Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0003141 · New Journal of Physics

Error avoiding quantum codes and dynamical stabilization of Grover's algorithm

M. Mussinger, A. Delgado, G. Alber

Published Sep 12, 2000• 8 citations
arXivHTMLPDF

Abstract

Dynamical stabilization properties of error avoiding quantum codes are investigated beyond the perturbative regime. As an example Grover's search algorithm and its behaviour under a particular class of coherent errors are studied. Numerical examples which demonstrate that error avoiding quantum codes may be capable of stabilizing quantum algorithms well beyond the regime for which they were designed originally are presented.

DOI
10.1088/1367-2630/2/1/319

Related papers

Dynamical stabilization of Grover's algorithm with embedded quantum codes

G. Alber, M. Mussinger, A. Delgado

Stabilizing quantum algorithms against external perturbations and preserving quantum coherence are main challenges in the area of quantum information processing. In this contribution main ideas underlying a new class of recently proposed embedded error-correcting quantum codes are discussed. These detected-jump correcting quantum codes are capable of stabilizing distinguishable qubits against spontaneous decay provided these decay processes originate from couplings to statistically independent reservoirs. Exploiting the classical information about which qubit has been affected by the environment these embedded quantum codes minimize the number of required control measurements and recovery operations as well as redundancy. Their stabilizing properties are exemplified by applying them to Grover's quantum search algorithm.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 30, 2001•2 citations

Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation

G. Alber, T. Beth, Ch. Charnes +3 more

The recently introduced detected-jump-correcting quantum codes are capable of stabilizing qubit systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one-detected-jump-error-correcting quantum codes is shown and the optimal redundancy, encoding, and recovery as well as general properties of detected-jump-error-correcting quantum codes are discussed. By the use of design theory, multiple-jump-error-correcting quantum codes can be constructed. The performance of one-jump-error-correcting quantum codes under nonideal conditions is studied numerically by simulating a quantum memory and Grover's algorithm.

arXiv:quant-ph/0208140•Physical Review A•Jul 21, 2003•23 citations

Studies Of A Quantum Scheduling Algorithm And On Quantum Error Correction

Feng Lu

Quantum computation has been a rich field of study for decades because it promises possible spectacular advances, some of which may run counter to our classically rooted intuitions. At the same time, quantum computation is still in its infancy in both theoretical and practical areas. Efficient quantum algorithms are very limited in number and scope; no real breakthrough has yet been achieved in physical implementations. Grover's search algorithm can be applied to a wide range of problems; even problems not generally regarded as searching problems can be reformulated to take advantage of quantum parallelism and entanglement leading to algorithms which show a square root speedup over their classical counterparts. This dissertation discusses a systematic way to formulate such problems and gives as an example a quantum scheduling algorithm for an R||C_max problem. This thesis shows that quantum solution to such problems is not only feasible but in some cases advantageous. The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting only a single error per error correction cycle. Yet, time-correlated errors are common for physical implementations of quantum systems; an error corrected during a certain cycle may reoccur in a later cycle due to physical processes specific to each physical implementation of the qubits. This dissertation discusses quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm proposed allows the correction of two errors per error correction cycle, provided that one of them is time-correlated. The algorithm can be applied to any stabilizer code, perfect or non-perfect, and simplified the circuit complexity significantly comparing to the classic quantum error correction codes.

Journal of International Crisis and Risk Communication Research•Jan 1, 2007•1 citation

Simulating a Quantum Computer : Grover's Search Algorithm with Error Correction

Martin Sjöborg, Hanna Linn

Classical simulations of quantum computers give us an insight into various things such as what quantum algorithms can achieve, whether it is possible to verify that they function as postulated, the difficulties that have to be overcome before quantum computers can be realized, as well as how we can handle these difficulties. We build a Python library to simulate a quantum computer that can perform all quantum algorithms by defining an universal gate set, from which all quantum gates – and thereby all circuits – can be constructed. We implement two algorithms that both highlight two important aspects of quantum computing: Grover’s quantum search algorithm, which demonstrates the efficiency of quantum algorithms and their superiority over their classical counterparts by searching an unsorted list quadratically faster; and an error correcting code, the Shor code, which highlights the cost of correcting the possible errors in a quantum computer.We test the rigidity of Grover’s algorithm by introducing errors without correction, and find that the algorithm shows resilience to smaller 1-qubit Pauli errors, but looses its efficiency under larger errors and thus the need for Pauli error correction arise.

•Jan 1, 2018•0 citations

Family of Grover’s quantum-searching algorithms

Alberto Galindo, Miguel Angel Martin-Delgado

We introduce the concepts of Grover operators and Grover kernels to systematically analyze Grover's searching algorithms. Then we investigate a one-parameter family of quantum searching algorithms of Grover type and we show that the standard Grover algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order $O(\sqrt{N}),$ with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.

arXiv:quant-ph/0009086•Physical Review A•Nov 8, 2000•50 citations

Codeword Stabilized Quantum Codes and Their Error Correction

Yunfan Li

Quantum decoherence and errors represent some of the major challenges arising in quantum computations. Quantum error correcting codes protect quantum states against these errors and make quantum computing more reliable. One of the main problems of quantum error correction is the design of feasible decoding algorithms that can simplify error-correction for general quantum codes. The dissertation addresses decoding of general Codeword Stabilized (CWS) codes. This class of quantum codes also includes some other important classes such as additive Stabilizer codes and non-additive Union Stabilizer (USt) codes.We first design a generic error-correcting algorithm for CWS codes and analyze the number of decoding measurements and quantum gates. This algorithm performs exhaustive screening of different error patterns, similar to decoding of classical non-linear codes. For an n-qubit quantum code correcting up to t erroneous qubits, this brute-force approach consecutively tests all correctable error patterns and employs a separate n-qubit measurement in each test.The main result is a new error grouping technique that enables simultaneous testing of large groups of errors in a single measurement. To achieve this reduction, we first proceed with a new error-correction algorithm for the USt codes. Secondly, we design an algorithm that converts generic non-linear CWS codes into the simpler quasi-linear USt codes. Each decoding measurement can then either locate the actual error in a given group of errors or entirely eliminate this group. This technique yields a much simpler algorithm that exponentially reduces the number of measurements about 3^t times in any t-error-correcting CWS code.

