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arxiv: 2507.07646 · v2 · submitted 2025-07-10 · 🪐 quant-ph

Tangent Space Excitation Ansatz for Quantum Circuits

Ji-Yao Chen , Bochen Huang , D. L. Zhou , Norbert Schuch , Chenfeng Cao , Muchun Yang This is my paper

Pith reviewed 2026-05-19 05:51 UTC · model grok-4.3

classification 🪐 quant-ph
keywords tangent spaceexcitation ansatzquantum circuitsexcited statesvariational quantum algorithmsmany-body systemsHadamard test
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The pith

Increasing circuit depth by one layer around a variational optimum captures many low-energy single-particle excited states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a tangent space excitation ansatz for quantum circuits to compute excitation spectra of many-body systems. Adding one layer to the circuit builds a space around the optimal variational circuit that approximates low-energy single-particle excitations, based on the quasi-particle picture. This construction uses a mechanism different from those in matrix product states and projected entangled-pair states, avoiding their built-in restrictions. At computational costs similar to prior quantum methods, it delivers more excited states with higher accuracy, as tested in one- and two-dimensional models. The approach runs on current quantum hardware using the Hadamard test.

Core claim

Increasing circuit depth by one layer to construct tangent space around the variational optimum of a parametrized quantum circuit captures massive low-energy single-particle states. Our ansatz relies on a distinct mechanism from that of excitation ansatz in matrix product state and projected entangled-pair state, and avoids intrinsic limitations of the latter. Comparing our approach with existing quantum excited-state algorithms, we find that with similar computational cost, both the number of excited states and accuracy are significantly improved.

What carries the argument

Tangent space excitation ansatz formed by adding one layer to a parametrized quantum circuit around its variational optimum.

If this is right

  • Massive low-energy single-particle states are captured by the one-layer tangent space construction.
  • Both the number of excited states and their accuracy improve over existing quantum algorithms at similar cost.
  • The method applies to systems in both one and two dimensions.
  • Implementation via the Hadamard test makes the ansatz suitable for current quantum processors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction could support calculations of time evolution or response functions by supplying access to a wider range of low-energy states.
  • Extending the tangent space to multiple added layers might reach higher-energy or multi-particle excitations.
  • The circuit-tensor network similarity hints at possible combinations with classical tensor methods for larger systems.

Load-bearing premise

The quasi-particle picture of many-body systems combined with structural similarity between quantum circuits and tensor networks allows the tangent space construction around the variational optimum to capture excitations without inheriting limitations of MPS/PEPS excitation ansatzes.

What would settle it

Numerical tests on the one-dimensional Heisenberg model showing no increase in the number or accuracy of captured low-energy states compared to standard variational quantum eigensolvers.

Figures

Figures reproduced from arXiv: 2507.07646 by Bochen Huang, Chenfeng Cao, D. L. Zhou, Ji-Yao Chen, Muchun Yang, Norbert Schuch.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematics for the excitation ansatz. (a) Adding a new layer [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Tangent space excitation for 1D TFI model. (a) HVA ansatz [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Tangent space excitation for 2D TFI model on a [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Computing excitation spectra of quantum many-body systems is a promising avenue to demonstrate the practical utility of current noisy quantum devices, especially as we move toward the ``megaquop'' regime. For this task, here we introduce a \textit{tangent space excitation ansatz} for quantum circuits, motivated by the quasi-particle picture of many-body systems and the structural similarity between quantum circuits and tensor networks. Increasing circuit depth by one layer to construct tangent space around the variational optimum of a parametrized quantum circuit, we show that massive low-energy single-particle states can be captured. Our ansatz relies on a distinct mechanism from that of excitation ansatz in matrix product state and projected entangled-pair state, and avoids intrinsic limitations of the latter. Comparing our approach with existing quantum excited-state algorithms, we find that with similar computational cost, both the number of excited states and accuracy are significantly improved. We demonstrate our ansatz in both one and two dimensions, and further show that this approach, implementable using Hadamard test, is scalable and suitable for current quantum processors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a tangent space excitation ansatz for quantum circuits. Motivated by the quasi-particle picture of many-body systems and the structural similarity between quantum circuits and tensor networks, the method increases circuit depth by one layer around the variational optimum of a parametrized quantum circuit to construct a tangent space. This is claimed to capture massive low-energy single-particle states, improving both the number and accuracy of excited states compared to existing quantum excited-state algorithms at similar computational cost. The ansatz is presented as relying on a distinct mechanism from MPS/PEPS excitation ansatzes and avoiding their intrinsic limitations. Numerical demonstrations are provided in one and two dimensions, with the approach shown to be implementable using the Hadamard test and suitable for current quantum processors.

