Time-Dependent Variational Principle

• Time-Dependent Variational Principle is a framework for approximating quantum dynamics by projecting evolution onto variational manifolds with locally optimal trajectories.
• It incorporates both pure and mixed states, leveraging metrics and techniques like tensor networks, Gaussian states, and neural networks to simulate closed and open systems.
• TDVP enables efficient simulation in models such as the Fermi-Hubbard chain by using Gaussification and metric-independent projections to maintain accuracy in dynamics.

The time-dependent variational principle (TDVP) is a foundational framework for approximating the dynamics of quantum many-body systems by optimally projecting time evolution onto a chosen variational manifold. TDVP has been extended to encompass both pure and mixed states, closed and open systems, and various parameterizations (including tensor networks, Gaussian/Slater determinants, neural networks, and more). Its formal structure provides a unified approach to real-time, imaginary-time, and finite-temperature simulations.

1. Mathematical Structure of the Time-Dependent Variational Principle

The core of TDVP is a dynamical projection that restricts quantum evolution to a variational manifold, ensuring the local optimality of the trajectory in some information-theoretic or metric sense.

Pure-State TDVP

Given a family of normalized pure states $|\psi(x)\rangle \in \mathcal{H}$, parameterized smoothly by real coordinates $x = (x^1, ..., x^D)$, the TDVP starts from the exact Schrödinger equation,

$$i \ \partial_t |\psi(x)\rangle = H |\psi(x)\rangle,$$

and projects $-i H |\psi(x)\rangle$ orthogonally onto the tangent space $T_{|\psi\rangle} V$ of the variational manifold $V$. This yields the McLachlan/Dirac–Frenkel variational principle,

$$\delta \|\dot{x}^j \partial_j |\psi(x)\rangle + i H |\psi(x)\rangle \|^2 = 0,$$

which reduces to the projected equation

$$G_{jk} \dot{x}^k = -\mathrm{Im} \langle \partial_j \psi(x) | H | \psi(x) \rangle,$$

where $$G_{jk} = \mathrm{Re} \langle \partial_j \psi | \partial_k \psi \rangle$$ is the Fubini–Study metric (Kraus et al., 2012).

Mixed-State TDVP for Lindblad Dynamics

For mixed states, $\rho \in \mathcal{D}_n$ (full-rank, trace-one, Hermitian $n\times n$ matrices), dynamics are governed by the Lindblad master equation,

$\partial_t \rho = \mathcal{L}(\rho),$

where $\mathcal{L}$ is a Lindbladian superoperator. The mixed-state extension requires a Riemannian metric $M_\rho(\cdot, \cdot)$ on the density manifold. For monotone metrics, any such $M_\rho$ may be represented as

$$M_\rho(A, B) = \mathrm{tr}[A^\dagger \Omega_\rho(B)],$$

with $\Omega_\rho$ a positive, operator-monotone superoperator. On a variational manifold $V = \{\rho(x)\}$ spanned by tangent vectors $\partial_j \rho$, the projected evolution seeks $A = v^j \partial_j \rho$ such that

$$\min_{A \in T_\rho V} M_\rho(A - \mathcal{L}(\rho),\ A - \mathcal{L}(\rho)),$$

yielding the normal equations

$G_{jk}(\rho) v^k = l_j(\rho),$

with

$$G_{jk} = \mathrm{tr} [\partial_j \rho\ \Omega_\rho(\partial_k \rho)], \quad l_j = \mathrm{tr} [\partial_j \rho\ \Omega_\rho (\mathcal{L}(\rho))].$$

Hence, the evolution in parameter space is $\dot{x}^j = (G^{-1})^{jk} l_k$ (Kraus et al., 2012).

2. Choice of Information Metrics and the Family of Trajectories

In contrast to pure-state TDVP, where the Fubini–Study metric is canonical, no unique monotone metric exists for mixed states; the family of possible trajectories is indexed by the chosen monotone Riemannian metric. The Petz monotone metrics form a natural, operationally motivated family, parameterized by a real parameter $\alpha \in [0,1]$:

$$\Omega_\rho^{(\alpha)}(\sigma) = \tfrac{1}{2} \left( \rho^{-\alpha} \sigma\, \rho^{\alpha-1} + \rho^{\alpha-1} \sigma\, \rho^{-\alpha} \right),$$

which encompasses quantum Fisher, Bures, and Kubo–Mori metrics as particular cases. For each such $\alpha$-metric, the TDVP defines a distinct, locally optimal dissipative trajectory (Kraus et al., 2012).

However, in certain cases—most importantly, for fermionic Gaussian states (see below)—the TDVP dynamics are independent of the chosen monotone metric.

