Projective Cluster-Additive Transformation (PCAT)

• Projective Cluster-Additive Transformation is a projection-based method that defines effective Hamiltonians by block-diagonalizing sectors and removing lower-energy components.
• It subtracts projections onto lower-energy subspaces before orthonormalization, ensuring additive behavior on disconnected clusters vital for linked-cluster expansions.
• PCAT has been benchmarked in NLCE and quantum algorithms, providing accurate one–quasi-particle dispersions and reliable performance in various quantum lattice model regimes.

Projective Cluster-Additive Transformation (PCAT) is a projection-based, cluster-additive transformation for quantum lattice models that block-diagonalizes sectors of $\mathcal{H}=\mathcal{H}_0+V$ or $\mathcal{H}=\mathcal{H}_0+\lambda V$ into subspaces adiabatically connected to eigenspaces of $\mathcal{H}_0$. Its defining purpose is to produce effective Hamiltonians that remain additive on disconnected clusters, which is the prerequisite for perturbative and non-perturbative linked-cluster expansions. In recent numerical linked-cluster expansion (NLCE) workflows, including hybrid quantum-classical pipelines, PCAT serves as the post-processing step that converts finite-cluster low-energy information into intensive, cluster-additive one–quasi-particle effective Hamiltonians whose Fourier transforms yield thermodynamic-limit dispersions (Hörmann et al., 2023, Sumeet et al., 10 Nov 2025, Marti et al., 27 May 2026).

1. Formal setting and conceptual scope

PCAT is formulated for local lattice Hamiltonians whose unperturbed part admits a block structure in quasi-particle number or, more generally, in degenerate eigenspaces. In the foundational formulation, the Hilbert space decomposes as

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$

with an ordering of unperturbed energies $e_0^m\leq e_0^n$ for $m\leq n$. The objective is to block-diagonalize the full Hamiltonian into sectors adiabatically connected to these subspaces, so that

$$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$

The relevant projectors satisfy

$\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$

where $P_n$ projects onto an eigenspace of $\mathcal{H}_0$ and $\mathcal{H}=\mathcal{H}_0+\lambda V$0 onto the adiabatically connected eigenspace of $\mathcal{H}=\mathcal{H}_0+\lambda V$1 (Hörmann et al., 2023).

In this setting, “projective” refers to constructions that use projectors onto eigenspaces of $\mathcal{H}=\mathcal{H}_0+\lambda V$2 and the adiabatically connected eigenspaces of $\mathcal{H}=\mathcal{H}_0+\lambda V$3, while “cluster-additive” refers to the requirement that the effective Hamiltonian respects additivity on disconnected clusters. For disconnected clusters $\mathcal{H}=\mathcal{H}_0+\lambda V$4 and $\mathcal{H}=\mathcal{H}_0+\lambda V$5,

$\mathcal{H}=\mathcal{H}_0+\lambda V$6

This property is not an auxiliary convenience. It is the structural condition that allows linked-cluster expansions to be formulated exclusively on connected subgraphs.

The later one–quasi-particle formulations adopt the same logic in a more specialized setting. There one considers a finite connected cluster $\mathcal{H}=\mathcal{H}_0+\lambda V$7 with cluster Hamiltonian $\mathcal{H}=\mathcal{H}_0+\lambda V$8, solves for a low-energy effective Hamiltonian that is block-diagonal in quasi-particle number, enforces cluster additivity on the transformation, and then embeds and Fourier-transforms the one–quasi-particle block to obtain the dispersion $\mathcal{H}=\mathcal{H}_0+\lambda V$9 or $\mathcal{H}_0$0 in the thermodynamic limit. This suggests that PCAT is best understood as a compatibility condition between block-diagonalization and linked-cluster summation rather than as a stand-alone diagonalization scheme.

2. Construction of the transformation

The defining modification introduced by PCAT is the subtraction of all projections onto lower-energy subspaces before orthonormalization. For the single-particle sector in the foundational construction,

$\mathcal{H}_0$1

and

$\mathcal{H}_0$2

which implies

$\mathcal{H}_0$3

For general target sector $\mathcal{H}_0$4, one introduces

$\mathcal{H}_0$5

and, assuming a non-singular subtraction operator $\mathcal{H}_0$6, defines modified states by subtracting lower-energy components so that

$\mathcal{H}_0$7

In the quantum-algorithm formulation this same idea is written directly in projector form as

$\mathcal{H}_0$8

with $\mathcal{H}_0$9 (Hörmann et al., 2023, Sumeet et al., 10 Nov 2025).

