
Accepted for publication in Theoretical Computer Science.
On Quantware Library
http://www.quantware.ups-tlse.fr/EDIQIP/index.html
G. Benenti, S. Felloni and G. Strini, Simulation of the quantum privacy
amplification protocol (QPA) with a noisy apparatus, code for numerical
studies in (1), 2005.
QNR3
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 21/22
Main References
References
[1] G. Benenti, G. Casati, G. Strini, Principles of Quantum Computation and Information, Volume I: Basic Concepts, World Scientific, 2004.
[2] G. Benenti, G. Casati, G. Strini, Principles of Quantum Computation and Information, Volume II: Basic Tools And Special Topics, World Scientific,
2007.
[3] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
[4] C.P. Williams, S.H. Clearwater, Explorations in Quantum Computing, Springer Verlag, 1998.
[5] S. Loepp, W. Wootters, Protecting Information. From Classical Error Correction to Quantum Cryptography, Cambridge University Press, 2006.
[6] V. Vedral, Introduction to Quantum Information Science, Oxford University Press, 2006.
[7] D. Bruss et al., Lectures on Quantum Information, John Wiley and Sons Inc, 2006.
[8] G. Strini, A. Carati, S. Vicari, Algoritmi quantistici per la risoluzione dell’equazione di Schroedinger, dipartimento di Matematica, facoltà di Scienze
Matematiche, Fisiche e Naturali, Università degli Studi di Milano, a.a. 2004/2005.
[9] G. Strini, A. Carati, R. Suardi, Sensibilità agli errori di alcuni algoritmi quantistici, dipartimento di Matematica, facoltà di Scienze Matematiche, Fisiche
e Naturali, Università degli Studi di Milano, a.a. 2004/2005.
Sara Felloni, August 19, 2007
Decoherence, Inaccuracy and Errors in Quantum Information Processing - p. 22/22