Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
Conclusions
References
All Topics
Mathematics
Probability

Scalable quantum field simulations of conditioned systems

Profile image of Andre CarvalhoAndre Carvalho

2009, Physical Review A

https://doi.org/10.1103/PHYSREVA.80.013606
visibility

…

description

5 pages

link

1 file

Abstract

We demonstrate a technique for performing stochastic simulations of conditional master equations. The method is scalable for many quantum field problems and therefore allows first-principles simulations of multimode bosonic fields undergoing continuous measurement, such as those controlled by measurement-based feedback. As examples, we demonstrate a 53-fold speed increase for the simulation of the feedback cooling of a single trapped particle, and the feedback cooling of a quantum field with 32 modes, which would be impractical using previous brute force methods.

Related papers

Quantum Simulation of Dissipative Processes without Reservoir Engineering
A. Del Campo

We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of our method providing specific error bounds that quantify its accuracy.

View PDFchevron_right
Simulation of open quantum system dynamics based on the generalized quantum master equation on quantum computing devices
Sabre Kais

arXiv (Cornell University), 2022

We present a quantum algorithm based on the Generalized Quantum Master Equation (GQME) approach to simulate open quantum system dynamics on noisy intermediatescale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining 1

View PDFchevron_right
STOCHASTIC GAUGE: A NEW TECHNIQUE FOR QUANTUM SIMULATIONS
Peter Drummond

Laser Spectroscopy - Proceedings of the XVI International Conference, 2004

We review progress towards direct simulation of quantum dynamics in many-body systems, using recently developed stochastic gauge techniques. We consider master equations, canonical ensemble calculations and reversible quantum dynamics are compared, as well the general question of strategies for choosing the gauge.

View PDFchevron_right
Quantum dynamics with stochastic gauge simulations
Peter Drummond

Journal of Optics B: Quantum and Semiclassical Optics, 2003

The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite stochastic time-evolution equations, equivalent to master equations, for many systems including quantum time evolution. The method is illustrated with a variety of simple examples ranging from astrophysical molecular hydrogen production, through to the topical problem of Bose-Einstein condensation in an optical trap and the resulting quantum dynamics.

View PDFchevron_right
FAST TRACK COMMUNICATION: Deterministic constant-temperature dynamics for dissipative quantum systems
Alessandro Sergi

Journal of Physics A-mathematical and Theoretical, 2007

A novel method is introduced in order to treat the dissipative dynamics of quantum systems interacting with a bath of classical degrees of freedom. The method is based upon an extension of the Nos\`e-Hoover chain (constant temperature) dynamics to quantum-classical systems. Both adiabatic and nonadiabatic numerical calculations on the relaxation dynamics of the spin-boson model show that the quantum-classical Nos\`e-Hoover chain dynamics represents the thermal noise of the bath in an accurate and simple way. Numerical comparisons, both with the constant energy calculation and with the quantum-classical Brownian motion treatment of the bath, show that the quantum-classical Nos\`e-Hoover Chain dynamics can be used to introduce dissipation in the evolution of a quantum subsystem even with just one degree of freedom for the bath. The algorithm can be computationally advantageous in modeling, within computer simulation, the dynamics of a quantum subsystem interacting with complex molecular environments.

View PDFchevron_right
A C++ library using quantum trajectories to solve quantum master equations
Todd Brun

Computer physics communications, 1997

Quantum trajectory methods can be used for a wide range of open quantum systems to solve the master equation by unraveling the density operator evolution into individual stochastic trajectories in Hilbert space. This C++ class library offers a choice of integration algorithms for three important unravelings of the master equation. Different physical systems are modeled by different Hamiltonians and environment operators. The program achieves flexibility and user friendliness, without sacrificing execution speed, through the way it represents operators and states in Hilbert space. Primary operators, implemented in the form of simple routines acting on single degrees of freedom, can be used to build up arbitrarily complex operators in product Hilbert spaces with arbitrary numbers of components. Standard algebraic notation is used to build operators and to perform arithmetic operations on operators and states. States can be represented in a local moving basis, often leading to dramatic savings of computing resources. The state and operator classes are very general and can be used independently of the quantum trajectory algorithms. Only a rudimentary knowledge of C++ is required to use this package. *

