Q-SENSE: Seniority-Based Quantum Subspace Expansion

• Q-SENSE is a hybrid quantum-classical method that uses seniority symmetry to compress electronic structure simulations by restricting initial sampling to the seniority-zero sector.
• It employs a DOCI-QSCI protocol to generate compact determinant sets and expands these via a Cartesian product to include essential seniority-breaking configurations.
• The method integrates quantum sampling with classical post-processing such as ph-AFQMC, achieving high accuracy in benchmark systems while reducing qubit requirements.

Searching arXiv for papers on Q-SENSE and related seniority-based quantum subspace methods. Quantum SENiority-based Subspace Expansion (Q-SENSE) denotes a seniority-structured family of hybrid quantum-classical methods for electronic-structure simulation in which the quantum device is used on a compressed paired-electron representation and the resulting information is expanded classically into a larger determinant space. In the 2026 realization "Doubling the size of quantum selected configuration interaction based on seniority-zero space and its application to QC-QSCI-AFQMC," Q-SENSE is implemented as three coupled ideas: sampling in the seniority-zero sector through DOCI-QSCI, expanding the sampled configurations to seniority-breaking determinants through a Cartesian product of spin strings, and then using the expanded multi-determinant state as the trial wave function in phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) to recover dynamical correlation in the full orbital space (Yoshida et al., 8 Feb 2026). Related preprints use the same label for closely allied seniority-constrained subspace-expansion and state-preparation strategies, but the common principle is the use of seniority symmetry to reduce circuit depth or qubit count while retaining a systematically improvable classical post-processing stage (Patel et al., 1 Sep 2025, Halder et al., 28 Apr 2025).

1. Seniority as the organizing principle

Seniority is the number of unpaired particles. In molecular electronic structure, for spatial orbital $i$ with spin occupations $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$, the seniority operator used in the 2026 Q-SENSE realization is

$$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$

A Slater determinant has $\Omega=0$ when each spatial orbital is either empty or doubly occupied, so that no singly occupied orbitals appear (Yoshida et al., 8 Feb 2026).

This modern electronic-structure usage is consistent with the older many-body notion of seniority as the number of particles not paired to angular momentum $J=0$. In the single-shell formalism, seniority labels blocks of the many-body Hilbert space and can be expressed through quasi-spin algebra; that earlier literature also established the importance of exact and partial conservation of seniority in pairing-dominated Hamiltonians (Isacker et al., 2014). This historical connection is significant because Q-SENSE exploits the same structural fact: low-seniority sectors often capture the dominant static or pairing correlation in a much smaller subspace than the full configuration space.

Doubly occupied configuration interaction (DOCI) is the CI problem restricted to the $\Omega=0$ sector,

$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$

with determinants $|\Phi_I\rangle$ spanning the seniority-zero subspace. The central attraction of DOCI is that it captures pair correlation efficiently and uses far fewer configurations than full CI (Yoshida et al., 8 Feb 2026). Q-SENSE inherits this compactness, but it is not identical to DOCI. Its distinctive step is to treat seniority-zero as a compressed sampling manifold rather than as the final variational space.

A separate but related formalism appears in the orthogonal subspace-expansion variant of Q-SENSE, where orbital seniority is represented by commuting operators $\hat{\Omega}_i$ or, under Jordan–Wigner, by stabilizers $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$. In that setting, basis states from different seniority sectors are orthogonal by construction (Patel et al., 1 Sep 2025). This suggests that “Q-SENSE” is best understood not as one immutable algorithm but as a seniority-centered design pattern spanning several hybrid methods.

2. Seniority-zero compression and the DOCI-QSCI construction

The 2026 implementation begins from quantum selected configuration interaction (QSCI), which constructs a classically diagonalizable effective Hamiltonian on a sampled subspace $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$0,

$n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$1

The computational-basis bitstrings $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$2 are obtained by sampling a prepared quantum state in the computational basis, retaining $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$3 distinct bitstrings (Yoshida et al., 8 Feb 2026).

