arxiv: 2604.13735 · v1 · submitted 2026-04-15 · 🪐 quant-ph · cs.CC· cs.ET· cs.LG

Recognition: unknown

Reachability Constraints in Variational Quantum Circuits: Optimization within Polynomial Group Module

Yun-Tak Oh , Dongsoo Lee , Jungyoul Park , Kyung Chul Jeong , Panjin Kim

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Pith reviewed 2026-05-10 13:57 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.ETcs.LG

keywords variational quantum circuitsreachability constraintsgroup modulesmatchgate circuitsMaxCutclassical simulabilityquantum optimization

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The pith

Variational quantum circuits reach the exact ground state only if the norms of projections onto each group module match between input and target states.

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

The paper establishes a necessary condition that any variational quantum circuit must satisfy to reach the exact ground state of a Hamiltonian. The projection norms of the initial state and the ground state onto every module in the group-module decomposition must be equal. This forces prior knowledge of the ground state's module weights before optimization can succeed. Matchgate circuits on problems with classical bit-string solutions automatically satisfy the condition because all basis states carry identical module weights. When this is combined with classical simulability of observables in a low-dimensional subspace, certain problems such as MaxCut become solvable by a classical surrogate algorithm whose each iteration costs O(n^5) time.

Core claim

This work identifies a necessary condition for any variational quantum approach to reach the exact ground state: the norms of the projections of the input and the ground state onto each group module must match, implying that module weights of the solution state have to be known in advance in order to reach the exact ground state. An exemplary case is provided by matchgate circuits applied to problems whose solutions are classical bit strings, since all computational basis states share the same module-wise weights. Combined with the known classical simulability of quantum circuits for which observables lie in a small linear subspace, this implies that certain problems admit a classical surro

What carries the argument

Group-module decomposition of the Hilbert space, which imposes that the projection norms of the input state and ground state onto each module must coincide for the variational circuit to be able to reach the target state.

Load-bearing premise

The variational quantum circuit and the problem Hamiltonian admit a decomposition into the same group-module structure used to derive the projection-norm condition.

What would settle it

An explicit variational circuit and Hamiltonian pair in which the ground state is reached exactly even though the projection norms onto the modules differ, or a MaxCut instance where the classical O(n^5) surrogate fails to recover the optimal cut.

Figures

Figures reproduced from arXiv: 2604.13735 by Dongsoo Lee, Jungyoul Park, Kyung Chul Jeong, Panjin Kim, Yun-Tak Oh.

**Figure 1.** Figure 1: (a) Schematic projection of ρ Ψ and ρ g onto two representative modules Bκ. The projected components are shown as vectors in the corresponding subspaces (blue and orange). (b) Within each module, unitary transformations act as rotations that preserve the magnitude of the projected component. If ∥ρ Ψ κ ∥HS ̸= ∥ρ g κ∥HS, the two vectors cannot be matched by such rotations. 2.1 Parameterized quantum circuit a… view at source ↗

**Figure 2.** Figure 2: Running time as a function of n (the number of vertices) for evaluating the expectation value of HMaxCut using a matchgate circuit. In Pennylane simulation, n is the number of qubits. Each marker represents an average value of 100 trials [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic of the matchgate circuit employed in the optimization simulations. For an n‑qubit system the circuit consists of n blocks; each block contains n single‑qubit gates and n − 1 two‑qubit gates acting on adjacent qubit pairs. where the generator γi is independently sampled for each block from the set { Xi⊗Xi+1, Xi⊗Yi+1, Yi⊗Xi+1, Yi⊗Yi+1 }. (12) Thus, while the geometric placement of the two‑qubit gat… view at source ↗

**Figure 4.** Figure 4: Convergence of the cost for an n = 60 instance that the projected method reached the exact ground state. The lowest eigenvalue of the instance is −72 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

**Figure 5.** Figure 5: Convergence of the cost for n = 60, p = 0.5 Erdős-Rényi random graph instance g05_60.0 from Biq Mac Library [21]. Out of nine trials, the best one we obtained was near −185 corresponding to the cut size of 535, whereas the optimal cut is 536 corresponding to −187 in cost. B Overparameterization in quantum circuits from a Fisher information perspective Fisher information quantifies the sensitivity of a prob… view at source ↗

read the original abstract

This work identifies a necessary condition for any variational quantum approach to reach the exact ground state. Briefly, the norms of the projections of the input and the ground state onto each group module must match, implying that module weights of the solution state have to be known in advance in order to reach the exact ground state. An exemplary case is provided by matchgate circuits applied to problems whose solutions are classical bit strings, since all computational basis states share the same module-wise weights. Combined with the known classical simulability of quantum circuits for which observables lie in a small linear subspace, this implies that certain problems admit a classical surrogate for exact solution with each step taking $O(n^5)$ time. The Maximum Cut problem serves as an illustrative example.

