Sample-based quantum diagonalization (SQD) offers a powerful route to accurate quantum chemistry on noisy intermediate-scale quantum (NISQ) devices by combining quantum sampling with classical diagonalization. Here we introduce HSQD, a novel half-qubit SQD approach that halves the qubit requirement for simulating a chemical system and drastically reduces overall circuit depth and gate counts, suppressing hardware noise. When modeling the dissociation of the nitrogen molecule with a (10e, 26o) active space, HSQD matches the accuracy of SQD on IBM quantum hardware using only half the number of qubits and 40% fewer measurements. We further enhance HSQD with a heat-bath configuration interaction (HCI) inspired selection of the samples, forming HCI-HSQD. This yields sub-millihartree accuracy across the N2 potential energy surface and produces subspaces up to 39% smaller than those from classical HCI, showing a significant improvement in the compactness of the ground-state representation. Finally, we demonstrate the scalability of HCI-HSQD using iron-sulfur clusters, reaching active spaces of up to (54e, 36o) while using only half as many qubits as the original SQD. For these systems, HCI-HSQD reduces SQD energy errors by up to 76% for [2Fe-2S] and 26% for [4Fe-4S], while also reducing subspace sizes, halving measurement requirements, and eliminating expensive post-processing. Together, these results establish half-qubit SQD as a noise-resilient and resource-efficient pathway toward practical quantum advantage in strongly correlated chemistry.
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Sample-based quantum diagonalization (SQD) uses quantum-generated configurations to define a classically diagonalized electronic Hamiltonian. This paper asks whether its sampling circuits and classical subspace construction can be reduced without sacrificing energy accuracy.
Half-qubit SQD (HSQD) samples single-spin configurations using a modified LUCJ circuit on $M/2$ rather than $M$ qubits. It remaps opposite-spin Jastrow gates to same-spin gates, then combines sampled configurations classically to recover the full spin-orbital configuration space. This is a sampling heuristic, not an exact encoding of the electronic wavefunction into half the qubits. HCI-HSQD additionally uses an HCI-inspired deterministic selection to construct a subspace without explicitly forming the full tensor product.
IBM hardware experiments cover nitrogen dissociation and two iron–sulfur clusters, reaching $(54e,36o)$ with 36 qubits. Transpiled two-qubit gate counts fall from 4,660 to 1,777 for the largest cluster. For nitrogen, HCI-HSQD obtains errors below $1$ mHa relative to variational HCI across the dissociation curve, with subspaces 18–39% smaller. Relative to reported SQD results, iron–sulfur energy errors against DMRG decrease by 76% and 26%, with subspaces 58% and 88% smaller, respectively, using half the qubits and measurements. A noniterative sample correction replaces self-consistent recovery for these clusters.
The paper states that inconsistent classical reference energies prevent a direct nitrogen comparison with prior SQD. Demonstrations use spin-symmetric systems; other cases would require separate spin-sector circuits. HCI comparisons exclude perturbative corrections, and the reported benchmarks do not establish quantum advantage.
Some signs of AI-generated textOverclaimsMissing comparison
Central claim may not hold: The abstract claims matched nitrogen accuracy against SQD, although Methods states that differing classical references prevented a direct HSQD–SQD comparison.
Nitrogen's reported sub-millihartree accuracy is established only relative to variational HCI, whose error relative to the exact ground-state energy is not bounded in the paper.
The printed HCI selection criterion H_ij c_i ≥ epsilon_1 lacks the absolute value needed for invariance under arbitrary determinant sign conventions, although an implementation error is not established.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Sample-based quantum diagonalization (SQD) offers a powerful route to accurate quantum chemistry on noisy intermediate-scale quantum (NISQ) devices by combining quantum sampling with classical diagonalization. Here we introduce HSQD, a novel half-qubit SQD approach that halves the qubit requirement for simulating a chemical system and drastically reduces overall circuit depth and gate counts, suppressing hardware noise. When modeling the dissociation of the nitrogen molecule with a (10e, 26o) active space, HSQD matches the accuracy of SQD on IBM quantum hardware using only half the number of qubits and 40% fewer measurements. We further enhance HSQD with a heat-bath configuration interaction (HCI) inspired selection of the samples, forming HCI-HSQD. This yields sub-millihartree accuracy across the N2 potential energy surface and produces subspaces up to 39% smaller than those from classical HCI, showing a significant improvement in the compactness of the ground-state representation. Finally, we demonstrate the scalability of HCI-HSQD using iron-sulfur clusters, reaching active spaces of up to (54e, 36o) while using only half as many qubits as the original SQD. For these systems, HCI-HSQD reduces SQD energy errors by up to 76% for [2Fe-2S] and 26% for [4Fe-4S], while also reducing subspace sizes, halving measurement requirements, and eliminating expensive post-processing. Together, these results establish half-qubit SQD as a noise-resilient and resource-efficient pathway toward practical quantum advantage in strongly correlated chemistry.
