k-UpCCGSD: Unitary Pair Coupled-Cluster Ansatz

• k-UpCCGSD is a variational ansatz that stacks repeated paired double excitation layers to approach the full unitary manifold.
• It retains efficiency by restricting doubles to seniority-zero paired excitations while leveraging generalized singles over the active space.
• Benchmark studies show that increasing k systematically improves accuracy and noise robustness, making it competitive in VQE workflows.

Searching arXiv for papers on k-UpCCGSD and foundational references. k-UpCCGSD, short for $k$-unitary pair coupled-cluster generalized singles and doubles, is a layered unitary coupled-cluster ansatz used in variational quantum eigensolver (VQE) workflows for electronic-structure problems. It combines generalized single excitations with a restricted class of paired double excitations and repeats the corresponding unitary block $k$ times, yielding a variational family that is systematically improvable in $k$ while remaining substantially more compact than full generalized doubles constructions (Lee et al., 2018). The ansatz has been studied as a near-term quantum computing wavefunction for ground and excited states, spin-state energetics, transition-metal oxides, reaction pathways, molecular property optimization, and larger active-space simulations (Skosana et al., 11 Apr 2025).

1. Formal definition and operator content

In its standard form, k-UpCCGSD starts from a reference Slater determinant $|\Phi_0\rangle$, usually Hartree–Fock or a restricted/open-shell Hartree–Fock determinant in an active space, and applies $k$ successive anti-Hermitian cluster layers. A commonly used expression is

$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$

where $\hat T_S$ is a generalized single-excitation operator and $\hat T_\text{pCCD}$ is a pair-double operator (Lee et al., 2018). In later active-space formulations the same structure is written as

$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$

with $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$ (Skosana et al., 11 Apr 2025).

The defining restriction is on the doubles sector. Standard UCC doubles allow all two-electron excitations, whereas UpCCGSD retains only paired moves between spatial orbitals. In the language used across the cited studies, these are described as seniority-zero double excitations, pair coupled-cluster doubles, or simultaneous excitation of an $k$0 electron pair from one spatial orbital to another (Farag et al., 2022). Generalized singles remain unrestricted over the active space.

The parameter $k$1 counts how many pair-GSD layers are stacked. As $k$2 increases, the ansatz becomes more expressive; one study states that in the limit $k$3, and in the absence of Trotterization error, it spans the full unitary manifold generated by pair-GSD (Skosana et al., 11 Apr 2025). This makes $k$4 the principal systematic-improvement parameter.

2. Relation to UCCSD and UCCGSD

k-UpCCGSD is typically positioned between UCCSD and UCCGSD. UCCSD includes singles and all occupied-to-virtual doubles; UCCGSD generalizes both singles and doubles over all orbitals; k-UpCCGSD retains generalized singles but restricts doubles to pair excitations (Lee et al., 2018). The resulting compression is the central reason for its use in VQE.

Ansatz	Doubles sector	Reported asymptotic cost
UCCSD	all doubles	depth $k$5
UCCGSD	generalized all-orbital doubles	depth $k$6, amplitudes $k$7
k-UpCCGSD	paired doubles + generalized singles	depth $k$8, amplitudes $k$9

The foundational benchmark paper reports that k-UpCCGSD requires circuit depth $k$0, compared with $k$1 for UCCGSD and $k$2 for UCCSD, where $k$3 is the number of spin orbitals and $k$4 is the number of electrons (Lee et al., 2018). In the heme-related study, the same scaling is expressed in spatial-orbital notation: per layer, generalized singles contribute $k$5 parameters, paired doubles contribute $k$6, total parameters are $k$7, and depth scales as $k$8 for $k$9 spatial orbitals (Skosana et al., 11 Apr 2025).

System-specific resource counts illustrate the same trend. For the $|\Phi_0\rangle$0 spin-orbital active space used for Li$|\Phi_0\rangle$1Co$|\Phi_0\rangle$2O$|\Phi_0\rangle$3/Co$|\Phi_0\rangle$4O$|\Phi_0\rangle$5, singles plus paired doubles give 135 parameters per layer, so k-UpCCGSD has 405 parameters for $|\Phi_0\rangle$6 and 675 for $|\Phi_0\rangle$7, with 5,850 and 9,750 CNOTs respectively; the same study reports 20,950 CNOTs for UCCSD and 78,150 for UCCGSD (Farag et al., 2022). This suggests that the ansatz is designed as an accuracy–resource compromise rather than as a replacement for the full expressibility of UCCGSD.