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.

Desk Editor's Note private letter to a colleague

The paper states a necessary projection-norm condition on group modules for a VQC to reach an exact ground state and shows this yields an O(n^5) classical surrogate for MaxCut under matchgates.

read the letter

The main thing to know is that any variational circuit reaches the exact ground state only when the norms of the projections of the input and target states onto each group module are identical. This means the module weights of the solution must be known in advance. The paper then notes that matchgate circuits satisfy this automatically for any classical bit-string solution because all basis states carry the same module weights, and it combines that fact with prior classical-simulability results to claim an O(n^5) classical stand-in per optimization step for MaxCut. That combination is the concrete new piece. The necessary condition itself looks like a clean incremental observation that had not been written down in exactly this form. The MaxCut illustration is straightforward and shows how the constraint can turn a quantum heuristic into a classical polynomial-time procedure for certain instances. The logic on the page is internally consistent and does not rely on hidden circularity once the earlier simulability theorem is granted. The main soft spot is that the abstract and the supplied description do not include the full derivation of the module decomposition or the precise counting that produces O(n^5), so it is still possible that the bound depends on post-hoc choices or on the observables staying inside a low-dimensional subspace in a way that is not automatic for every problem. If the full text supplies the missing steps without extra assumptions, the claim holds; otherwise the time bound may need adjustment. This work is aimed at people who study the reachability limits of variational quantum algorithms and the boundary between quantum and classical methods for optimization. It is worth sending to referees because it supplies a usable necessary condition and a practical classical alternative rather than another heuristic. I would bring it to a reading group to walk through the module decomposition and verify the MaxCut counting.

Referee Report

2 major / 2 minor

Summary. The manuscript identifies a necessary reachability condition for variational quantum circuits to prepare the exact ground state of a Hamiltonian: the norms of the projections of the input state and the ground state onto each group module in a polynomial group-module decomposition must match. This implies that the module weights of the target state must be known in advance. The condition is illustrated for matchgate circuits on problems whose solutions are classical bit strings (all computational-basis states share identical module weights). Combined with known classical simulability results for circuits whose observables lie in a low-dimensional linear subspace, the approach yields a classical surrogate for exact optimization with O(n^5) time per step, using MaxCut as an illustrative example.

Significance. If the group-module decomposition and projection-norm condition are rigorously justified and independent of prior simulability results, the work would establish a concrete expressivity limitation for restricted VQCs and demonstrate that certain classically simulable ansatze admit efficient classical surrogates for exact ground-state search. The MaxCut example, if fully derived, would provide a concrete polynomial-time classical benchmark against which variational quantum performance can be compared.

major comments (2)

[Abstract] The O(n^5) complexity claim for the classical surrogate (abstract) requires an explicit derivation showing how the low-dimensional observable subspace dimension, the number of group modules, and the matchgate structure combine to produce this scaling; without it, the per-step cost cannot be verified as polynomial and independent of the specific MaxCut instance size.
[Abstract] The central projection-norm condition (abstract) is presented as necessary for any VQC, but the manuscript must demonstrate that the chosen group-module decomposition is preserved by the circuit evolution and applies uniformly to both the variational ansatz and the problem Hamiltonian; otherwise the reachability implication does not hold for arbitrary VQCs.

minor comments (2)

Clarify whether the classical-simulability result invoked is independent of the group-module construction or whether it is applied post-hoc; an explicit citation and statement of the subspace dimension for the MaxCut observables would strengthen the argument.
Ensure consistent use of terminology between the title ('Polynomial Group Module') and the abstract ('group module'); define the module decomposition formally at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We have revised the paper to provide the requested explicit derivation of the complexity and to clarify the preservation and applicability of the group-module decomposition. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] The O(n^5) complexity claim for the classical surrogate (abstract) requires an explicit derivation showing how the low-dimensional observable subspace dimension, the number of group modules, and the matchgate structure combine to produce this scaling; without it, the per-step cost cannot be verified as polynomial and independent of the specific MaxCut instance size.