The seniority-zero restriction changes the qubit accounting. Conventional spin-orbital mappings use one qubit per spin orbital, so a device with $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$4 qubits represents $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$5 spin orbitals, or $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$6 spatial orbitals. In seniority-zero, each spatial orbital is binary—empty or doubly occupied—so one qubit per spatial orbital suffices. With $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$7 spatial orbitals, the seniority-zero mapping therefore uses $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$8 qubits instead of $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$9, and the accessible spatial-orbital space doubles at fixed device size. In the 2026 work, sampling is performed directly on $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$0 qubits representing spatial-orbital pair occupancies (Yoshida et al., 8 Feb 2026).

State preparation for this compressed sampling stage uses a spinless local unitary cluster Jastrow (LUCJ) ansatz with one repetition and Jastrow couplings restricted to adjacent orbitals or qubits. Circuit parameters are initialized from frozen-core CCSD amplitudes in the chosen active space. Sampling is carried out either on the IBM Quantum device ibm_kobe or on a noiseless simulator, and samples that violate the target electron number are discarded (Yoshida et al., 8 Feb 2026).

The restriction to $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$1 is advantageous but incomplete. The 2026 paper states directly that this sector restriction can compromise quantitative accuracy. The reason is that the $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$2 sector efficiently represents pair or static correlation, whereas much of the dynamical correlation resides in seniority-breaking configurations. Q-SENSE addresses this deficiency by expanding the sampled seniority-zero configurations into a larger spinful determinant set rather than by increasing the quantum-state complexity at the sampling stage (Yoshida et al., 8 Feb 2026).

3. Cartesian-product subspace expansion

The defining expansion step in the 2026 realization starts from $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$3 distinct $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$4-bit seniority-zero strings in the pair-occupation representation,

$$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$5

Each bitstring is interpreted as both an $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$6-spin string and a $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$7-spin string. Because DOCI imposes identical $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$8 and $$\Omega = \sum_i \big(n_{i\uparrow} + n_{i\downarrow} - 2 n_{i\uparrow} n_{i\downarrow}\big).$$9 occupations, the pools are

$\Omega=0$0

Q-SENSE then constructs a larger determinant set in the spinful space through the Cartesian product

$\Omega=0$1

which yields up to $\Omega=0$2 spin-resolved Slater determinants (Yoshida et al., 8 Feb 2026).

The crucial observation is that when $\Omega=0$3, the $\Omega=0$4 and $\Omega=0$5 strings differ, so singly occupied orbitals appear and the determinant has $\Omega=0$6. The effective Hamiltonian is therefore built on a space that contains both seniority-zero and seniority-breaking determinants, even though the quantum sampling cost was incurred only in the compact seniority-zero representation (Yoshida et al., 8 Feb 2026).

An optional heat-bath-like enlargement can then be applied classically. Starting from $\Omega=0$7, an external determinant $\Omega=0$8 is added if

$\Omega=0$9

where $J=0$0, $J=0$1 are the current CI coefficients, and $J=0$2 is a user-set threshold implemented as select_cutoff in PySCF (Yoshida et al., 8 Feb 2026). This step follows a conventional selected-CI growth and is conceptually separate from the seniority-based Cartesian expansion itself.

The computational consequence is a deliberate shift of cost from the quantum to the classical side. If seniority-zero sampling returns $J=0$3 bitstrings, the determinant count grows from $J=0$4 to $J=0$5 after the Cartesian product. The qubit count remains $J=0$6 at the sampling stage, but the expanded Hamiltonian must be built and diagonalized classically. This suggests a characteristic Q-SENSE trade-off: qubit compression and shallow sampling circuits are obtained at the price of a larger post-sampling linear-algebra problem (Yoshida et al., 8 Feb 2026).

4. End-to-end workflow and ph-AFQMC recovery of dynamical correlation

In the 2026 workflow, the electronic Hamiltonian in the active space is

$J=0$7

with standard one-electron integrals $J=0$8 and antisymmetrized two-electron integrals $J=0$9 (Yoshida et al., 8 Feb 2026).