3. Circuit realization and VQE workflows

In practical implementations, fermionic excitation generators are mapped to qubit operators, commonly with the Jordan–Wigner transformation. The heme-related study states that each $|\Phi_0\rangle$8 and each paired double becomes a Pauli string under Jordan–Wigner, and that each layer $|\Phi_0\rangle$9 is Trotter-approximated with a single Trotter step,

$k$0

with each small exponential expanded analytically as a $k$1 or $k$2 rotation (Skosana et al., 11 Apr 2025).

A complementary circuit description appears in the dibenzothiophene study. There, PennyLane’s k-UpCCGSD template maps each single-excitation generator to an $k$3 rotation on a weight-two Pauli string and each paired-double generator to rotations on weight-four Pauli strings. Gate synthesis is described as basis rotation, a single-qubit $k$4, and reversal of the basis change; a weight-$k$5 Pauli string thus costs roughly $k$6 CNOTs plus one $k$7 (Tailor, 3 Dec 2025).

The ansatz has also been embedded in more elaborate optimization loops. For spin-state energetics in a heme-related model, it was used in a state-averaged, orbital-optimized VQE targeting singlet, triplet, and quintet states simultaneously through

$k$8

with equal weights $k$9, ADAM for the $$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$0 parameters, and a final orbital update by PySCF’s classical orbital optimizer using 1- and 2-RDMs from the last VQE step (Skosana et al., 11 Apr 2025). Initial states were either single-reference ROHF determinants for each spin state or, for the triplet only, a small two-determinant multi-reference superposition denoted T1.

A different extension couples k-UpCCGSD to a two-phase optimization of the one-particle reduced density matrix. In that setting, the first phase minimizes the energy, while the second adds an RMSD-based 1-RDM penalty to the loss. For an active space $$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$1 and $$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$2, the study reports 22 parameters and a CNOT depth per layer on the order of twice the number of parameters, approximately 50–60 gates; energy changes are modest, but electron density, dipole moments, and atomic charges improve substantially (Lima et al., 10 Jul 2025). This indicates that k-UpCCGSD can serve not only as an energy ansatz but also as a source of reduced-density-matrix observables.

4. Canonical molecular benchmarks

The original comparative benchmarks on H$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$3, H$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$4O, and N$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$5 established the main empirical profile of k-UpCCGSD. For H$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$6 in STO-3G with $$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$7, 1-UpCCGSD uses 72 amplitudes and has a ground-state non-parallelity error $$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$8 m$$|\Psi_{k\text{-UpCCGSD}}\rangle =\prod_{\ell=1}^k \exp\!\Bigl[ \bigl(\hat T_\text{pCCD}^{(\ell)}+\hat T_S^{(\ell)}\bigr)-\mathrm{H.c.} \Bigr]\, |\Phi_0\rangle,$$9, while 2-UpCCGSD uses 144 amplitudes and reaches $\hat T_S$0; UCCGSD also reaches $\hat T_S$1 but with 3,136 amplitudes. For H$\hat T_S$2 in 6-31G with $\hat T_S$3, 2-UpCCGSD gives $\hat T_S$4, 3-UpCCGSD gives $\hat T_S$5, and UCCGSD gives $\hat T_S$6. For H$\hat T_S$7O double dissociation in STO-3G, 2-UpCCGSD gives $\hat T_S$8 and 3-UpCCGSD gives $\hat T_S$9. For N$\hat T_\text{pCCD}$0 dissociation in STO-3G, 4-UpCCGSD gives $\hat T_\text{pCCD}$1, 5-UpCCGSD gives $\hat T_\text{pCCD}$2, and UCCGSD gives $\hat T_\text{pCCD}$3, with chemical accuracy stated as $\hat T_\text{pCCD}$4 m$\hat T_\text{pCCD}$5 (Lee et al., 2018).