View PDFchevron_right
Feedback cooling of atomic motion in cavity QED
Tanmoy Bhattacharya

Physical Review A, 2006

We consider the problem of controlling the motion of an atom trapped in an optical cavity using continuous feedback. In order to realize such a scheme experimentally, one must be able to perform state estimation of the atomic motion in real time. While in theory this estimate may be provided by a stochastic master equation describing the full dynamics of the observed system, integrating this equation in real time is impractical. Here we derive an approximate estimation equation for this purpose, and use it as a drive in a feedback algorithm designed to cool the motion of the atom. We examine the effectiveness of such a procedure using full simulations of the cavity QED system, including the quantized motion of the atom in one dimension.

View PDFchevron_right
Quantum limits of stochastic cooling of a bosonic gas
Denis Ivanov

Physical Review A, 2003

The quantum limits of stochastic cooling of trapped atoms are studied. The energy subtraction due to the applied feedback is shown to contain an additional noise term due to atom-number fluctuations in the feedback region. This novel effect is shown to dominate the cooling efficiency near the condensation point. Furthermore, we show first results that indicate that Bose-Einstein condensation could be reached via stochastic cooling.

View PDFchevron_right
Quantum simulation of quantum field theory using continuous variables
George Siopsis

Physical Review A, 2015

Much progress has been made in the field of quantum computing using continuous variables over the last couple of years. This includes the generation of extremely large entangled cluster states (10,000 modes, in fact) as well as a fault tolerant architecture. This has led to the point that continuous-variable quantum computing can indeed be thought of as a viable alternative for universal quantum computing. With that in mind, we present a new algorithm for continuousvariable quantum computers which gives an exponential speedup over the best known classical methods. Specifically, this relates to efficiently calculating the scattering amplitudes in scalar bosonic quantum field theory, a problem that is believed to be hard using a classical computer. Building on this, we give an experimental implementation based on cluster states that is feasible with today's technology.

View PDFchevron_right
Continuously decoupling single-qubit operations from a perturbing thermal bath of scalar bosons
Reginaldo Napolitano

Physical Review A, 2007

We investigate the use of continuously-applied external fields to maximize the fidelity of quantum logic operations performed on a decohering qubit. Assuming a known error operator and an environment represented by a scalar boson field at a finite temperature, we show how decoherence during logical operations can be efficiently reduced by applying a superposition of two external vector fields: one rotating orthogonally to the direction of the other, which remains static. The required field directions, frequency of rotation and amplitudes to decouple noise dynamically are determined by the coupling constants and the desired logical operation. We illustrate these findings numerically for a Hadamard quantum gate and an environment with ohmic spectral density.

View PDFchevron_right

References (33)

