Tangent-Space Excitation Ansatz

• Tangent-Space Excitation Ansatz is a variational framework that represents low-lying excitations as momentum-projected tangent vectors around an optimized many-body state.
• It leverages tensor-network structures (e.g., uMPS and PEPS) by replacing local tensors to form an effective eigenvalue problem for extracting dispersion relations.
• Recent adaptations in quantum-circuit and dissipative-system settings highlight its flexibility in capturing quasiparticle dispersions while addressing computational challenges.

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 (Vanderstraeten et al., 2018, Vanderstraeten et al., 2015, Chen et al., 10 Jul 2025, Takeuchi et al., 2011).

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 (Vanderstraeten et al., 2018).

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 (Chen et al., 10 Jul 2025).

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 (White, 7 Sep 2025).

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$0, where $B$1 spans the null space orthogonal to $B$2. In that parametrization, the norm simplifies to

$B$3

so the effective norm matrix becomes the identity (Vanderstraeten et al., 2018).

The excitation energy is obtained variationally by minimizing the Rayleigh quotient

$B$4

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

$B$5

The lowest few eigenvalues at each momentum $B$6 define the approximate dispersion relations $B$7 (Vanderstraeten et al., 2018).

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

$B$8

whereas explicit formation of the full matrix would cost $B$9. This makes iterative eigensolvers such as Lanczos or Arnoldi the natural numerical strategy (Vanderstraeten et al., 2018).

The same formalism supports several extensions. Larger $A$0-site blocks replace the one-site perturbation by a block tensor $A$1 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 (Vanderstraeten et al., 2018).

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,

$A$2

where $A$3 denotes insertion of $A$4 at site $A$5 into an otherwise uniform iMPS. SBEA replaces the momentum-dependent optimization of a full tensor $A$6 by a small basis expansion,

$A$7

The basis tensors $A$8 are obtained from a single-site effective Hamiltonian diagonalization, analogous to a one-site DMRG step but targeting multiple low-lying states (White, 7 Sep 2025).

The construction begins from an iMPS ground state produced by a finite-system DMRG route: run finite open-chain DMRG on a long chain $A$9, insert a new site at the center, optimize it with Lanczos, convert the central tensor into a uniform tensor $A$0, 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 $A$1,

$A$2

Once the basis is fixed, one computes overlap and Hamiltonian kernels,

$A$3

whose exponential decay in a gapped system implies exponentially banded Fourier-transformed kernels. The excitation problem at momentum $A$4 is then

$A$5

a small generalized eigenvalue problem explicitly compared to non-orthogonal band theory (White, 7 Sep 2025).

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 (White, 7 Sep 2025).

The benchmark is the $A$6 Heisenberg chain

$A$7

With a small basis, SBEA reproduces the one-magnon dispersion accurately; with $A$8, the dispersion is reported as excellent over the whole magnon branch accessible before the two-magnon continuum. At $A$9, the result

$B$0

is compared with the essentially exact

$B$1

and the branch enters the two-magnon continuum around

$B$2

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 (White, 7 Sep 2025).

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 $B$3, and the excitation ansatz replaces one tensor by $B$4 at lattice position $B$5, then superposes all translated insertions with a momentum phase: $B$6 The variational problem becomes the generalized eigenvalue problem

$B$7

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 (Vanderstraeten et al., 2015).

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

$B$8

This gives direct access to gaps, dispersion relations, and spectral weights in the thermodynamic limit (Vanderstraeten et al., 2015).

The square-lattice AKLT model is a representative application. A single-mode approximation gives a gap

$B$9

whereas the full tangent-space ansatz with a $$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$0 perturbation block gives

$$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$1

Near the minimum, the lowest magnon carries about $$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$2 of the spectral-weight sum rule, and the transfer-matrix correlation length is

$$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$3

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 $$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$4. Along the $$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$5 axis, the flux gap closes at

$$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$6

while the charge sector remains gapped at that point (Vanderstraeten et al., 2015).