•Jan 1, 2010•0 citations
Dynamical stabilization properties of error avoiding quantum codes are investigated beyond the perturbative regime. As an example Grover's search algorithm and its behaviour under a particular class of coherent errors are studied. Numerical examples which demonstrate that error avoiding quantum codes may be capable of stabilizing quantum algorithms well beyond the regime for which they were designed originally are presented.
Stabilizing quantum algorithms against external perturbations and preserving quantum coherence are main challenges in the area of quantum information processing. In this contribution main ideas underlying a new class of recently proposed embedded error-correcting quantum codes are discussed. These detected-jump correcting quantum codes are capable of stabilizing distinguishable qubits against spontaneous decay provided these decay processes originate from couplings to statistically independent reservoirs. Exploiting the classical information about which qubit has been affected by the environment these embedded quantum codes minimize the number of required control measurements and recovery operations as well as redundancy. Their stabilizing properties are exemplified by applying them to Grover's quantum search algorithm.
The recently introduced detected-jump-correcting quantum codes are capable of stabilizing qubit systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one-detected-jump-error-correcting quantum codes is shown and the optimal redundancy, encoding, and recovery as well as general properties of detected-jump-error-correcting quantum codes are discussed. By the use of design theory, multiple-jump-error-correcting quantum codes can be constructed. The performance of one-jump-error-correcting quantum codes under nonideal conditions is studied numerically by simulating a quantum memory and Grover's algorithm.
Quantum computation has been a rich field of study for decades because it promises possible spectacular advances, some of which may run counter to our classically rooted intuitions. At the same time, quantum computation is still in its infancy in both theoretical and practical areas. Efficient quantum algorithms are very limited in number and scope; no real breakthrough has yet been achieved in physical implementations. Grover's search algorithm can be applied to a wide range of problems; even problems not generally regarded as searching problems can be reformulated to take advantage of quantum parallelism and entanglement leading to algorithms which show a square root speedup over their classical counterparts. This dissertation discusses a systematic way to formulate such problems and gives as an example a quantum scheduling algorithm for an R||C_max problem. This thesis shows that quantum solution to such problems is not only feasible but in some cases advantageous. The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting only a single error per error correction cycle. Yet, time-correlated errors are common for physical implementations of quantum systems; an error corrected during a certain cycle may reoccur in a later cycle due to physical processes specific to each physical implementation of the qubits. This dissertation discusses quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm proposed allows the correction of two errors per error correction cycle, provided that one of them is time-correlated. The algorithm can be applied to any stabilizer code, perfect or non-perfect, and simplified the circuit complexity significantly comparing to the classic quantum error correction codes.
Classical simulations of quantum computers give us an insight into various things such as what quantum algorithms can achieve, whether it is possible to verify that they function as postulated, the difficulties that have to be overcome before quantum computers can be realized, as well as how we can handle these difficulties. We build a Python library to simulate a quantum computer that can perform all quantum algorithms by defining an universal gate set, from which all quantum gates – and thereby all circuits – can be constructed. We implement two algorithms that both highlight two important aspects of quantum computing: Grover’s quantum search algorithm, which demonstrates the efficiency of quantum algorithms and their superiority over their classical counterparts by searching an unsorted list quadratically faster; and an error correcting code, the Shor code, which highlights the cost of correcting the possible errors in a quantum computer.We test the rigidity of Grover’s algorithm by introducing errors without correction, and find that the algorithm shows resilience to smaller 1-qubit Pauli errors, but looses its efficiency under larger errors and thus the need for Pauli error correction arise.
We introduce the concepts of Grover operators and Grover kernels to systematically analyze Grover's searching algorithms. Then we investigate a one-parameter family of quantum searching algorithms of Grover type and we show that the standard Grover algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order O(N),O(\sqrt{N}),O(N​), with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.
Quantum decoherence and errors represent some of the major challenges arising in quantum computations. Quantum error correcting codes protect quantum states against these errors and make quantum computing more reliable. One of the main problems of quantum error correction is the design of feasible decoding algorithms that can simplify error-correction for general quantum codes. The dissertation addresses decoding of general Codeword Stabilized (CWS) codes. This class of quantum codes also includes some other important classes such as additive Stabilizer codes and non-additive Union Stabilizer (USt) codes.We first design a generic error-correcting algorithm for CWS codes and analyze the number of decoding measurements and quantum gates. This algorithm performs exhaustive screening of different error patterns, similar to decoding of classical non-linear codes. For an n-qubit quantum code correcting up to t erroneous qubits, this brute-force approach consecutively tests all correctable error patterns and employs a separate n-qubit measurement in each test.The main result is a new error grouping technique that enables simultaneous testing of large groups of errors in a single measurement. To achieve this reduction, we first proceed with a new error-correction algorithm for the USt codes. Secondly, we design an algorithm that converts generic non-linear CWS codes into the simpler quasi-linear USt codes. Each decoding measurement can then either locate the actual error in a given group of errors or entirely eliminate this group. This technique yields a much simpler algorithm that exponentially reduces the number of measurements about 3^t times in any t-error-correcting CWS code.