The same work also examined excited states with orthogonally constrained VQE. Typical first-excited-state NPEs for k-UpCCGSD were 0.0 for H$\hat T_\text{pCCD}$6/STO-3G at 2-UpCCGSD and higher, approximately 0.9 m$\hat T_\text{pCCD}$7 for H$\hat T_\text{pCCD}$8O/STO-3G at 2-UpCCGSD and approximately 0.0 at 3-UpCCGSD, and approximately 2.6 m$\hat T_\text{pCCD}$9 and 1.1 m$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$0 for N$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$1/STO-3G at 4-UpCCGSD and 5-UpCCGSD, respectively (Lee et al., 2018).

A notable result in the excited-state setting is that a specialized multi-determinantal reference obtained from classical linear-response calculations improved excited-state energetics. In the N$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$2 $$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$3 example, improving the OC-VQE reference to a four-determinant state reduced the UCCGSD error at $$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$4 Å from approximately $$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$5 m$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$6 to approximately $$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$7 m$$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$8 (Lee et al., 2018). Although that example concerns UCCGSD, it is directly relevant to k-UpCCGSD because the same paper frames multi-determinantal reference construction as a mechanism for improving excited-state calculations within constrained VQE.

5. Strong correlation, transition metals, and application domains

The heme-related spin-state study is one of the most detailed examinations of k-UpCCGSD in a strongly correlated transition-metal context. Using an in-house statevector simulator and single- and multi-reference trial wavefunctions, it computed singlet, triplet, and quintet energetics for active spaces from 5 to 10 spatial orbitals, equivalent to 10–20 qubits. For $$|\Psi_k(\boldsymbol{\theta})\rangle = \prod_{i=1}^k \exp\!\bigl(\hat T^{(i)}(\boldsymbol{\theta})-\hat T^{(i)\dagger}(\boldsymbol{\theta})\bigr) |\Phi_0\rangle,$$9, the VQE spin-state energetics were found to agree with CASSCF to within 1–4 kcal/mol; for 5- and 6-orbital active spaces, all three spin energies and their gaps agreed with CASSCF to within 1 kcal/mol. For 7–9 orbitals, the single-reference T0 ansatz drifted up to approximately 5 kcal/mol in the worst case, whereas the multi-reference triplet T1 kept most errors within 2 kcal/mol and the singlet–quintet and triplet–quintet gaps within 4 kcal/mol. In the 10-orbital run, limited to $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$0 and 32-bit precision, the singlet error reached up to 15 kcal/mol for T1, although the quintet–triplet gap remained within chemical accuracy (Skosana et al., 11 Apr 2025).

That study also reported multi-reference diagnostics $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$1. The singlet states had $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$2–0.29, triplets lay near or above $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$3 for most $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$4, and quintets had $$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$5. The interpretation given there is that singlets show strong static correlation, triplets moderate multi-reference character, and quintets are essentially single-reference (Skosana et al., 11 Apr 2025). These results delimit a regime in which k-UpCCGSD is capable of reproducing spin-state energetics of strongly correlated systems but remains sensitive to reference quality and resource limits.

In transition-metal oxides, VQE simulations of Li$$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$6Co$$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$7O$$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$8 and Co$$\hat T^{(i)}=\hat T_1^{(i)}+\hat T_{2,\mathrm{pair}}^{(i)}$$9O$k$00 found that k-UpCCGSD with $k$01 produces results similar to UCCSD but at a lower cost. For the Li$k$02Co$k$03O$k$04Co$k$05O$k$06 energy difference relative to CASCI, the reported values were $k$07 kcal/mol for UCCSD, $k$08 for UCCGSD, $k$09 for k-UpCCGSD($k$10), and $k$11 for k-UpCCGSD($k$12). The same work reports absolute ground-state errors versus CASCI of approximately $k$13 kcal/mol at $k$14 and $k$15 kcal/mol at $k$16, with near-equilibrium Co$k$17O$k$18 potential-energy-curve agreement within approximately 0.5 kcal/mol relative to CCSD for both UCCSD and $k$19, but errors larger than 1 kcal/mol at stretched geometries beyond $k$20 Å (Farag et al., 2022).