  1. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).
  2. G. J. Milburn, Schrödinger Machines (W. H. Freeman, 1997).
  3. V. P. Belavkin, in Information Complexity and Control in Quantum Systems, edited by A. Blaquiere, S. Diner, and G. Lochack (1987), pp. 311-329.
  4. H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993).
  5. H. M. Wiseman, Phys. Rev. A 49, 2133 (1994).
  6. A. C. Doherty and K. Jacobs, Phys. Rev. A 60, 2700 (1999).
  7. R. van Handel, S. J. K., and H. Mabuchi, J. Opt. B: Quantum Semiclass. Opt. 7, S179 (2005).
  8. N. V. Morrow, S. K. Dutta, and G. Raithel, Phys. Rev. Lett. 88, 093003 (2002).
  9. M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, Phys. Rev. Lett. 89, 133602 (2002).
  10. J. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, Phys. Rev. Lett. 91, 250801 (2003).
  11. J. E. Reiner, W. P. Smith, L. A. Orozco, H. M. Wiseman, and J. Gambetta, Phys. Rev. A 70, 023819 (2004).
  12. P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, Phys. Rev. Lett. 96, 043003 (2006).
  13. D. Felinto, C. W. Chou, J. Laurat, E. W. Schomburg, H. De Riedmatten, and H. J. Kimble, Nature Physics 2, 844 (2006).
  14. C. Gardiner, Handbook of Stochastic Methods (Springer- Verlag, 1983).
  15. C. Gardiner, Quantum Noise (Springer-Verlag, 1991).
  16. H. M. Wiseman and L. K. Thomsen, Phys. Rev. Lett. 86, 1143 (2001).
  17. P.D.Drummond and S.J.Carter, J. Opt. Soc. Am. B 4, 1565 (1997).
  18. J. J. Hope, Phys. Rev. A 64, 053608 (2001).
  19. J. Corney and P. Drummond, Phys. Rev. B 73, 125112 (2006).
  20. L. Bouten, R. van Handel, and M. James, SIAM J. Con- trol Optim. 46, 2199 (2007).
  21. H. Carmichael, An open systems approach to quantum optics, Lecture Notes in Physics m 18 (Springer-Verlag, Berlin, 1993).
  22. J. Gambetta and H. M. Wiseman, J. Opt. B: Quantum Semiclass. Opt. 7, S250 (2005).
  23. T. P. McGarty, Stochastic systems and state estimation (Wiley, New York, 1974).
  24. P. Deuar and P. D. Drummond, Phys. Rev. A 66, 6902 (2002).
  25. A. Barchielli and V. P. Belavkin, J. Phys. A: Math. Gen. 24, 1495 (1991).
  26. N. Gisin, Phys. Rev. Lett. 52, 1657 (1984).
  27. L. Disi, Physics Letters A 129, 419 (1988), ISSN 0375-9601, URL http://www.sciencedirect. com/science/article/B6TVM-46MDDKN-1J7/2/ 6eb51e0b57b3d36dd6099f84c736496b.
  28. L. Disi, Physics Letters A 114, 451 (1986), ISSN 0375-9601, URL http://www.sciencedirect. com/science/article/B6TVM-46VPYW4-2W/2/ 6af1d4b1b9a9df802aee32035bef8f3a.
  29. H. M. Wiseman and G. J. Milburn, Phys. Rev. A 47, 642 (1993).
  30. S. D. Wilson, A. R. R. Carvalho, J. J. Hope, and M. R. James, Physical Review A 76, 013610 (2007).
  31. N. Trivedi and D. M. Ceperley, Phys. Rev. B 41, 4552 (1990).
  32. M. J. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond, S. M. Tan, M. J. Collett, D. F. Walls, and R. Graham, Phys. Rev. A 58, 4824 (1998).
  33. Project website http://www.xmds.org.

Related papers

Algorithmic Cooling of a Quantum Simulator
Jacob Taylor

2012

Controlled quantum mechanical devices provide a means of simulating more complex quantum systems exponentially faster than classical computers. Such "quantum simulators" rely heavily upon being able to prepare the ground state of Hamiltonians, whose properties can be used to calculate correlation functions or even the solution to certain classical computations. While adiabatic preparation remains the primary means of producing such ground states, here we provide a different avenue of preparation: cooling to the ground state via simulated dissipation. This is in direct analogy to contemporary efforts to realize generalized forms of simulated annealing in quantum systems.

View PDFchevron_right
Number-phase Wigner representation for scalable stochastic simulations of controlled quantum systems
Andre Carvalho

Physical Review A, 2012

Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as BEC and atom lasers, full quantum field simulations must rely on scalable stochastic methods whose convergence time is restricted by the use of representations based on coherent states. Here we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stochastic method based on the NPW can converge over an order of magnitude longer and more precisely than its coherent equivalent. This opens the possibility of realistic simulations of controlled multi-mode quantum systems.