The algorithm proceeds in a fixed sequence. One first builds an $\Omega=0$0-qubit spinless LUCJ circuit, initializes parameters from frozen-core CCSD, executes $\Omega=0$1 shots, discards electron-number-violating samples, and collects $\Omega=0$2 unique seniority-zero bitstrings. One then forms the $\Omega=0$3- and $\Omega=0$4-pools, generates all Cartesian-product determinants, assembles the effective Hamiltonian $\Omega=0$5 in that basis, and solves

$\Omega=0$6

classically for the ground-state eigenpair. Optional selected-CI enlargement repeats the importance-based growth and rediagonalization. The resulting multi-determinant QSCI state,

$\Omega=0$7

is then used as the trial state $\Omega=0$8 in ph-AFQMC (Yoshida et al., 8 Feb 2026).

The AFQMC stage is not an auxiliary refinement but a central component of the Q-SENSE realization. The local energy for a walker Slater determinant $\Omega=0$9 is

$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$0

In the phaseless formulation, the walker-weight update over time step $$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$1 is

$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$2

where $$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$3 is a reference energy and $$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$4 is the phase of the overlap ratio (Yoshida et al., 8 Feb 2026). The expanded QSCI trial is intended to mitigate phaseless bias by supplying a trial wave function that already contains a broad, explicitly multi-determinant description of seniority-breaking configurations.

The implementation details are concrete. Qiskit v2.2.3 is used for the sampling circuits; ipie v0.6.2 is used for AFQMC; the default AFQMC time step is $$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$5; and the standard settings are 50 steps and 3000 blocks unless otherwise noted. For hardware runs on ibm_kobe, no additional error mitigation is applied beyond postselection on valid electron number (Yoshida et al., 8 Feb 2026). This is important because it shows that the reported behavior is not contingent on a large error-mitigation stack.

5. Benchmark systems and reported performance

The method is evaluated on three classes of systems: the H$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$6 linear chain dissociation, N$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$7 dissociation, and the addition of singlet O$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$8 to a BODIPY dye. The H$$|\Psi_{\mathrm{DOCI}}\rangle = \sum_I c_I |\Phi_I\rangle,$$9 calculations use a cc-pVTZ active-space basis, ibm_kobe hardware and a noiseless simulator, $|\Phi_I\rangle$0 shots per geometry point, and 400 AFQMC walkers. The N$|\Phi_I\rangle$1 calculations use cc-pVQZ, $|\Phi_I\rangle$2 shots per point, and 640 walkers, with comparison to reduced multireference CCSD and CCSD(T). The BODIPY–O$|\Phi_I\rangle$3 calculations use an RB3LYP/6-31G(d)-optimized reaction path and IRC, energies at 6-31G(d,p), QSCI trials built in natural orbitals from CISD, active spaces up to $|\Phi_I\rangle$4, $|\Phi_I\rangle$5 shots, and AFQMC with 512 walkers and 1000 blocks (Yoshida et al., 8 Feb 2026).

For the H$|\Phi_I\rangle$6 chain, DOCI-QSCI-AFQMC on ibm_kobe reproduces CASCI-AFQMC and matches HCI references across the dissociation coordinate, within chemical accuracy except for slight deviations at three pre-minimum points. Simulator runs without enlargement may undersample because the distribution is skewed toward HF, but adding selected-CI expansion recovers high accuracy. DOCI-AFQMC, which uses a seniority-zero trial without Cartesian expansion, is reported to be significantly less accurate (Yoshida et al., 8 Feb 2026).

For N$|\Phi_I\rangle$7, a fixed subspace without enlargement causes both DOCI-QSCI and DOCI-QSCI-AFQMC to fail to reproduce dissociation quantitatively because of insufficient sampling. After enlarged-subspace construction, DOCI-QSCI qualitatively captures dissociation, while DOCI-QSCI-AFQMC closely matches RMR-CCSD and RMR-CCSD(T) over the full curve. By contrast, single-reference RCCSD and RCCSD(T) are qualitatively incorrect in the stretched regime (Yoshida et al., 8 Feb 2026). The comparison is important because it isolates the regime in which seniority-structured multireference information matters.