Once the lower-energy projections have been removed, PCAT constructs the blockwise correction from modified overlaps. In the one–quasi-particle sector,

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$0

and the correction is the symmetric Löwdin orthonormalization

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$1

Equivalently, in block $$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$2,

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$3

The resulting effective Hamiltonian is obtained by conjugation with the PCAT unitary or partial isometry. In the one–quasi-particle case,

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$4

or, cluster by cluster,

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$5

A central structural result is that PCAT depends only on the target sector and all lower-energy eigenspaces. The 2023 construction states this explicitly as a minimal basis requirement relative to other linked-cluster block-diagonalization schemes. The 2025 quantum-algorithm paper reformulates the same point operationally: PCAT uses only low-energy eigenspace information, namely energies, overlaps with unperturbed states, and low-energy Hamiltonian matrix elements, without tomography of higher sectors. A plausible implication is that PCAT separates the difficulty of preparing a low-energy subspace from the distinct difficulty of enforcing cluster additivity on that subspace.

3. Cluster additivity and linked-cluster expansions

The reason PCAT is necessary appears most transparently on disconnected clusters. If $$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$6 and $$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$7 do not share sites or bonds, then the microscopic Hamiltonian factorizes, and exact eigenstates satisfy the product structure

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$8

Approximate block-diagonalization can spoil this structure by leaking lower-sector components into the target manifold. PCAT removes those components, and on disconnected clusters the modified product states factorize:

$$\mathscr{H}=\bigoplus_{n=0}^N \mathscr{H}_0^n, \qquad \mathcal{H}_0=\bigoplus_{n=0}^N \mathcal{H}_0^n,$$9

This is the core mechanism by which PCAT restores cluster additivity (Sumeet et al., 10 Nov 2025).

For one–quasi-particle quantities, cluster additivity also requires subtraction of the ground-state energy. The effective one–quasi-particle Hamiltonian is defined as

$e_0^m\leq e_0^n$0

The subtraction is necessary to make the one–quasi-particle block intensive and additive: on $e_0^m\leq e_0^n$1, matrix elements describing an excitation on one disconnected component would otherwise carry the ground-state energy of the other component. In the earlier linked-cluster formulation this same point appears as the additive decomposition of the ground-state-energy-reduced one-particle effective Hamiltonian, following the observation that $e_0^m\leq e_0^n$2 for non-degenerate ground states (Hörmann et al., 2023).

Once cluster additivity holds, NLCE proceeds by inclusion–exclusion. For a property $e_0^m\leq e_0^n$3 evaluated on clusters,

$e_0^m\leq e_0^n$4

In the excited-state pipeline, $e_0^m\leq e_0^n$5 is the real-space matrix of the one–quasi-particle effective Hamiltonian. Effective hoppings are extracted from its matrix elements and embedded across the lattice. Their Fourier transform yields the thermodynamic-limit dispersion. On rectangular clusters of the square lattice,

$e_0^m\leq e_0^n$6

while in one dimension the rectangular expansion reduces to subtracting the contributions of consecutive cluster sizes (Marti et al., 27 May 2026).

The failure mode is equally explicit: if the transformation is not cluster-additive, spurious inter-cluster matrix elements appear, unphysical hopping between disconnected clusters is generated, and NLCE diverges. This point is emphasized both in the abstract mathematical construction and in later one–quasi-particle benchmarks.

4. Relation to established effective-Hamiltonian methods

PCAT was introduced as a generalization of the minimal transformation known under several names: Takahashi’s transformation, Schrieffer–Wolff transformation, des Cloizeaux effective Hamiltonian, canonical van Vleck effective Hamiltonian, and the two-block orthogonalization method. The minimal projective transformation is written as

$e_0^m\leq e_0^n$7

In the two-block setting this is equivalent, order by order, to the Schrieffer–Wolff direct rotation. It has minimal norm, but the 2023 analysis shows that it fails to be cluster-additive for excitations in general (Hörmann et al., 2023).

