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/0110101 · Physical Review A

Decoherence in Grover’s quantum algorithm: Perturbative approach

Hiroo Azuma

Published Apr 1, 2002• 27 citations
arXivHTMLPDF

Abstract

In this paper, we study decoherence in Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers ${\ensuremath{\sigma}}_{z}$ error with probability $p(0<~p<~1)$ independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of $2Mnp$ and derive its matrix element order by order under the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ limit. (In this large-n limit, we assume p is small enough, so that $2Mnp(>~0)$ can take any real positive value or $0.)$ This approach gives us an interpretation about creation of new modes caused by ${\ensuremath{\sigma}}_{z}$ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of $2Mnp$ (perturbative parameter), where the algorithm finds the correct item with a threshold of probability ${P}_{\mathrm{th}}$ or more. It satisfies $2Mnp<(8/5)(1\ensuremath{-}{P}_{\mathrm{th}})$ around $2Mnp\ensuremath{\simeq}0$ and ${P}_{\mathrm{th}}\ensuremath{\simeq}1,$ and this linear relation is applied to a wide range of ${P}_{\mathrm{th}}$ approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.

DOI
10.1103/physreva.65.042311

Related papers

Higher-order perturbation theory for decoherence in Grover’s algorithm

Hiroo Azuma

In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability $p$ independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an $n$-qubit density operator to which Grover's iterative operation is applied $M$ times, we expand it in powers of $2Mnp$ and derive its matrix element order by order under the large-$n$ limit. [In this large-$n$ limit, we assume $p$ is small enough, so that $2Mnp$ can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter $x$ at which the algorithm finds a correct item with a given threshold of probability ${P}_{\mathrm{th}}$ or more. (We refer to this maximum value of $x$ as ${x}_{\mathrm{c}}$, a critical point of $x$.) We obtain a curve of ${x}_{\mathrm{c}}$ as a function of ${P}_{\mathrm{th}}$ by repeating this numerical calculation for many points of ${P}_{\mathrm{th}}$ and find the following facts: a tangent of the obtained curve at ${P}_{\mathrm{th}}=1$ is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}>\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near ${P}_{\mathrm{th}}=0$.

arXiv:quant-ph/0504033•Physical Review A•Oct 4, 2005•6 citations

Comment on a classical limit of Grover's algorithm

Kazuo Fujikawa, Koichiro Umetsu

A classical limit of Grover's algorithm is discussed by assuming a very rapid decoherence (or dephasing) between consecutive Grover's unitary operations, which leads pure quantum states to completely decohered mixed states. One can identify a specific element among $N$ unsorted elements by a probability of the order of unity after $k\sim N/4$ steps of classical amplification defined by the decohered mixed states, in contrast to Grover's $k\sim \pi \sqrt{N}/4$ steps in quantum mechanical amplification. This difference is caused by the loss of quantum coherence with or without the loss of entanglement depending on each case.

•Apr 26, 2018•0 citations

A classical limit of Grover’s algorithm induced by dephasing: Coherence versus entanglement

Kazuo Fujikawa, C. H. Oh, Koichiro Umetsu

A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.

arXiv:1804.10082•Modern Physics Letters A•Mar 1, 2019•6 citations

On Quantum Simulators and Adiabatic Quantum Algorithms

Sarah Mostame

This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.

•Oct 2, 2008•0 citations

Effects of decoherence and imperfections for quantum algorithms

A. A. Pomeransky, O. V. Zhirov, Dima L. Shepelyansky

We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength $ε$. The critical border $ε_c$ between two phases drops polynomially with the number of qubits $n_q$ as $ε_c \sim n_q^{-3/2}$. In the regular phase $(ε< ε_c)$ the algorithm remains robust against imperfections showing the efficiency gain $ε_c / ε$ for $ε> 2^{-n_q/2}$. In the chaotic phase $(ε> ε_c)$ the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.

arXiv:quant-ph/0407264•Jul 30, 2004•0 citations

Quantum Error Correction Protects Quantum Search Algorithms Against Decoherence

Panagiotis Botsinis, Zunaira Babar, Dimitrios Alanis +4 more

Abstract When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.

