---
title: Tangent-Space Excitation Ansatz
url: https://www.emergentmind.com/topics/tangent-space-excitation-ansatz
type: topic
---

# Tangent-Space Excitation Ansatz

Tangent-Space Excitation Ansatz denotes a class of variational constructions in which low-lying excited states are represented as momentum-resolved tangent vectors around an optimized many-body reference state. In the tensor-network setting, the reference state is typically a uniform matrix product state (uMPS) or a projected entangled-pair state (PEPS), and the excitation is built by replacing one local tensor \(A\) by a variational tensor \(B\), then superposing all translated insertions with a plane-wave phase. In recent work on parametrized quantum circuits, the same geometric idea is implemented by inserting one additional local gate at a chosen depth and projecting onto symmetry sectors. Across these settings, the ansatz converts excited-state estimation into an effective eigenvalue problem on a tangent-space subspace. A related but distinct line of work on dissipative systems provides a broader tangent-space interpretation, identifying dynamically relevant tangent directions that are hyperbolically isolated from rapidly decaying ones [1810.07006] [1507.02151] [2507.07646] [1107.2567].

## 1. Geometric premise and terminology

The starting point is that the variational family under consideration is a manifold rather than a linear subspace of Hilbert space. For a uniform MPS, the state is parameterized by a single local tensor \(A\), and a tangent vector is obtained by differentiating with respect to those tensor parameters. In explicit lattice form, the tangent vector is a sum over all lattice positions where one tensor \(A\) is replaced by a tensor \(B\):
\[
|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .
\]
The excitation ansatz is then obtained by momentum projection,
\[
|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,
\]
so that the variational state is a translation eigenstate in the thermodynamic limit [1810.07006].

This construction is explicitly connected to the quasiparticle picture. The relevant intuition is close to the single-mode approximation, in which a local operator is translated over the system to create a momentum eigenstate, but the tangent-space formulation is more expressive because the perturbation acts on the optimized variational background itself. In the circuit setting, the same logic is described as a tangent-space construction because varying one local gate while keeping the rest of the optimized circuit fixed generates a derivative-like direction on the circuit manifold [2507.07646].

A distinct terminological issue appears in the MPS literature. The “excitation ansatz” and the “tangent-space excitation ansatz” are often structurally the same construction, but the single-site excitation space used in some formulations is related to, and not identical with, the tangent space used in time-dependent variational principle (TDVP). In particular, tangent-space vectors typically share the ground state’s quantum numbers, whereas the excitation space is designed specifically for physical excitations and can include different quantum numbers [2509.06241].

## 2. Uniform MPS formulation

For uniform MPS, gauge freedom is central. The tensor transformation \(A^s \rightarrow X^{-1}A^sX\) leaves the state invariant, and the tangent tensor inherits an overcomplete representation. In the excitation problem, the variational tensor \(B\) obeys the gauge redundancy
\[
B \rightarrow B + Y A_R - e^{ip} A_L Y .
\]
To remove this redundancy, one imposes a gauge-fixing condition, typically a left gauge, and parameterizes the admissible tangent tensors in a reduced basis \(B = V_L X\), where \(V_L\) spans the null space orthogonal to \(A_L\). In that parametrization, the norm simplifies to
\[
\langle \Phi_{p'}(X')|\Phi_p(X)\rangle = 2\pi\delta(p-p')\,(X'^\dagger X),
\]
so the effective norm matrix becomes the identity [1810.07006].

The excitation energy is obtained variationally by minimizing the Rayleigh quotient
\[
\omega(p)= \frac{\langle \Phi_p(B)|H|\Phi_p(B)\rangle} {\langle \Phi_p(B)|\Phi_p(B)\rangle}.
\]
Because numerator and denominator are quadratic in the reduced parameters, the problem reduces to an effective eigenvalue problem. In the gauge-fixed uMPS formulation this becomes
\[
H_{\mathrm{eff}(p)}X=\omega X.
\]
The lowest few eigenvalues at each momentum \(p\) define the approximate dispersion relations \(\omega_\gamma(p)\) [1810.07006].

The computational structure is a key part of the method. The effective Hamiltonian is assembled from infinite-network contractions that resum geometric series of transfer matrices. The required matrix-vector action can be implemented with cost
\[
\mathcal{O}(D^3),
\]
whereas explicit formation of the full matrix would cost \(\mathcal{O}(D^6)\). This makes iterative eigensolvers such as Lanczos or Arnoldi the natural numerical strategy [1810.07006].