PCAT resolves that failure by modifying the underlying eigenvectors before the Löwdin-type orthonormalization. The added ingredients relative to the minimal projective transformation are projectors onto lower-energy eigenspaces. This is the sense in which PCAT retains a minimal-basis character while restoring linked-cluster additivity. The 2023 paper contrasts this with the multi-block orthogonalization method and perturbative continuous unitary transformations, which are genuine linked-cluster methods but typically require a larger operator basis (Hörmann et al., 2023).

The later one–quasi-particle literature rephrases the same issue in physically transparent terms. In the pure transverse-field Ising model, the global $e_0^m\leq e_0^n$8 parity separates ground-state and one–quasi-particle sectors, so

$e_0^m\leq e_0^n$9

In that special case the PCAT correction in the one–quasi-particle sector vanishes, and a good single-unitary block-diagonalization already has a clean block structure. By contrast, adding a longitudinal field breaks parity, mixes the ground-state and one–quasi-particle sectors, and causes canonical two-block decoupling to violate cluster additivity; the resulting effective Hamiltonian can exhibit unphysical hopping between disconnected clusters (Sumeet et al., 10 Nov 2025). A common misconception is therefore that any accurate low-energy block-diagonalization is sufficient for NLCE. The published constructions show that accuracy of the low-energy subspace and cluster additivity of the transformation are distinct requirements.

5. Variational, adiabatic, and hardware realizations

In the NLCE+VQE formulation, each cluster $m\leq n$0 is assigned a single unitary $m\leq n$1 that approximately block-diagonalizes the low-energy subspace containing the ground state and the one–quasi-particle sector. For the transverse-field Ising model with or without longitudinal field, the Hamiltonian variational ansatz (HVA) is chosen to mirror the Hamiltonian structure:

$m\leq n$2

with XX–X–Z ordering. Benchmarks used $m\leq n$3 or $m\leq n$4 layers. Two cost-function families were employed. The trace-minimization cost promotes the correct low-energy block without forcing internal diagonalization, while the variance-based cost vanishes iff the subspace is an invariant eigenspace and thus directly penalizes couplings from the transformed one–quasi-particle sector to the rest of the Hilbert space (Sumeet et al., 10 Nov 2025).

After optimization, the cluster workflow measures three $m\leq n$5 matrices: overlaps with unperturbed states, overlaps among prepared low-energy states, and the Hamiltonian matrix in the prepared basis. These measurements scale as $m\leq n$6 per cluster. Classical post-processing then orthogonalizes the subspace when necessary through a generalized eigenvalue problem, constructs the modified overlap, and applies the PCAT correction to the one–quasi-particle block. The same paper emphasizes that the PCAT post-processing is independent of the cluster solver and applies to VQE, phase estimation, adiabatic preparation, and other methods that can prepare or characterize the low-energy subspace (Sumeet et al., 10 Nov 2025).

The hardware realization of NLCE+QA implements the same logic on a 20-qubit trapped-ion quantum processing unit. It considers both adiabatic state preparation (ASP) and VQE trained on a classical device, while the final expectation values are obtained from the QPU. A distinctive feature is the CX-test, introduced as an alternative to the Hadamard test for overlap estimation. In PCAT, the global prefactor $m\leq n$7 cancels in $m\leq n$8, so it need not be measured. Ground-state energies are obtained via sample-based quantum diagonalization (SQD). The one–quasi-particle matrices remain $m\leq n$9, the measurement cost remains $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$0, and the paper gives explicit circuit counts such as

$$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$1

for a one-dimensional TFIM chain, with additional circuits in the longitudinal-field case (Marti et al., 27 May 2026).

6. Benchmarks, regimes of validity, and limitations

The original PCAT paper benchmarks the method in the low-field ordered phase of the square-lattice transverse-field Ising model for single spin-flips and two spin-flip bound states. It reports perturbative linked-cluster series for the single spin-flip gap $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$2 up to order $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$3 and the two spin-flip bound-state gap $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$4 up to order $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$5. In its non-perturbative implementation, exact diagonalization is performed on connected clusters and Wynn extrapolations are used across cluster sizes. The NLCE for $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$6 converges up to $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$7, Wynn extrapolations extend slightly further, but both break down before $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$8. Finite-size scaling at $$\mathcal{H}_{\mathrm{eff}}=\bigoplus_{n=0}^N \mathcal{H}_{\mathrm{eff}}^n.$$9 indicates algebraic behavior with exponent $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$0 (Hörmann et al., 2023).