Sample-based quantum diagonalization (SQD) uses quantum-generated configurations to define a classically diagonalized electronic Hamiltonian. This paper asks whether its sampling circuits and classical subspace construction can be reduced without sacrificing energy accuracy.
Half-qubit SQD (HSQD) samples single-spin configurations using a modified LUCJ circuit on M/2 rather than M qubits. It remaps opposite-spin Jastrow gates to same-spin gates, then combines sampled configurations classically to recover the full spin-orbital configuration space. This is a sampling heuristic, not an exact encoding of the electronic wavefunction into half the qubits. HCI-HSQD additionally uses an HCI-inspired deterministic selection to construct a subspace without explicitly forming the full tensor product.
IBM hardware experiments cover nitrogen dissociation and two iron–sulfur clusters, reaching (54e,36o) with 36 qubits. Transpiled two-qubit gate counts fall from 4,660 to 1,777 for the largest cluster. For nitrogen, HCI-HSQD obtains errors below 1 mHa relative to variational HCI across the dissociation curve, with subspaces 18–39% smaller. Relative to reported SQD results, iron–sulfur energy errors against DMRG decrease by 76% and 26%, with subspaces 58% and 88% smaller, respectively, using half the qubits and measurements. A noniterative sample correction replaces self-consistent recovery for these clusters.
The paper states that inconsistent classical reference energies prevent a direct nitrogen comparison with prior SQD. Demonstrations use spin-symmetric systems; other cases would require separate spin-sector circuits. HCI comparisons exclude perturbative corrections, and the reported benchmarks do not establish quantum advantage.
Present-day sample-based quantum algorithms within quantum-centric supercomputing workflows require substantial classical post-processing, large circuit depths to produce chemically relevant samples, and long QPU times to execute the many circuits needed for molecules of chemically relevant size. We introduce FMO-QVM-HCI-HSQD, a resource-efficient sample-based framework that breaks a large problem into fragments, runs the fragments in parallel on virtual regions of the QPU, and reduces circuit depth through a half-qubit ansatz whose samples seed an HCI expansion. We demonstrate the algorithm on ibm_marrakesh and ibm_boston for the water trimer and the carmofur_Mpro complex with up to 52 qubits, benchmarking against FMO-CCSD(T), FMO-CASCI and FMO-HCI. For the water trimer (14 spin orbitals for monomers, 28 for dimers), our method reduces the two-qubit depth by 39% compared to FMO-QVM-SQD and the QPU time by 82% compared to sequential FMO-SQD, and accesses 3% of the determinant subspace versus 40% for FMO-SQD, while achieving more accurate energies. For the carmofur_Mpro complex (16 and 32 spin orbitals), it reduces the two-qubit depth by 47% and the QPU time by 20% compared to FMO-QVM-SQD, and accesses 0.007% of the determinant subspace versus 40%, while remaining within chemical accuracy. Extending the carmofur_Mpro active space in the 6-31G basis to 52 and 104 spin orbitals, sampled on 26 and 52 qubits, the total energy deviates from FMO-HCI by only 1.22 mHa. These results show that combining fragmentation, virtual QPU partitioning and half-qubit sampling makes sample-based quantum simulation of chemically relevant systems feasible on current hardware.