For BODIPY–O$|\Phi_I\rangle$8, the reported activation and reaction energies in kcal/mol are: DOCI-QSCI$|\Phi_I\rangle$9-AFQMC, $\hat{\Omega}_i$0, $\hat{\Omega}_i$1; DOCI-QSCI$\hat{\Omega}_i$2-AFQMC, $\hat{\Omega}_i$3, $\hat{\Omega}_i$4; DOCI-QSCI$\hat{\Omega}_i$5, $\hat{\Omega}_i$6, $\hat{\Omega}_i$7; DOCI-QSCI$\hat{\Omega}_i$8, $\hat{\Omega}_i$9, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$0; RCCSD, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$1, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$2; RCCSD(T), $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$3, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$4; SCF, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$5, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$6; RB3LYP, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$7, $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$8 (Yoshida et al., 8 Feb 2026). The same study reports $\hat{S}_i = \hat{Z}_{2i}\hat{Z}_{2i+1}$9 diagnostics of 0.012 for the reactant, 0.068 for the transition state, and 0.012 for the product, with transition-state $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$00 indicating single-reference CC breakdown. This provides the stated rationale for why CCSD and CCSD(T) disagree qualitatively on the barrier while DOCI-QSCI-AFQMC yields reasonable reaction energetics (Yoshida et al., 8 Feb 2026).

Taken together, the benchmarks support three specific claims made in the 2026 paper: DOCI-QSCI doubles the orbital space accessible to conventional QSCI at fixed device size, the Cartesian expansion is necessary to inject seniority-breaking determinants, and subsequent ph-AFQMC post-processing delivers reasonably high accuracy (Yoshida et al., 8 Feb 2026).

6. Broader formulations, misconceptions, and limitations

The broader literature shows that Q-SENSE is not yet a single standardized protocol. One formulation presents it as an orthogonal, seniority-symmetry-based subspace method that interpolates between VQE and CI by constructing basis states in distinct seniority sectors, measuring $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$01, and, because orthogonality is enforced by construction, solving a standard Hermitian eigenvalue problem with $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$02 rather than a generalized one (Patel et al., 1 Sep 2025). Another formulation describes a seniority-driven operator-selection and state-preparation framework in which compact ansätze are built from ordered seniority-zero pair excitations and sparsely admitted rank-one exchanges, with subspace-expansion and unitary-ansatz realizations treated as equivalent viewpoints (Halder et al., 28 Apr 2025). A review of quantum subspace methods supplies the general Rayleigh–Ritz and generalized-eigenvalue machinery into which seniority-adapted constructions naturally fit, even though that review does not explicitly define Q-SENSE (Motta et al., 2023).

A common misconception is that Q-SENSE requires exact seniority conservation by the Hamiltonian. The literature does not support that claim. The 2026 DOCI-QSCI-AFQMC realization explicitly uses a seniority-zero sampling stage precisely because the full problem contains important seniority-breaking contributions, and the orthogonal-subspace formulation states that the full Coulombic electronic Hamiltonian generally does not commute with total seniority (Yoshida et al., 8 Feb 2026, Patel et al., 1 Sep 2025). Q-SENSE is therefore better understood as exploiting seniority as a compression and selection principle, not as assuming an exact symmetry of the full electronic Hamiltonian.

The limitations are also explicit. Restricting the initial sampling to $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$03 biases determinant selection toward pair-occupied patterns and makes open-shell and high-spin targets inaccessible within that sampling scheme. Finite-shot sampling can skew the determinant distribution, for example by overweighting HF, which degrades trial quality unless the subspace is enlarged. The Cartesian product produces $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$04 growth in the determinant count, and the remaining dynamical correlation may still include subtle high-seniority contributions not fully recovered by the initial expansion (Yoshida et al., 8 Feb 2026). In the broader variants, the gains from seniority structure diminish when seniority breaking is strong enough that many inter-sector couplings must be retained (Patel et al., 1 Sep 2025).

The natural directions for further development are stated in the sources. The 2026 paper identifies adaptive expansion strategies, improved ansätze or ADAPTively optimized sampling preparations, nonorthogonal multi-reference trials, more sophisticated selection criteria beyond $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$05, tighter integration with tailored CC or perturbative corrections, and extensions beyond $n_{i\uparrow}, n_{i\downarrow} \in \{0,1\}$06 for open-shell targets (Yoshida et al., 8 Feb 2026). This suggests that Q-SENSE is evolving toward a modular framework in which seniority organizes the quantum stage, while increasingly flexible classical or projector-based corrections restore the parts of correlation that seniority-zero sampling cannot represent on its own.