3. Specialization: Fermionic Gaussian States and Metric Independence

When the variational manifold $V_G$ comprises all fermionic Gaussian states (fGS) on $N$ modes, every state is characterized by an antisymmetric covariance matrix $\Gamma_{kl} = (i/2) \mathrm{tr} ([c_k, c_l]\ \rho)$, where $\{ c_k \}$ are $2N$ Majorana operators. Gaussification is defined as the unique Gaussian state matching a given covariance matrix.

The principal result asserts that the optimal TDVP evolution of a general Lindblad equation, projected on $V_G$, is independent of the chosen Petz monotone metric: The projection at each step corresponds precisely to Gaussification:

$$\rho(t + \delta t) \simeq \mathcal{G}[ \rho(t) + \delta t\ \mathcal{L}(\rho(t)) ],$$

where $\mathcal{G}$ denotes the Gaussification map. This equivalence arises because the Gram matrix and the driving vector, computed per the monotone metric, enforce the covariance-matching condition dictated by Wick’s theorem (Kraus et al., 2012). Thus, for fGS, the TDVP projected dissipative dynamics are geometrically and physically canonical.

4. Physical Application: Hubbard Model with Spin Decoherence

The practical utility of the mixed-state TDVP is demonstrated for a 1D Fermi-Hubbard chain subject to spin-flip (decoherence) Lindblad operators. The combined Hamiltonian-dissipator dynamics read:

$$\partial_t \rho = -i [H, \rho] + \kappa \sum_x \left( j_x \rho j_x^\dagger - \tfrac{1}{2} \{ j_x^\dagger j_x, \rho \} \right), \quad j_x = a_{x,\uparrow}^\dagger a_{x,\downarrow},$$

where $H$ is the standard Hubbard model with hopping $J$, interaction $U$, and chemical potential $\mu$.

At each time step, the fGS covariance matrix $\Gamma(t)$ evolves via:

$$\partial_t \Gamma = A(\Gamma) + B(\Gamma) \Gamma + \Gamma B(\Gamma)^\mathsf{T},$$

with $A$ and $B$ determined by contractions of $H$ and the Lindblad operators. The Gaussification at each step ensures closure of the equations.

Physical observables—such as the purity $\mathrm{tr}\,\rho^2$, spin magnetization $\langle n_\uparrow-n_\downarrow \rangle$, and double occupancy $\langle n_\uparrow n_\downarrow \rangle$—relax on timescales set by dissipation rate $\kappa$, with interaction $U$ generating nontrivial time-dependent many-body correlations manifest in $\Gamma(t)$. Crucially, after initial transient (non-Gaussian) dynamics, the TDVP-Gaussified trajectory closely matches the exact evolution both in covariance space and in fidelity for finite system sizes; all $\alpha$-metrics yield coincident results in this setting (Kraus et al., 2012).

5. Broader Impact and Computational Implications

The time-dependent variational principle provides a systematic, variationally optimal, and geometrically consistent methodology for both unitary and dissipative quantum dynamics, particularly when formulated with monotone Riemannian metrics for mixed states. Its formalism naturally incorporates both closed and open quantum system dynamics, enables efficient simulation within tractable variational manifolds (such as fGS), and allows for rigorous control of approximation when the variational space is physically motivated.

The availability of a family of possible metrics for dissipative systems raises important questions about the physical interpretability of the various TDVP flows in general non-Gaussian mixed-state manifolds. However, for variational classes with high physical relevance and symmetry (e.g., fermionic Gaussian, certain bosonic Gaussian), the TDVP flow becomes effectively unique and operationally robust.

Moreover, the practical success of the TDVP-Gaussification approach in benchmark problems such as spin-dephased Hubbard chains, as well as its flexibility to other models and variational classes, underscores its central role in the modern simulation toolbox for quantum nonequilibrium and open-system dynamics.

6. Connections to Related Variational Methods

TDVP forms the backbone of a large class of algorithms across quantum dynamics simulation:

• Tensor network evolution: Projection of dynamics onto the manifold of matrix product states, tree tensor networks, or projected entangled-pair states.
• Gaussian and Slater determinants: Application to both fermionic and bosonic Gaussian states, yielding tractable nonlinear ODEs for covariance dynamics (Kraus et al., 2012).
• Neural network wavefunctions: Generalization to unrestricted variational classes, including autoregressive neural network parametrizations, and various stochastic and metric-based projections for mixed-state dynamics.
• Finite-temperature and imaginary-time evolution: Analytic continuation formalism and thermal dynamics via imaginary-time TDVP or related path-integral constructions.

The unifying feature is the use of geometric projection—often but not always with monotone metrics—to distill quantum evolution into a locally optimal, efficiently simulatable form within a given variational class.

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References:

• "A time-dependent variational principle for dissipative dynamics" (Kraus et al., 2012)