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:2205.05251 · Physical Review A

Gradient-based reconstruction of molecular Hamiltonians and density matrices from time-dependent quantum observables

Wucheng Zhang, Ilia Tutunnikov, Ilya Sh. Averbukh, Roman V. Krems

Published Jun 21, 2022• 1 citation
arXivHTMLPDF

Abstract

We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters $\ensuremath{\alpha}$. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters $P$. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable $\mathcal{O}(t)$. We show that it is possible to obtain closed-form expressions for the gradients of the distance between $\mathcal{O}(t)$ and a calculated observable with respect to $\ensuremath{\alpha}, P$, and all elements of the system density matrix, whether for pure or mixed states. These gradients can be used in projected gradient descent to infer $\ensuremath{\alpha}, P$, and the relevant density matrix from dynamical observables. We combine this approach with random phase wave function approximation to obtain closed-form expressions for gradients that can be used to infer population distributions from averaged time-dependent observables in problems with a large number of quantum states participating in dynamics. The approach is illustrated by determining the temperature of molecular gas (initially, in thermal equilibrium at room temperature) from laser-induced time-dependent molecular alignment.

DOI
10.1103/physreva.105.l060802

Related papers

Efficient Quantum Optimization via Dynamical Simulation

Ahmet Burak Çatlı, Sophia Simon, Nathan Wiebe

We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f : R N → R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O ( N 2 κ 2 / h x 2 ϵ ) queries to a phase oracle to find an ϵ -approximate local optimum, where κ is the condition number of the Hessian matrix and h x is the discretization spacing. In contrast, we show that methods based on gradient descent require O ( N 3 / 2 ( 1 / ϵ ) κ log ( 3 ) / 4 ) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

arXiv:2502.04285•PRX Quantum•May 14, 2026•1 citation

Evolved quantum Boltzmann machines

Michele Minervini, Dhrumil Patel, Mark M. Wilde

We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians $G(θ)$ and $H(ϕ)$, an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian $G(θ)$ followed by unitary evolution according to the second Hamiltonian $H(ϕ)$. Alternatively, one can think of it as first realizing imaginary time evolution according to $G(θ)$ followed by real time evolution according to $H(ϕ)$. After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.

arXiv:2501.03367•Physical Review A•Jan 21, 2026•2 citations

Gradient algorithm for Hamiltonian identification of open quantum systems

Shibei Xue, Re-Bing Wu, Shan Ma +2 more

In this paper, we present a gradient algorithm for identifying unknown parameters in an open quantum system from the measurements of time traces of local observables. The open system dynamics is described by a general Markovian master equation based on which the Hamiltonian identification problem can be formulated as minimizing the distance between the real time traces of the observables and those predicted by the master equation. The unknown parameters can then be learned with a gradient descent algorithm from the measurement data. We verify the effectiveness of our algorithm in a circuit QED system described by a Jaynes-Cummings model whose Hamiltonian identification has been rarely considered. We also show that our gradient algorithm can learn the spectrum of a non-Markovian environment based on an augmented system model.

arXiv:1905.09990•Physical Review A•Feb 10, 2021•18 citations

Beyond quantum microcanonical statistics

Barbara Fresch, Giorgio J. Moro

Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.

arXiv:1011.1858•The Journal of Chemical Physics•Feb 3, 2011•10 citations

Exact Work Characteristic Functions for Time-Dependent Bosonic Quadratic Hamiltonians

F. Nicacio, R. N. P. Maia

We derive an exact analytical expression for the characteristic function of work in closed many-body bosonic systems governed by general time-dependent quadratic Hamiltonians. This is made possible by adopting a unified prescription in which all operators appearing in the characteristic function are treated as elements of the inhomogeneous Metaplectic group. The esulting expression involves a single symplectic matrix, a single phase-space displacement, and a single accumulated phase. It holds for any number of degrees of freedom, arbitrary time dependence of the quadratic Hamiltonian, and any initial thermal equilibrium state of the system. The formalism trades the explicit time-ordering problem associated with the unitary evolution for the determination of a symplectic path in phase space, from which the corresponding driving Hamiltonian can be reconstructed. We also discuss the generalization to arbitrary nonequilibrium initial states within the two-projective-measurement scheme for quantum work. The same construction yields Loschmidt echoes and mixed-state total phases. Using the derived characteristic function, we obtain formulas for the Helmholtz free energy, the Jarzynski equality, the mean work, the work variance, and the nonequilibrium lag. Undisplaced Hamiltonians, sudden quenches, constant-Hessian protocols, and the adiabatic limit follow as special cases of the general result. As an illustration, we analyze a one-dimensional system constructed from an unconventional symplectic path and exhibiting physically relevant limiting cases. We then show how affine and purely symplectic contributions affect work statistics and irreversibility.

quant-pharXiv:2609.16435•Sep 16, 2026•0 citations

Quantum mechanical density matrix

J. W. Halleý

For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.