The same formalism supports several extensions. Larger \(M\)-site blocks replace the one-site perturbation by a block tensor \(B\) to improve expressiveness for broad excitations. In symmetry-broken phases, topological excitations can be represented by an ansatz that interpolates between different asymptotic MPS ground states on the left and right. The notes also emphasize an important limitation: not every eigenvector of the variational problem is physically meaningful, and many states in continua are not well captured by a one-site quasiparticle ansatz [1810.07006].

## 3. Reduced bases, nonorthogonal formulations, and the site-basis variant

A recent variation is the Site Basis Excitation Ansatz (SBEA), formulated for one-dimensional quantum lattice systems described by infinite MPS. In the standard excitation ansatz for a one-site unit cell,
\[
|\Psi_k(B)\rangle = \sum_j e^{ikj} |B\rangle_j ,
\]
where \(|B\rangle_j\) denotes insertion of \(B\) at site \(j\) into an otherwise uniform iMPS. SBEA replaces the momentum-dependent optimization of a full tensor \(B(k)\) by a small basis expansion,
\[
B(k) = \sum_{\alpha=1}^{N_\alpha} c_\alpha^k B_\alpha .
\]
The basis tensors \(B_\alpha\) are obtained from a single-site effective Hamiltonian diagonalization, analogous to a one-site DMRG step but targeting multiple low-lying states [2509.06241].

The construction begins from an iMPS ground state produced by a finite-system DMRG route: run finite open-chain DMRG on a long chain \(N \gg \xi\), insert a new site at the center, optimize it with Lanczos, convert the central tensor into a uniform tensor \(A\), and optionally canonicalize it using the Orús–Vidal procedure. The excitation basis is then generated by a single-site Lanczos diagonalization in an effective environment. To reduce cost, the method introduces a Schmidt-value-based pre-truncation with an isometry \(U\),
\[
B' = U^\dagger B U, \qquad B = U B' U^\dagger .
\]
Once the basis is fixed, one computes overlap and Hamiltonian kernels,
\[
O_{\alpha\alpha'}(j'-j) = \langle j B_\alpha | B_{\alpha'} \rangle_{j'}, \qquad
H_{\alpha\alpha'}(j-j') = \langle j B_\alpha |H| B_{\alpha'} \rangle_{j'},
\]
whose exponential decay in a gapped system implies exponentially banded Fourier-transformed kernels. The excitation problem at momentum \(k\) is then
\[
\tilde H(k)c = E(k)\tilde O(k)c,
\]
a small generalized eigenvalue problem explicitly compared to non-orthogonal band theory [2509.06241].

A notable methodological conclusion concerns gauge choice. In standard tangent-space EA, a left gauge is often imposed so that excitations at different positions are orthogonal. For SBEA, however, not imposing a gauge condition and leaving the basis nonorthogonal is reported to be crucial, whereas imposing a left-orthonormal gauge severely hampers convergence. The stated reason is that overlap among translated local states is an essential part of efficiently reconstructing low-energy band states, rather than a nuisance to be removed [2509.06241].

The benchmark is the \(S=1\) Heisenberg chain
\[
H = J \sum_j \mathbf{S}_j \cdot \mathbf{S}_{j+1}, \qquad J=1.
\]
With a small basis, SBEA reproduces the one-magnon dispersion accurately; with \(N_\alpha=7\), the dispersion is reported as excellent over the whole magnon branch accessible before the two-magnon continuum. At \(k=\pi\), the result
\[
\Delta/J = 0.4107
\]
is compared with the essentially exact
\[
\Delta/J = 0.410479\ldots
\]
and the branch enters the two-magnon continuum around
\[
k \approx 0.72\pi .
\]
The same framework also yields Wannier excitations: one localized Wannier excitation, translated across all sites, can reconstruct the single-magnon modes exactly for all momenta [2509.06241].

