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:1706.05243 · Physical Review A

Pechukas-Yukawa approach to the evolution of the quantum state of a parametrically perturbed system

Mumnuna A. Qureshi, Johnny Q. Zhong, Zihad Qureshi, Peter Mason, Joseph J. Betouras, Alexandre M. Zagoskin

Published Mar 16, 2018• 3 citations
arXivHTMLPDF

Abstract

We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.

DOI
10.1103/physreva.97.032117

Related papers

The evolution of Liouville von Neumann master equations in the Pechukas–Yukawa framework

Mumnuna A. Qureshi

This paper presents a novel formalism for the out-of-equilibrium dynamics of the density matrix, capable of describing highly entangled many-body interactions. The evolution of quantum states is evaluated via eigenvalue dynamics of a general Hamiltonian system, perturbed by a parametrically evolving variable λ(t) that carries the time-dependence. This is achieved using the Pechukas–Yukawa mapping of the evolution of the energy levels governed by their initial conditions on a generalized Calogero–Sutherland model of a 1D classical gas. As such, quantum systems can be described exactly in their entirety from eigenvalue dynamics. Under this description, we provide an improved understanding of the relationship between nonequilibrium quantum phase transitions and decoherence, which has significant impacts on a wide range of applications.

arXiv:2510.21939•Low Temperature Physics•Dec 1, 2025•0 citations

Level dynamics of a non-equilibrium adiabatically evolving system using the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

Mumnuna A. Qureshi, Johnny Q. Zhong, Joseph J. Betouras +1 more

Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).

•Oct 7, 2016•0 citations

Bogoliubov-Born-Green-Kirkwood-Yvon chain and kinetic equations for the level dynamics in an externally perturbed quantum system

Mumnuna A. Qureshi, Johnny Q. Zhong, Joseph J. Betouras +1 more

Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as $1/N$, where $N$ is the number of the ``Pechukas gas particles'' (i.e., energy levels in the system).

arXiv:1610.02380•Physical Review A•Mar 22, 2017•4 citations

Modeling an adiabatic quantum computer

Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than |\dotλ|^γ, where |\dotλ| is the adiabaticity parameter. The scaling exponent for the escape probability is γ= 1/2 for all levels, except the edge (bottom and top) ones, where γ

arXiv:quant-ph/0701161•Jan 22, 2007•1 citation

Modeling an Adiabatic Quantum Computer via an Exact Map to a Gas of Particles

Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}{|}^{\ensuremath{\gamma}}$, where $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}|$ is the adiabaticity parameter. The scaling exponent for the escape probability is $\ensuremath{\gamma}=\frac{1}{2}$ for all levels, except the edge (bottom and top) ones, where $\ensuremath{\gamma}\ensuremath{\lesssim}\frac{1}{3}$. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.

Physical Review Letters•Mar 23, 2007•27 citations

Pechukas-Yukawa formalism for Landau-Zener transitions in the presence of external noise

Mumnuna A. Qureshi, Johnny Q. Zhong, Peter Mason +2 more

Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.

arXiv:1803.05034•Physical Review A•Jul 23, 2018•3 citations
We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter \ensuremathλ(t)\ensuremath{\lambda}(t)\ensuremathλ(t). Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of d\ensuremathλ/dtd\ensuremath{\lambda}/dtd\ensuremathλ/dt and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.
This paper presents a novel formalism for the out-of-equilibrium dynamics of the density matrix, capable of describing highly entangled many-body interactions. The evolution of quantum states is evaluated via eigenvalue dynamics of a general Hamiltonian system, perturbed by a parametrically evolving variable λ(t) that carries the time-dependence. This is achieved using the Pechukas–Yukawa mapping of the evolution of the energy levels governed by their initial conditions on a generalized Calogero–Sutherland model of a 1D classical gas. As such, quantum systems can be described exactly in their entirety from eigenvalue dynamics. Under this description, we provide an improved understanding of the relationship between nonequilibrium quantum phase transitions and decoherence, which has significant impacts on a wide range of applications.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N1/N1/N, where NNN is the number of the ``Pechukas gas particles'' (i.e., energy levels in the system).
We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than |\dotλ|^γ, where |\dotλ| is the adiabaticity parameter. The scaling exponent for the escape probability is γ= 1/2 for all levels, except the edge (bottom and top) ones, where γ
We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}{|}^{\ensuremath{\gamma}}$, where $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}|$ is the adiabaticity parameter. The scaling exponent for the escape probability is $\ensuremath{\gamma}=\frac{1}{2}$ for all levels, except the edge (bottom and top) ones, where $\ensuremath{\gamma}\ensuremath{\lesssim}\frac{1}{3}$. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.
Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.