For a reaction-pathway application, a study of chloride attack on chloromethane used a 4-qubit HOMO–LUMO active-space model and found that in noiseless simulations UCCSD and k-UpCCGSD both reproduced the full configuration interaction potential-energy surface within chemical accuracy. The reported RMSEs relative to FCI were $k$21 kcal/mol for UCCSD and $k$22, $k$23, $k$24, $k$25, and $k$26 kcal/mol for $k$27 k-UpCCGSD, respectively. Under a Qulacs arbitrary noise model, UCCSD had energy-error bounds of approximately 1.26 to 1.54 mHa, whereas k-UpCCGSD gave smaller error ranges, with maxima from 0.889 to 0.696 mHa over $k$28 to $k$29 and 0.721 mHa at $k$30 (Lim et al., 2021). This is one of the clearest demonstrations in the literature that the reduced structure of k-UpCCGSD can translate into lower noise sensitivity.

At larger scale, the dibenzothiophene study used a 14-qubit $k$31 active space and reported 252 total variational parameters, circuit depth 9,398 layers, and 15,375 total parameterized gates. The final ground-state energy was $k$32 Ha after 114 VQE iterations and 112.4 s wall time on the state-vector simulator, corresponding to a recovered correlation energy of $k$33 Ha relative to the Hartree–Fock reference $k$34 Ha. The same paper contrasts this with ADAPT-VQE, which reached $k$35 Ha with a 41-layer circuit and concludes that the k-UpCCGSD circuit, although chemically accurate in simulation, is infeasible for hardware execution (Tailor, 3 Dec 2025).

6. Limitations, misconceptions, and research directions

A recurring limitation is that the favorable asymptotic scaling of k-UpCCGSD does not imply universally shallow hardware circuits. The dibenzothiophene case, with a 9,398-layer circuit and 15,375 parameterized gates, is an explicit counterexample: the ansatz remained viable in simulation but was judged infeasible for hardware execution (Tailor, 3 Dec 2025). A common misconception is therefore to read the $k$36 or $k$37 depth statements as guarantees of near-term executability; the application studies show that prefactors and active-space size remain decisive.

Another limitation is expressibility in strongly correlated regimes. The heme study notes that a single-layer Trotter step introduces approximation error, although it is variationally suppressed, and reports some overstabilization of high-spin states by up to approximately 2.5 kcal/mol in a few cases (Skosana et al., 11 Apr 2025). The transition-metal-oxide study states that the ansatz misses higher-order three- and higher-body excitations and that a single-reference HF state may have insufficient overlap in strongly correlated regimes such as stretched bonds or open-shell Co$k$38 states (Farag et al., 2022). The SN2 study likewise emphasizes that larger $k$39 may be required for very strongly correlated or larger active spaces, but that this can exceed NISQ coherence times (Lim et al., 2021).

The literature also shows that reference preparation matters. In the heme calculations, a small two-determinant multi-reference triplet initial state improved convergence relative to the single-reference T0 setup (Skosana et al., 11 Apr 2025). In the foundational excited-state benchmarks, a specialized multi-determinantal reference improved constrained-VQE energetics (Lee et al., 2018). This suggests that the restriction to paired doubles does not eliminate the need for careful state preparation when the target state has substantial multi-reference character.

Several research directions are identified explicitly in the cited works. The heme-related study recommends choosing $k$40 as large as resources permit, enforcing spin symmetry by using pair-GSD excitation operators that commute with $k$41 and $k$42, and considers adaptive methods such as ADAPT-VQE, memory-streaming of cluster operators, perturbative MRPT(2) corrections on top of k-UpCCGSD, and near-term hardware experiments on strongly correlated transition-metal active spaces as future improvements (Skosana et al., 11 Apr 2025). The 1-RDM optimization study adds a different direction: when energies are already near classical references, augmenting the VQE objective with a density-matrix penalty can substantially improve molecular properties without changing the ansatz class itself (Lima et al., 10 Jul 2025).

Taken together, these results place k-UpCCGSD in a specific methodological niche. It is not the most expressive coupled-cluster ansatz and not always the most hardware-efficient in absolute terms, but it is systematically improvable, compact relative to UCCGSD, often more noise-robust than UCCSD, and already capable of chemically accurate results in several benchmark and application settings when $k$43, the active space, and the reference state are chosen appropriately (Lee et al., 2018).