## 4. Two-dimensional PEPS generalization and automatic differentiation

For PEPS, the tangent-space excitation ansatz extends the one-dimensional construction to infinite two-dimensional tensor networks. A translation-invariant PEPS ground state is specified by a local tensor \(A\), and the excitation ansatz replaces one tensor by \(B\) at lattice position \((m,n)\), then superposes all translated insertions with a momentum phase:
\[
|\Phi_{\kappa_x\kappa_y}(B)\rangle = \sum_{m,n} e^{i(\kappa_x m+\kappa_y n)} \; |\Psi_{m,n}(B)\rangle .
\]
The variational problem becomes the generalized eigenvalue problem
\[
\mathsf{H}^{\kappa_x\kappa_y}_{\mathrm{eff}} B = \omega\, \mathsf{N}^{\kappa_x\kappa_y}_{\mathrm{eff}} B.
\]
As in the MPS case, gauge-like null modes and the component parallel to the ground-state tensor must be projected out because they produce zero-norm directions [1507.02151].

The principal computational difficulty is the evaluation of momentum-transformed two-point and three-point functions on the PEPS background. The proposed solution is a corner/channel contraction scheme that combines MPS boundary methods with corner transfer matrices. In this framework, infinite sums generated by momentum projection are resummed through channel inverses such as
\[
(1 - e^{i\kappa} \mathcal{B})^{-1} = \sum_{m=0}^\infty e^{i\kappa m}\mathcal{B}^m.
\]
This gives direct access to gaps, dispersion relations, and spectral weights in the thermodynamic limit [1507.02151].

The square-lattice AKLT model is a representative application. A single-mode approximation gives a gap
\[
\Delta_{\mathrm{SMA}} = 0.0199,
\]
whereas the full tangent-space ansatz with a \(2\times 2\) perturbation block gives
\[
\Delta_{\mathrm{var}} = 0.0147, \qquad v_{\mathrm{var}} = 0.04115.
\]
Near the minimum, the lowest magnon carries about \(99.6\%\) of the spectral-weight sum rule, and the transfer-matrix correlation length is
\[
\xi_{\mathrm{AKLT}} = 2.06491 .
\]
The same formalism is extended to topological sectors in a perturbed toric code by attaching a half-infinite virtual matrix-product-operator string to the local tensor \(B\). Along the \(\beta_x=0\) axis, the flux gap closes at
\[
\beta_z=\log(\sqrt2+1)\approx 0.88,
\]
while the charge sector remains gapped at that point [1507.02151].

Automatic differentiation (AD) provides a further simplification of the PEPS excitation framework. Instead of manually differentiating the excitation contractions, one treats the contraction pipeline as a computational graph
\[
f:(\mathbf{B}^\dagger,\mathbf{B}) \to \tilde E,
\]
with
\[
\mathbb{H}_{\mathbf{k}}\mathbf{B} = \partial_{\mathbf{B}^\dagger} f.
\]
CTM-based boundary tensors carrying excitation insertions are built and differentiated through reverse-mode AD. A fixed-point AD variant reduces memory by differentiating through one converged CTM step rather than through an unrolled sequence of iterations. In the half-filled Hubbard model, this approach is implemented with a \(2\times2\) unit cell, bond dimensions \(D=4,5,6\), and sufficiently large CTM boundary dimensions \(\chi\); the reported charge-gap results agree well with QMC for \(U/t \le 8\), and the method remains practical at larger \(U/t\), where QMC becomes exponentially hard [2107.03399].

## 5. Quantum-circuit tangent spaces

The quantum-circuit version translates the tensor-network idea into the language of parametrized circuits. An optimized variational ground state is written as
\[
|\Psi\rangle = U_D U_{D-1}\cdots U_1 |\Psi_0\rangle,
\]
and excitations are generated by adding one extra local gate \(G\) of support size \(n\) between two consecutive layers. In the one-dimensional translationally invariant case, the basis state is
\[
|\phi(G)\rangle = \sum_{j=0}^{N-1} e^{-ij\frac{2\pi m}{N} } T^j U_D\cdots U_{l+1}\, G\, U_l\cdots U_1|\Psi_0\rangle .
\]
Collecting all such states over gate choices and insertion layers yields an overcomplete variational subspace \(\mathbb V\), described as the tangent space
\[
\mathbb{V}=T_{|\Psi(U)\rangle}\mathcal{M}\equiv \{G\,\partial_U|\Psi(U)\rangle\}.
\]
Excitation energies follow from
\[
\mathbf{H}\mathbf{v}=E\,\mathbf{N}\mathbf{v},
\]
or, when necessary, from diagonalization of \(N^{-1}H\) using a pseudoinverse [2507.07646].