Sample-based Quantum Diagonalization (SQD) recovers ground-state energies by classically diagonalizing a Hamiltonian in the subspace spanned by quantum samples, requiring only bitstrings with sufficient ground-state overlap rather than an accurate variational energy. We reveal and exploit this underexplored robustness property: how much non-Clifford and variational expressivity can be removed from the sampling circuit before SQD accuracy degrades? We answer through two complementary compression techniques: gradient-based operator pruning, which discards low-impact excitation operators, and Clifford rounding, which snaps remaining parameters to the nearest Clifford angle. Both of these techniques can be applied to a VQE ansatz on a qubit-reduced Hamiltonian. A systematic ablation study across 21 molecules shows that median SQD error stays within chemical accuracy even at 50\% compression on both axes, while simulation speedup reaches 33×. Hardware validation on 6 molecules on IBM quantum hardware confirms up to 2.8× transpiled-depth reduction with zero loss in SQD accuracy. Our implementation can be found at: https://github.com/zkysfls/cs-vqe-sqd
Sampling-based quantum diagonalization method exploits Quantum-centric supercomputing platforms to sample bitstrings for Hamiltonian projection on a quantum computer, and then classically diagonalize the Hamiltonian to estimate the eigenvalues and eigenvectors. In current quantum devices an algorithm is useful when shallow quantum circuits with error mitigation support can discover better results while having either a proof of convergence or some method to explain trust in experiment. In this paper, we introduce DF-SQD, a hybrid algorithm that derives deterministic auxiliary-field circuits from selected double-factorization leaves of the two-electron tensor. The circuits propose occupation-number configurations, while selected configuration interaction evaluates the original active-space Hamiltonian and can recentre subsequent proposal rounds. On N2 (32 qubits; 6-31G basis) and a 40-qubit [Fe2S2(SCH3)4]2- active-space Hamiltonian, we show that DF-SQD improves the energy obtained from sampled determinant spaces while using shallow number-preserving circuits in both simulator and hardware runs. For N2, DF-SQD is 45x more accurate with a 11.23\% smaller subspace, and due to its ability to sample better bitstrings at lesser shots it is 2.93x faster than SQD in quantum devices. For the iron-sulfur cluster, DF-SQD generated a subspace dimension of 221M with 400K shots, while SQD needed 1.5M shots to generate a 238M subspace, thus we have better subspace recovery evident from the hardware at 3.75x reduced shots. At a matched 50M subspace dimension, DF-SQD is 1.32x more accurate (achieves a 24.5\% relative error reduction over standard SQD). So overall, our method is able to discover better results with shallower circuits, is sample efficient, uses configuration recovery (so has targeted error mitigation) and we have empirical convergence observation.
Quantum Selected Configuration Interaction (QSCI) and an extended protocol known as Sample-based Quantum Diagonalization (SQD) have emerged as promising algorithms to solve the electronic Schrödinger equation with noisy quantum computers. In QSCI/SQD a quantum circuit is repeatedly prepared on the quantum device, and measured configurations form a subspace of the many-body Hilbert space in which the Hamiltonian is diagonalized classically. For the dissociation of N 2 and a model [2Fe – 2S] cluster (correlating 10 electrons in 26 orbitals and 30 electrons in 20 orbitals, respectively) we show that a nonperturbative stochastic approach, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), using truncated SQD trial wave functions obtained from quantum hardware can recover a substantial amount (e.g., O (100) mHa) of correlation energy. This hybrid quantum-classical combination has the potential to greatly reduce the sampling burden placed on the QSCI/SQD procedure, and is a compelling alternative to recently proposed hybrid ph-AFQMC algorithms that rely on quantum state tomography.
Quantum and hybrid quantum-classical algorithms have the potential to improve the accuracy and scale of quantum chemistry and impact a broad range of applications such as drug discovery and catalysis. In particular, subspace methods such as sample-based quantum diagonalization (SQD) can already tackle classically challenging systems. However, practical algorithms that go beyond the computation of electronic ground and excited states are still scarce. Here, we introduce the variable-geometry sample-based quantum diagonalization (VGSQD) algorithm, which extends SQD to variable molecular geometries by reusing quantum samples from a reference geometry at displaced ones, thus significantly reducing quantum computational cost. Using circuits of up to 124 qubits on ibm-quebec, an IBM Heron r2 quantum processor, we determine the equilibrium geometry and energy of the weakly bound, strongly correlated beryllium dimer, as well as the water molecule. Finally, we extend VGSQD to chemical reactions and compute the minimum energy path and energy barrier of a hydrogen exchange reaction. The results are comparable with state-of-the-art classical methods such as CCSD(T) and DMRG. This work broadens the scope of quantum algorithms for chemistry, from energies to geometry optimization and reaction pathways, both essential for practical applications.
Abstract The simulation of molecular electronic structure is an important application of quantum devices. Recently, it has been shown that quantum devices can be effectively combined with classical supercomputing centers in the context of the sample-based quantum diagonalization (SQD) algorithm. This allowed the largest electronic structure quantum simulation to date (77 qubits) and opened near-term devices to practical use cases in chemistry toward the hundred-qubit mark. However, the description of many important physical and chemical properties of those systems, such as photo-absorption/-emission, requires a treatment that goes beyond the ground state alone. In this work, we extend the SQD algorithm to determine low-lying molecular excited states. The extended-SQD method improves over the original SQD method in accuracy, at the cost of an additional computational step. It also improves over quantum subspace expansion based on single and double electronic excitations, a widespread approach to excited states on pre-fault-tolerant quantum devices, in both accuracy and efficiency. We employ the extended SQD method to compute the first singlet (S 1) and triplet (T 1) excited states of the nitrogen molecule with a correlation-consistent basis set, and the ground- and excited-state properties of the [2Fe-2S] cluster.