Authors: We agree that an explicit derivation is needed for verifiability. In the revised manuscript we have added a dedicated paragraph (new Section 4.2) that derives the O(n^5) per-step cost: the observable subspace for matchgate circuits has dimension O(n^2), the number of group modules is O(n), and each classical surrogate step (projection-norm matching plus low-dimensional linear algebra) therefore scales as O(n^5) using standard matrix operations. The scaling depends only on qubit number n and is independent of any particular MaxCut instance. revision: yes
Referee: [Abstract] The central projection-norm condition (abstract) is presented as necessary for any VQC, but the manuscript must demonstrate that the chosen group-module decomposition is preserved by the circuit evolution and applies uniformly to both the variational ansatz and the problem Hamiltonian; otherwise the reachability implication does not hold for arbitrary VQCs.

Authors: We thank the referee for highlighting this point. The group-module decomposition is defined relative to the polynomial group generated by the variational circuit family; by construction the modules are invariant under the group action, so the decomposition is preserved under circuit evolution. We have expanded the text (revised Section 3) to show explicitly that the same decomposition applies to the input state, all states reachable by the ansatz, and the Hamiltonian observables (which lie in the low-dimensional subspace). While the concrete modules are circuit-class dependent, the necessary projection-norm matching condition holds for any VQC once its associated group-module decomposition is identified. This clarification keeps the claim within the paper's scope of restricted VQCs. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation

full rationale

The central derivation establishes a necessary reachability condition by equating projection norms of the input state and target ground state onto each group module; this follows algebraically from the assumed shared decomposition into the polynomial group module and does not reduce to a fitted parameter or self-definition. The specialization to matchgate circuits uses the independent structural fact that all computational-basis states carry identical module weights, which is a property of the circuit class rather than a constructed prediction. Invocation of classical simulability for observables in a low-dimensional subspace is presented as a 'known' external result rather than a self-citation whose validity depends on the present paper. No load-bearing step collapses to its own inputs by construction, and the MaxCut illustration remains non-essential to the general claim. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a group-module decomposition for both the variational circuit and the problem Hamiltonian, plus the applicability of a known classical-simulability theorem; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Variational quantum circuits and the target Hamiltonian admit a common decomposition into group modules whose projections determine reachability.
This decomposition is the framework used to state the necessary condition.
standard math Observables lying in a small linear subspace of the circuit algebra are classically simulable.
Cited as previously known; used to obtain the O(n^5) classical surrogate.

pith-pipeline@v0.9.0 · 5440 in / 1462 out tokens · 57358 ms · 2026-05-10T13:57:50.150796+00:00 · methodology

discussion (0)

Reference graph

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State initialization : Construct the initial state jψ0i (either all-zero j0i⊗n or single-qubit flipped X1j0i⊗n) and compute its representation in the Ma- jorana basis. 19
[24]

F orward pass: Apply the matchgate circuit U (θ) =Q q U (θq) to the initial state by sequentially applying each gate’s effective matrix M(θq) to the coeﬀicient vector
[25]

Expectation value : Compute the inner product hHi =hψ(θ)jHjψ(θ)i us- ing the evolved state coeﬀicients and the Hamiltonian representation
[26]

Backward pass : Perform automatic differentiation to compute gradients rθhHi, with optional gradient checkpointing
[27]

Parameter update: Apply Adam optimizer to update θ θη Adam(rθhHi)
[28]

A.8 Convergence curves 61 211 261 311 361 .71 .51 .31 1 .83 Ovncfs pg jufsbujpot Dptu Fig

Check convergence: Evaluate the improvement criterion; trigger learning rate decay or termination if conditions are met. A.8 Convergence curves 61 211 261 311 361 .71 .51 .31 1 .83 Ovncfs pg jufsbujpot Dptu Fig. 4. Convergence of the cost for an n = 60 instance that the projected method reached the exact ground state. The lowest eigenvalue of the instance...

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