In the high-field polarized phase, the 2025 NLCE+VQE study benchmarks one–quasi-particle dispersions for the transverse-field Ising model in one and two dimensions, and with longitudinal field. In one dimension at $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$1, the exact dispersion is

$\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$2

At $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$3, NLCE+VQE with $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$4 reproduces NLCE+ED and the exact dispersion, including the closing gap at $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$5, with rapid convergence up to $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$6. In two dimensions on the square lattice, at $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$7, NLCE+VQE with $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$8 matches NLCE+ED and high-order series benchmarks from pCUT; at $\mathds{1}=\sum_n P_n=\sum_n \bar P_n,$9 the difference between NLCE+VQE and NLCE+ED is $P_n$0, and all four tested cost functions converge identically within $P_n$1 variations, below NLCE truncation errors of $P_n$2. For the one-dimensional TFIM with longitudinal field at $P_n$3 and $P_n$4, Schrieffer–Wolff-only effective Hamiltonians show unphysical oscillations and non-additive long-range hopping, whereas the PCAT correction restores smooth dispersions and NLCE convergence. The same benchmarks report optimizer trapping for variance-based costs under ground-state initialization, with residual $P_n$5 and $P_n$6–$P_n$7 weight in $P_n$8QP/$P_n$9QP sectors; near-zero initialization yields $\mathcal{H}_0$0 and subspace infidelity $\mathcal{H}_0$1, while trace-based costs remain robust (Sumeet et al., 10 Nov 2025).

The trapped-ion NLCE+QA study addresses the separate question of whether current hardware can supply expectation values accurate enough to survive PCAT’s non-linear classical post-processing. PCAT involves matrix inversion and matrix square roots, and these operations amplify measurement noise. The paper quantifies this by Monte Carlo propagation with $\mathcal{H}_0$2 samples. Experiments used $\mathcal{H}_0$3 shots per matrix element, totaling $\mathcal{H}_0$4 shots across all runs on a $\mathcal{H}_0$5-qubit trapped-ion QPU with single-/two-qubit gate fidelities $\mathcal{H}_0$6. For one-dimensional TFIM at $\mathcal{H}_0$7, statevector NLCE+VQE overlaps perfectly with the analytic dispersion and QPU NLCE+VQE approaches it closely; at $\mathcal{H}_0$8 and especially $\mathcal{H}_0$9, noise amplification becomes more visible. In the one-dimensional TFIM with longitudinal field at $\mathcal{H}=\mathcal{H}_0+\lambda V$00, $\mathcal{H}=\mathcal{H}_0+\lambda V$01, the modified-state correction is mathematically required, but numerically $\mathcal{H}=\mathcal{H}_0+\lambda V$02, below the sampling-noise floor at $\mathcal{H}=\mathcal{H}_0+\lambda V$03 shots per element and therefore negligible in practice. The ladder benchmark at $\mathcal{H}=\mathcal{H}_0+\lambda V$04 and $\mathcal{H}=\mathcal{H}_0+\lambda V$05 resolves two one–quasi-particle bands with moderate upward bias. The paper attributes the characteristic upward shift and vertical compression of the dispersion to effective depolarization rates satisfying $\mathcal{H}=\mathcal{H}_0+\lambda V$06 with $\mathcal{H}=\mathcal{H}_0+\lambda V$07, and concludes that ASP requires one to two orders of magnitude lower gate depolarization rates than current reference values to be reliably usable without strong mitigation (Marti et al., 27 May 2026).

Across these formulations, the domain of validity is consistently stated. PCAT is most natural in gapped phases with well-defined quasi-particles and adiabatic connectivity of the target sector. It requires a non-singular subtraction operator in the general multiparticle construction. Near criticality, under avoided level crossings, or when particle decay processes render the subtraction ill-conditioned, the relevant singular values can drop sharply and NLCE convergence can deteriorate. In quantum-algorithm settings, inaccurate identification of the low-energy subspace, optimizer trapping, and hardware noise can degrade decoupling quality and violate additivity. Extending the method to higher-particle sectors is possible in principle, but increases the subspace dimension and measurement overhead.