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

$$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$7

with

$$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$8

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 $$|\Phi(B;A)\rangle = \sum_n \cdots A A B A A \cdots .$$9 unit cell, bond dimensions $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$0, and sufficiently large CTM boundary dimensions $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$1; the reported charge-gap results agree well with QMC for $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$2, and the method remains practical at larger $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$3, where QMC becomes exponentially hard (Ponsioen et al., 2021).

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

$$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$4

and excitations are generated by adding one extra local gate $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$5 of support size $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$6 between two consecutive layers. In the one-dimensional translationally invariant case, the basis state is

$$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$7

Collecting all such states over gate choices and insertion layers yields an overcomplete variational subspace $$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$8, described as the tangent space

$$|\Phi_p(B)\rangle = \sum_n e^{ipn}\cdots A_L A_L B A_R A_R\cdots,$$9

Excitation energies follow from

$A^s \rightarrow X^{-1}A^sX$0

or, when necessary, from diagonalization of $A^s \rightarrow X^{-1}A^sX$1 using a pseudoinverse (Chen et al., 10 Jul 2025).

The implementation is explicitly hybrid. Overlap and Hamiltonian matrix elements are measured with the Hadamard test. For basis states $A^s \rightarrow X^{-1}A^sX$2, the real and imaginary parts of $A^s \rightarrow X^{-1}A^sX$3 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 $A^s \rightarrow X^{-1}A^sX$4 fuzzy, so a threshold is required when taking the pseudoinverse (Chen et al., 10 Jul 2025).

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 $A^s \rightarrow X^{-1}A^sX$5. The $A^s \rightarrow X^{-1}A^sX$6 ansatz captures some but not all fermionic excitations, while increasing the gate size to $A^s \rightarrow X^{-1}A^sX$7 recovers missing states. Basis states inserted in deeper layers substantially improve the results, especially near the critical point $A^s \rightarrow X^{-1}A^sX$8. In the two-dimensional square-lattice transverse-field Ising model on a $A^s \rightarrow X^{-1}A^sX$9 lattice, using an HVA ground state of depth $B$0, the reported ground-state energy errors are around $B$1–$B$2, and the tangent-space ansatz with $B$3 captures the lowest five levels well across the phase transition. On a 12-site kagome Heisenberg cluster with depth $B$4, the method reproduces exact-diagonalization degeneracies associated with SU(2) and lattice symmetries. For the spin-$B$5 Heisenberg chain, the same excited states are used to compute the dynamical spin structure factor, with good agreement at $B$6 and $B$7. The paper also reports that $B$8 samples per matrix element already yield a very accurate reconstructed spectrum, while $B$9 samples make the error negligible in an 8-site example (Chen et al., 10 Jul 2025).

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 $B \rightarrow B + Y A_R - e^{ip} A_L Y .$0, the best lower-bond-dimension approximation $B \rightarrow B + Y A_R - e^{ip} A_L Y .$1 is characterized by the vanishing of the tangent-space residual orthogonal to the current state,

$B \rightarrow B + Y A_R - e^{ip} A_L Y .$2

with the fixed-point consistency condition

$B \rightarrow B + Y A_R - e^{ip} A_L Y .$3

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 (Vanhecke et al., 2020).

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 $B \rightarrow B + Y A_R - e^{ip} A_L Y .$4, the threshold is identified at $B \rightarrow B + Y A_R - e^{ip} A_L Y .$5, with

$B \rightarrow B + Y A_R - e^{ip} A_L Y .$6

and in the two-dimensional Kuramoto–Sivashinsky equation the same diagnostics give $B \rightarrow B + Y A_R - e^{ip} A_L Y .$7. 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 (Takeuchi et al., 2011).

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 (Vanderstraeten et al., 2018, White, 7 Sep 2025, Chen et al., 10 Jul 2025, Takeuchi et al., 2011).