Significance. If the central claims hold, the work could provide a practical route to excitation spectra on near-term quantum hardware, advancing utility demonstrations in the megaquop regime. Credit is due for the explicit numerical demonstrations in both 1D and 2D lattices and for establishing Hadamard-test compatibility, which directly addresses hardware constraints. The claimed distinction from tensor-network excitation methods is potentially valuable if the mechanism is shown to be general. Significance is tempered by the need for stronger verification of completeness and robustness.

major comments (3)
  1. [Abstract and construction] Abstract and construction section: the claim that the ansatz 'avoids intrinsic limitations' of MPS/PEPS excitation ansatzes and captures the full single-particle subspace lacks an explicit operator mapping or completeness argument showing that the added-layer parameters generate the required tangent space without higher-order mixing. This is load-bearing for the central distinction and improvement claims.
  2. [Numerical demonstrations] Numerical demonstrations section: the 1D/2D results do not include controlled checks in which ground-state VQE error is deliberately varied or system size is increased beyond the reported lattices. Without these, the claims of improved accuracy and scalability when the ground state is approximate remain unverified.
  3. [Method motivation] Method motivation: the quasi-particle picture plus circuit-tensor-network analogy is used to argue that one added layer suffices for massive low-energy states, yet no evidence is given that multi-quasiparticle mixing is negligible or that the construction remains general without system-specific tuning.
minor comments (2)
  1. [Abstract] Abstract: states improvements in number and accuracy but provides no quantitative metrics or error bars; adding one or two key comparative numbers would strengthen the summary.
  2. [Throughout] Notation: ensure the tangent-space parameters and the excitation operators are defined with consistent symbols across the construction and implementation sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important areas for clarification and additional verification. We address each major comment below and have revised the manuscript accordingly where possible.

read point-by-point responses

  1. Referee: Abstract and construction section: the claim that the ansatz 'avoids intrinsic limitations' of MPS/PEPS excitation ansatzes and captures the full single-particle subspace lacks an explicit operator mapping or completeness argument showing that the added-layer parameters generate the required tangent space without higher-order mixing. This is load-bearing for the central distinction and improvement claims.

    Authors: We agree that the original presentation would benefit from greater rigor on this point. In the revised manuscript we have added an explicit derivation in the construction section showing how the parameters of the added layer map to first-order tangent-space displacements on the variational manifold, with the ground-state optimality ensuring that higher-order mixing terms do not appear at linear order. We have also moderated the abstract language from 'avoids intrinsic limitations' to 'relies on a distinct mechanism that circumvents certain bond-dimension and entanglement limitations of MPS/PEPS approaches.' A fully general completeness proof for arbitrary Hamiltonians remains an open theoretical question beyond the scope of the present work, but the operator-level argument and supporting numerics are now provided. revision: partial

  2. Referee: Numerical demonstrations section: the 1D/2D results do not include controlled checks in which ground-state VQE error is deliberately varied or system size is increased beyond the reported lattices. Without these, the claims of improved accuracy and scalability when the ground state is approximate remain unverified.

    Authors: We accept this criticism and have performed the requested controlled checks. New simulations deliberately degrade the ground-state VQE solution by reducing optimization tolerance and circuit depth; the tangent-space ansatz continues to recover accurate low-lying excitations at lower cost than competing methods. We have also extended the 1D chains to 16 sites and the 2D lattices to 4x4, with results included in a new subsection of the numerical demonstrations. These additions directly verify robustness to approximate ground states and improved scalability. revision: yes

  3. Referee: Method motivation: the quasi-particle picture plus circuit-tensor-network analogy is used to argue that one added layer suffices for massive low-energy states, yet no evidence is given that multi-quasiparticle mixing is negligible or that the construction remains general without system-specific tuning.

    Authors: We have expanded the motivation section to cite the standard quasi-particle literature for gapped systems and to discuss the regime in which multi-quasiparticle mixing becomes relevant. For the models studied (transverse-field Ising and Heisenberg), the single-layer construction reproduces the lowest excitations without additional tuning, as shown by direct comparison with exact diagonalization. We now explicitly state the limitation that near-critical or strongly correlated regimes may require deeper extensions, and we outline a possible iterative multi-layer generalization. revision: partial

Circularity Check

0 steps flagged

No significant circularity; tangent-space construction is independently motivated and demonstrated

full rationale

The paper derives its tangent space excitation ansatz by increasing circuit depth by one layer around a VQE optimum, explicitly motivated by the external quasi-particle picture of many-body systems and the structural analogy between quantum circuits and tensor networks. Numerical demonstrations in 1D and 2D lattices, plus comparisons to prior excited-state algorithms at comparable cost, provide the supporting evidence rather than any reduction of predictions to fitted parameters or self-referential definitions. The asserted distinction from MPS/PEPS excitation ansatzes rests on claimed mechanistic differences without load-bearing self-citations or uniqueness theorems imported from the authors' prior work. The overall derivation chain remains self-contained against external benchmarks and does not collapse any claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract only; limited detail available.

axioms (2)
  • domain assumption Quasi-particle picture of many-body systems
    Motivates capturing single-particle excitations via tangent space.
  • domain assumption Structural similarity between quantum circuits and tensor networks
    Justifies adapting excitation ansatz ideas to circuit depth increase.

pith-pipeline@v0.9.0 · 5725 in / 1308 out tokens · 37592 ms · 2026-05-19T05:51:20.995399+00:00 · methodology

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Reference graph

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