Cambridge University Press eBooks•Nov 16, 2006•0 citations
We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters \ensuremathα\ensuremath{\alpha}\ensuremathα. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters PPP. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable O(t)\mathcal{O}(t)O(t). We show that it is possible to obtain closed-form expressions for the gradients of the distance between O(t)\mathcal{O}(t)O(t) and a calculated observable with respect to \ensuremathα,P\ensuremath{\alpha}, P\ensuremathα,P, and all elements of the system density matrix, whether for pure or mixed states. These gradients can be used in projected gradient descent to infer \ensuremathα,P\ensuremath{\alpha}, P\ensuremathα,P, and the relevant density matrix from dynamical observables. We combine this approach with random phase wave function approximation to obtain closed-form expressions for gradients that can be used to infer population distributions from averaged time-dependent observables in problems with a large number of quantum states participating in dynamics. The approach is illustrated by determining the temperature of molecular gas (initially, in thermal equilibrium at room temperature) from laser-induced time-dependent molecular alignment.
We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function f : R N → R by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of O ( N 2 κ 2 / h x 2 ϵ ) queries to a phase oracle to find an ϵ -approximate local optimum, where κ is the condition number of the Hessian matrix and h x is the discretization spacing. In contrast, we show that methods based on gradient descent require O ( N 3 / 2 ( 1 / ϵ ) κ log ( 3 ) / 4 ) queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of f by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.
We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians G(θ)G(θ)G(θ) and H(ϕ)H(ϕ)H(ϕ), an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian G(θ)G(θ)G(θ) followed by unitary evolution according to the second Hamiltonian H(ϕ)H(ϕ)H(ϕ). Alternatively, one can think of it as first realizing imaginary time evolution according to G(θ)G(θ)G(θ) followed by real time evolution according to H(ϕ)H(ϕ)H(ϕ). After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.
In this paper, we present a gradient algorithm for identifying unknown parameters in an open quantum system from the measurements of time traces of local observables. The open system dynamics is described by a general Markovian master equation based on which the Hamiltonian identification problem can be formulated as minimizing the distance between the real time traces of the observables and those predicted by the master equation. The unknown parameters can then be learned with a gradient descent algorithm from the measurement data. We verify the effectiveness of our algorithm in a circuit QED system described by a Jaynes-Cummings model whose Hamiltonian identification has been rarely considered. We also show that our gradient algorithm can learn the spectrum of a non-Markovian environment based on an augmented system model.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.
We derive an exact analytical expression for the characteristic function of work in closed many-body bosonic systems governed by general time-dependent quadratic Hamiltonians. This is made possible by adopting a unified prescription in which all operators appearing in the characteristic function are treated as elements of the inhomogeneous Metaplectic group. The esulting expression involves a single symplectic matrix, a single phase-space displacement, and a single accumulated phase. It holds for any number of degrees of freedom, arbitrary time dependence of the quadratic Hamiltonian, and any initial thermal equilibrium state of the system. The formalism trades the explicit time-ordering problem associated with the unitary evolution for the determination of a symplectic path in phase space, from which the corresponding driving Hamiltonian can be reconstructed. We also discuss the generalization to arbitrary nonequilibrium initial states within the two-projective-measurement scheme for quantum work. The same construction yields Loschmidt echoes and mixed-state total phases. Using the derived characteristic function, we obtain formulas for the Helmholtz free energy, the Jarzynski equality, the mean work, the work variance, and the nonequilibrium lag. Undisplaced Hamiltonians, sudden quenches, constant-Hessian protocols, and the adiabatic limit follow as special cases of the general result. As an illustration, we analyze a one-dimensional system constructed from an unconventional symplectic path and exhibiting physically relevant limiting cases. We then show how affine and purely symplectic contributions affect work statistics and irreversibility.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.