Scientific Reports•Dec 7, 2016•31 citations
In this paper, we study decoherence in Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers \ensuremathσz{\ensuremath{\sigma}}_{z}\ensuremathσz​ error with probability p(0< p< 1)p(0<~p<~1)p(0< p< 1) independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of 2Mnp2Mnp2Mnp and derive its matrix element order by order under the n\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}n\ensuremath→\ensuremath∞ limit. (In this large-n limit, we assume p is small enough, so that 2Mnp(> 0)2Mnp(>~0)2Mnp(> 0) can take any real positive value or 0.)0.)0.) This approach gives us an interpretation about creation of new modes caused by \ensuremathσz{\ensuremath{\sigma}}_{z}\ensuremathσz​ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of 2Mnp2Mnp2Mnp (perturbative parameter), where the algorithm finds the correct item with a threshold of probability Pth{P}_{\mathrm{th}}Pth​ or more. It satisfies 2Mnp<(8/5)(1\ensuremath−Pth)2Mnp<(8/5)(1\ensuremath{-}{P}_{\mathrm{th}})2Mnp<(8/5)(1\ensuremath−Pth​) around 2Mnp\ensuremath≃02Mnp\ensuremath{\simeq}02Mnp\ensuremath≃0 and Pth\ensuremath≃1,{P}_{\mathrm{th}}\ensuremath{\simeq}1,Pth​\ensuremath≃1, and this linear relation is applied to a wide range of Pth{P}_{\mathrm{th}}Pth​ approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.
In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability ppp independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an nnn-qubit density operator to which Grover's iterative operation is applied MMM times, we expand it in powers of 2Mnp2Mnp2Mnp and derive its matrix element order by order under the large-nnn limit. [In this large-nnn limit, we assume ppp is small enough, so that 2Mnp2Mnp2Mnp can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter xxx at which the algorithm finds a correct item with a given threshold of probability Pth{P}_{\mathrm{th}}Pth​ or more. (We refer to this maximum value of xxx as xc{x}_{\mathrm{c}}xc​, a critical point of xxx.) We obtain a curve of xc{x}_{\mathrm{c}}xc​ as a function of Pth{P}_{\mathrm{th}}Pth​ by repeating this numerical calculation for many points of Pth{P}_{\mathrm{th}}Pth​ and find the following facts: a tangent of the obtained curve at Pth=1{P}_{\mathrm{th}}=1Pth​=1 is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}>\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near Pth=0{P}_{\mathrm{th}}=0Pth​=0.
A classical limit of Grover's algorithm is discussed by assuming a very rapid decoherence (or dephasing) between consecutive Grover's unitary operations, which leads pure quantum states to completely decohered mixed states. One can identify a specific element among NNN unsorted elements by a probability of the order of unity after k∼N/4k\sim N/4k∼N/4 steps of classical amplification defined by the decohered mixed states, in contrast to Grover's k∼πN/4k\sim \pi \sqrt{N}/4k∼πN​/4 steps in quantum mechanical amplification. This difference is caused by the loss of quantum coherence with or without the loss of entanglement depending on each case.
A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.
This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.
We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength εεε. The critical border εcε_cεc​ between two phases drops polynomially with the number of qubits nqn_qnq​ as εc∼nq−3/2ε_c \sim n_q^{-3/2}εc​∼nq−3/2​. In the regular phase (ε<εc)(ε< ε_c)(ε<εc​) the algorithm remains robust against imperfections showing the efficiency gain εc/εε_c / εεc​/ε for ε>2−nq/2ε> 2^{-n_q/2}ε>2−nq​/2. In the chaotic phase (ε>εc)(ε> ε_c)(ε>εc​) the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.
Abstract When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.