The implementation is explicitly hybrid. Overlap and Hamiltonian matrix elements are measured with the Hadamard test. For basis states \(|\phi_j\rangle = u_j |0\rangle^{\otimes N}\), the real and imaginary parts of \(\langle \phi_1|\phi_2\rangle\) are extracted from ancilla probabilities, and momentum-sector matrices are assembled from untranslated basis overlaps by a classical Fourier transform. Finite sampling makes zero modes of \(N\) fuzzy, so a threshold is required when taking the pseudoinverse [2507.07646].

The numerical demonstrations are broad. In the one-dimensional transverse-field Ising chain, the method gives good agreement with exact diagonalization for the lowest four levels on a 16-site chain over a wide range of \(g\). The \(n=1\) ansatz captures some but not all fermionic excitations, while increasing the gate size to \(n=4\) recovers missing states. Basis states inserted in deeper layers substantially improve the results, especially near the critical point \(g=1\). In the two-dimensional square-lattice transverse-field Ising model on a \(4\times4\) lattice, using an HVA ground state of depth \(D=5N/2\), the reported ground-state energy errors are around \(10^{-3}\)–\(10^{-4}\), and the tangent-space ansatz with \(n=2\) captures the lowest five levels well across the phase transition. On a 12-site kagome Heisenberg cluster with depth \(D=64\), the method reproduces exact-diagonalization degeneracies associated with SU(2) and lattice symmetries. For the spin-\(\tfrac12\) Heisenberg chain, the same excited states are used to compute the dynamical spin structure factor, with good agreement at \(N=16\) and \(K=\pi\). The paper also reports that \(10^6\) samples per matrix element already yield a very accurate reconstructed spectrum, while \(10^8\) samples make the error negligible in an 8-site example [2507.07646].

## 6. Interpretation, limits, and the dissipative-systems analogue

A recurrent interpretation is that the excitation ansatz selects the locally relevant directions around an optimized state while discarding redundant or dynamically irrelevant ones. In uniform-MPS truncation, the same geometry appears in a different optimization problem: given a target uniform MPS \(\ket{\Psi(M)}\), the best lower-bond-dimension approximation \(\ket{\Psi(A)}\) is characterized by the vanishing of the tangent-space residual orthogonal to the current state,
\[
\mathcal{P}_A \ket{\Psi(M)} = 0,
\]
with the fixed-point consistency condition
\[
A_C' = A_L C' = C' A_R .
\]
This is not an excitation algorithm, but it uses the same manifold structure—canonical forms, gauge-fixed tangent vectors, and a projector removing the component parallel to the reference state—and therefore clarifies the geometric basis on which excitation ansätze are built [2001.11882].

A common misconception is that the tangent-space construction provides a global coordinate chart for the full variational manifold. The cited works instead frame it as a local linear description. In the tensor-network setting, that locality is explicit in the single-site or finite-block insertion. In dissipative dynamical systems, an analogous conclusion is drawn from covariant Lyapunov vectors: the tangent space splits into a finite set of frequently entangled physical modes and a spurious sector of strongly decaying modes that are hyperbolically isolated from the physical subspace. In the one-dimensional Kuramoto–Sivashinsky equation at \(L=96\), the threshold is identified at \(N_{\rm ph}=43\), with
\[
D = N_{\rm ph} \approx 0.43 \times L,
\]
and in the two-dimensional Kuramoto–Sivashinsky equation the same diagnostics give \(N_{\rm ph}=121\). The authors conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor, while the spurious modes do not excite the physical ones [1107.2567].

This suggests a broader reading of the excitation ansatz: only a restricted tangent-space sector may be necessary for faithful reduced descriptions. At the same time, the literature is explicit about limitations. The method is best suited to isolated quasiparticles or low-energy discrete branches; continua, broad resonances, and strongly multiparticle states are less reliably represented. Orthogonality constraints are also not universally beneficial: the standard gauge-fixed uMPS formalism relies on them for a well-conditioned tangent metric, whereas SBEA reports that a nonorthogonal basis is crucial for efficient reconstruction of low-energy magnons. The ansatz is therefore better understood as a family of geometry-driven variational constructions than as a single universally optimal prescription [1810.07006] [2509.06241] [2507.07646] [1107.2567].

Source: https://www.emergentmind.com/topics/tangent-space-excitation-ansatz