View PDFchevron_right
Universal Cooling Scheme for Quantum Simulation
Tin-Lun Ho

At present, there is a worldwide effort to use cold atoms to simulate strongly correlated quantum many-body systems. It is hoped that these "simulations" will provide solutions to many unsolved problems. However, the relevant energy scales in most of these experiments are so small that one has to go to entropy regimes far below those achievable today. Here, we present a general scheme to extract entropy directly from the region of interest. The late stage of this process is equivalent to a continuous "evaporation", and is able to combat intrinsic heating of the system. For illustration, we show how to cool a weak coupling BCS superfluid (with $T_{c}\sim 10^{-10}$K) to $10^{-11}$K with this simple procedure, with entropy per particle as low as $5\times 10^{-4}k_{B}$ in the superfluid region. Comment: 4 pages, 5 figures

View PDFchevron_right
Stochastic master equation for a probed system in a cavity
Klaus Mølmer

Physical Review A, 2008

We present a detailed derivation of the stochastic master equation determining the time evolution of the state of a general quantum system, which is placed inside a cavity and subjected to indirect measurements by monitoring the state of electromagnetic radiation transmitted through the cavity. The derivation is based on the physics involved and the final result is stated in terms of the physical parameters of the setup. To illustrate the predictions contained in the equation, we solve it analytically for a specific system, and we demonstrate quantum jumps and freezing of the internal coherent dynamics of the system as a result of continued measurements.

View PDFchevron_right
Proposal of a Computational Approach for Simulating Thermal Bosonic Fields in Phase Space
Alessandro Sergi

Physics, 2019

When a quantum field is in contact with a thermal bath, the vacuum state of the field may be generalized to a thermal vacuum state, which takes into account the thermal noise. In thermo field dynamics, this is realized by doubling the dimensionality of the Fock space of the system. Interestingly, the representation of thermal noise by means of an augmented space is also found in a distinctly different approach based on the Wigner transform of both the field operators and density matrix, which we pursue here. Specifically, the thermal noise is introduced by augmenting the classical-like Wigner phase space by means of Nosé-Hoover chain thermostats, which can be readily simulated on a computer. In this paper, we illustrate how this may be achieved and discuss how non-equilibrium quantum thermal distributions of the field modes can be numerically simulated.

View PDFchevron_right

Related topics

  • Stochastic Process
  • Physics
  • Statistical Physics
  • Stochastic Simulation
  • Bose-Einstein Condensate
  • QUantum state engineering
  • Master Equation

    • Cited by

    Continuous measurement feedback control of a Bose-Einstein condensate using phase-contrast imaging
    J. Hope

    Physical Review A, 2009

    We consider the theory of feedback control of a Bose-Einstein condensate (BEC) confined in a harmonic trap under a continuous measurement constructed via non-destructive imaging. A filtering theory approach is used to derive a stochastic master equation (SME) for the system from a general Hamiltonian based upon system-bath coupling. Numerical solutions for this SME in the limit of a single atom show that the final steady state energy is dependent upon the measurement strength, the ratio of photon kinetic energy to atomic kinetic energy, and the feedback strength. Simulations indicate that for a weak measurement strength, feedback can be used to overcome heating introduced by the scattering of light, thereby allowing the atom to be driven towards the ground state.

    View PDFchevron_right
    Number-phase Wigner representation for efficient stochastic simulations
    Andre Carvalho

    Physical Review A, 2010

    Phase-space representations based on coherent states (P, Q, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high dimensional quantum systems. However many problems using these techniques remain intractable over long integrations times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase space representations. This process requires an effective sampling of a non-classical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.

    View PDFchevron_right
    Number-phase Wigner representation for scalable stochastic simulations of controlled quantum systems
    Andre Carvalho

    Physical Review A, 2012

    Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as BEC and atom lasers, full quantum field simulations must rely on scalable stochastic methods whose convergence time is restricted by the use of representations based on coherent states. Here we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stochastic method based on the NPW can converge over an order of magnitude longer and more precisely than its coherent equivalent. This opens the possibility of realistic simulations of controlled multi-mode quantum systems.

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved