Source-linked AI summary
Reduced Density Matrix Sampling: Self-consistent Embedding and Multiscale Electronic Structure on Current Generation Quantum Computers
Jules Tilly, P. V. Sriluckshmy, Akashkumar Patel, Enrico Fontana, Ivan Rungger, Edward Grant, Robert Anderson, Jonathan Tennyson, George H. Booth
TL;DR
The paper asks whether noisy current-generation quantum computers can support fully self-consistent multiscale electronic-structure calculations. It couples correlated active regions to classical environments through sampled RDMs, using error mitigation and compact measurement groupings. The resulting algorithms are reported as robust across quantum-chemistry and condensed-matter examples, including non-energetic observables.
Problem
Fully self-consistent quantum-classical multiscale algorithms remain technically challenging because noisy active-space RDM sampling can affect robust convergence.
Source: §I · p. 1Method
The paper couples correlated quantum regions to classical environments through sampled RDMs, compact tensor-product-basis groupings, and light-touch error mitigation.
Source: §I · pp. 1–2, §II · p. 4Results
The unified approach supports robust self-consistency in CASSCF and EwDMET, while RDMs also provide dipole, self-energy, and mass-renormalization information.
Source: §I · p. 2, §V · p. 11Takeaways & Limitations
The results support routine self-consistency as a way to optimize correlated subspaces in hybrid quantum-classical electronic-structure calculations.
Source: §I · p. 1, §V · p. 11
Abstract
from arXiv · showhide
We investigate fully self-consistent multiscale quantum-classical algorithms on current generation superconducting quantum computers, in a unified approach to tackle the correlated electronic structure of large systems in both quantum chemistry and condensed matter physics. In both of these contexts, a strongly correlated quantum region of the extended system is isolated and self-consistently coupled to its environment via the sampling of reduced density matrices. We analyze the viability of current generation quantum devices to provide the required fidelity of these objects for a robust and efficient optimization of this subspace. We show that with a simple error mitigation strategy and optimization of compact tensor product bases to minimize the number of terms to sample, these self-consistent algorithms are indeed highly robust, even in the presence of significant noises on quantum hardware. Furthermore, we demonstrate the use of these density matrices for the sampling of non-energetic properties, including dipole moments and Fermi liquid parameters in condensed phase systems, achieving a reliable accuracy with sparse sampling. It appears that uncertainties derived from the iterative optimization of these subspaces is smaller than variances in the energy for a single subspace optimization with current quantum hardware. This boosts the prospect for routine self-consistency to improve the choice of correlated subspaces in hybrid quantum-classical approaches to electronic structure for large systems in this multiscale fashion.
I. INTRODUCTION
The paper develops self-consistent quantum-classical multiscale methods that couple a small strongly correlated active space to a classical environment through sampled reduced density matrices. It demonstrates feasibility across quantum chemistry and condensed matter applications despite current-device noise.
Approach
- Reduced density matrices provide the low-rank quantum variables coupling a strongly entangled active region to weakly interacting external degrees of freedom.The external space is treated classically with polynomially scaling resources, while the active region is realized on the QPU.Source: §I · p. 1
Motivation
- Self-consistent quantum-classical algorithms remain technically challenging because active-space RDM sampling must tolerate inherent quantum noise.The paper identifies robust convergence under noisy sampling as a central practical requirement.Source: §I · p. 1
Methods
- Commuting-term grouping and light-touch error mitigation address two practical bottlenecks: RDM measurement cost and noisy self-consistency.The grouping strategy substantially reduces sampled terms, while mitigation supports convergence on small quantum devices.Source: §I · p. 2
Applications
- The unified framework applies self-consistent active-space optimization to both strongly correlated quantum chemistry and condensed matter problems.The two concrete approaches are CASSCF for molecules and EwDMET for extended systems.Source: §I · pp. 1–2
Quantum chemistry
- In carbon-monoxide CASSCF, mitigation enables reliable active-orbital convergence and supports dipole-moment evaluation beyond energetics.The QPU optimization remains robust despite sampling noise.Source: §I · p. 2
Condensed matter
- In Bethe-Hubbard EwDMET, QPU treatment opens Hubbard bands and yields excellent agreement for local density of states and Matsubara self-energy.The result uses error-mitigated active-space RDMs within the self-consistent embedding calculation.Source: §I · p. 2
II. SAMPLING REDUCED DENSITY MATRICES ON A QUANTUM COMPUTER
The section defines RDM-based quantum measurements, reduces their cost through Pauli-string and tensor-product-basis grouping, and tests sampling fidelity on simulated and current hardware. Error mitigation substantially improves hardware estimates, although grouping introduces hardware-depth trade-offs.
RDM formulation
- RDMs retain the m-particle distribution and entanglement information needed for observables whose operators have rank up to m.Dipole moments require the one-body RDM, whereas energies require the two-body RDM.Source: §II · p. 3
- Jordan-Wigner mapping converts fermionic operators into Pauli strings that can be measured on a QPU, with symmetry reducing the independent quantities required.The one- and two-body RDMs are sampled from these mapped operators.Source: §II · p. 3
Operator grouping
- Tensor Product Basis grouping reduces the number of measurements by combining commuting Pauli strings, and the resulting optimal term sets can be reused for systems with the same orbital count.The grouping is largely independent of Hamiltonian details for a fixed number of orbitals.Source: §II · p. 3
- Over two orders of magnitude fewer terms are required for a 16-orbital active space after Tensor Product Basis grouping.The reduction factor increases for larger active spaces, while higher-body groupings are explored for future coupling schemes.Source: §II · pp. 3–4
Limitations
- Tensor Product Basis grouping can increase covariance and circuit depth, so its sampling benefits may be outweighed by added noise on small current-generation QPUs.The additional basis-rotation circuit scales as O(N^2), motivating future use on larger or deeper circuits.Source: §II · p. 4
Hardware sampling
- Error mitigation substantially improves QPU RDM accuracy, bringing estimates nearly to simulator performance at an equivalent measurement count.The extrapolation method uses a binomial independent-error model without additional measurements.Source: §II · p. 4
III. QUANTUM CASSCF
Quantum CASSCF isolates strongly correlated active orbitals from core and virtual external spaces, then optimizes their partition self-consistently because mean-field orbital choices can change under correlation. Its practical scope is constrained by the factorial classical cost of representing larger active spaces, while faithful two-body RDM sampling is central to quantum self-consistency.
CASSCF orbital partition
- CASSCF partitions orbitals into core, active, and virtual subspaces, treating full quantum fluctuations only within the active space while excluding active–external entanglement.Core orbitals are treated as fully occupied and chemically inert, whereas virtual orbitals remain unoccupied.Source: §III · p. 5
Self-consistent active-space choice
- Mean-field orbital selection is insufficient because correlation can substantially change orbital occupations and entanglement, requiring variational rotations among core, active, and virtual orbitals.The orbital character is optimized by allowing arbitrary mixing between the three classes.Source: §III · p. 5
Active-space limitation
- Exact CASSCF is commonly limited to about 16 electrons in 16 orbitals because classical representation and optimization scale factorially with active-space size.Some calculations reach 20 electrons in 20 orbitals, but larger valence spaces may lack active-space convergence for desired properties.Source: §III · p. 5
Quantum implementation
- In quantum CASSCF, the active space is embedded in a mean-field environment, but robust self-consistency requires faithful sampling of the active-space two-body RDM.Orbital-rotation parameters are nonlinear functionals of sampled RDM elements, so unbiased sampling can still produce systematic final errors.Source: §III · p. 5
A. Fully self-consistent algorithm
The fully self-consistent algorithm alternates classical orbital optimization with quantum active-space optimization. After optimizing the active wave function for a rotated Hamiltonian, its two-body RDM is sampled and used to update orbital rotations until convergence.
Alternating optimization
- The algorithm alternates optimization of orbital rotations R and active-space coefficients c, using the active-space two-body RDM to support the classical rotation update.This separates the classical one-body unitary optimization from the quantum active-space state optimization.Source: A. Fully self-consistent algorithm · p. 6
Classical orbital update
- The orbital rotations are updated from energy gradients and Hessians using a quasi-second-order classical step with iterative subspace acceleration.The implementation uses the PySCF package for this classical optimization.Source: A. Fully self-consistent algorithm · p. 6
Active-space Hamiltonian
- For fixed R, core-electron Coulomb and exchange contributions are integrated out to construct an active-space Hamiltonian Hact(R) for the quantum calculation.The resulting Hamiltonian spans only the active degrees of freedom.Source: A. Fully self-consistent algorithm · p. 6
Quantum active-space step
- The active-space wave function is optimized through a VQE over circuit parameters θ, after which its two-body RDM elements are sampled for the next orbital update.Iterations continue until the orbital and active-space optimization converges.Source: A. Fully self-consistent algorithm · p. 6
B. Results
QPU-CASSCF orbital optimization remains stable under sampling and hardware noise, while light-touch error mitigation substantially improves energetic and dipole accuracy. The results also expose practical scaling boundaries: strong parallelization is possible, but higher-rank extensions require much larger RDM measurements.
CASSCF energy convergence
- Error-mitigated CASSCF converged within less than 10 mEh of the exact value, whereas unmitigated results retained approximately 60 mEh systematic error.Simulations converged strongly above 5,000 shots; at 500 shots, finite sampling prevented reaching within 10 mEh on average.Source: B. Results · p. 7, Figure 3 · p. 7
Error-source analysis
- Orbital optimization stayed accurate to within 10 mEh even without RDM error mitigation, while mitigation produced an almost exact CASSCF energy.Separating exact active-space energies from noisy orbital updates showed that most error came from VQE bias rather than orbital convergence or RDM sampling.Source: B. Results · pp. 7–8, Figure 4 · p. 8
Dipole moments
- Error-mitigated dipoles removed the systematic error and yielded fluctuations comparable to noiseless 5,000-shot simulations, while unmitigated results retained an approximately 0.6 D error.Correlation and orbital optimization also reversed the direction of the charge imbalance relative to the starting description.Source: B. Results · p. 8, Figure 5 · p. 8
Runtime and parallelization
- The full QPU-CASSCF calculation required approximately 14 QPU-hours, although the 8,000 measurements over 49 operators could have been parallelized to reduce runtime to well under a second.The authors caution that this parallelized estimate does not establish scaling or long-term viability.Source: B. Results · p. 8
Extensions and limitations
- CASSCF omits active–external-space contributions needed for quantitative accuracy, motivating perturbative or subspace extensions based on higher-rank RDMs.For a six-qubit four-body RDM, grouping reduced 440,154 Pauli strings to 3,182 commuting groups, but the extension remains future work.Source: B. Results · pp. 8–9
IV. QPU-ENHANCED ENERGY-WEIGHTED DENSITY MATRIX EMBEDDING
The QPU-enhanced EwDMET approach uses ground-state reduced density matrices to self-consistently couple a local correlated fragment to its environment through spectral moments. This provides a ground-state route toward dynamical embedding while keeping the correlated subspace size independent of the full system size.
Method rationale
- EwDMET combines density-matrix and dynamical mean-field perspectives while avoiding QPU Green’s-function calculations and explicit numerical fitting.It instead obtains one-particle spectral moments directly from ground-state RDMs and can systematically enlarge their number toward DMFT dynamics.Source: §IV · p. 9
Embedding construction
- The method begins from a local correlated fragment and self-consistently optimizes its hole and particle spectral moments to describe fluctuations between fragment and environment.The fragment may be a transition-metal d-shell or a cluster of lattice sites.Source: §IV · p. 9
Systematic limit
- As the maximum moment order tends to infinity, EwDMET reproduces effective DMFT dynamics, while low-order truncations target dominant low-energy fluctuations.The method maps the full system to a fragment-plus-bath problem and updates auxiliary and bath spaces until convergence.Source: §IV · p. 9
Scaling and self-consistency
- The correlated subspace size depends on fragment size and the requested spectral moments rather than the size of the full system.The QPU solves this subspace at each iteration, after which auxiliary states algebraically update the full-system one-particle description.Source: §IV · p. 9
A. Infinitely coordinated Bethe-Hubbard Lattice
EwDMET models the infinitely coordinated Bethe-Hubbard lattice using a single fragment site coupled to a bath, with self-consistency built from sampled RDM-derived spectral moments.
Model and cluster
- The infinitely coordinated Bethe-Hubbard model has site-local correlation-driven changes in one-particle properties and is an exact limit of EwDMET as nmom→∞.Its noninteracting spectrum is represented by a central fragment site with 200 additional degrees of freedom.Source: A. Infinitely coordinated… · p. 10
Self-consistency
- Truncating at nmom=1 constructs the self-consistent n=0 and n=1 fragment moments from one- and two-body RDMs.This makes the moments efficiently accessible from the VQE ground state of the cluster Hamiltonian.Source: A. Infinitely coordinated… · p. 10
- For nmom=1, the optimized particle and hole spectral centers capture local correlation effects, unlike single-site DMET self-consistency in translationally symmetric systems.The fragment moments are obtained after solving the VQE cluster and sampling the relevant RDMs.Source: A. Infinitely coordinated… · p. 10
Model and cluster
- At nmom=1, EwDMET uses a single fragment site and one bath orbital, producing a four-qubit cluster solved by VQE at each iteration.The cluster is coupled through self-consistently optimized spectral moments and uses the same error mitigation as the CASSCF calculations.Source: A. Infinitely coordinated… · p. 10
Results
- Figure 6 compares exact, finite-shot simulator, and IBMQ spectra for EwDMET at nmom=1, including hardware results with and without error mitigation.The caption specifies 5,000 RDM-sampling shots for both hardware experiments.Source: Figure 6 · p. 10
B. Results
The EwDMET calculations converge under finite-shot and hardware noise, while error mitigation substantially improves correlated spectra and self-energy estimates.
Spectral results
- All calculations converge within 8 iterations, and the final spectra reproduce Hubbard bands and a qualitatively correct Mott gap despite noisy RDM sampling.A small quasiparticle peak remains, so the nmom=1 result retains metallic character and does not yet represent a true Mott insulator.Source: B. Results · p. 11
- Error mitigation improves IBMQ spectra by resolving the Hubbard bands and low-energy peak more accurately than unmitigated results.Unmitigated results contain unrepresentative Hubbard bands because increasing RDM variance systematically underestimates the gap.Source: B. Results · p. 11
Sampling dependence
- At 10,000 shots, noiseless emulated spectra are almost indistinguishable from the exact benchmark at all energies.With fewer shots, increased RDM variance makes the Hubbard-band gap too small.Source: B. Results · p. 11
Self-energy
- Finite-shot self-energies approach exact results as shots increase, while error-mitigated hardware is more accurate than unmitigated hardware.Unmitigated hardware overestimates self-energy and produces larger effective mass and quasiparticle renormalization; higher nmom could improve resolution but requires higher-body RDMs.Source: B. Results · p. 11
- Figure 7 compares simulator self-energies against exact results and contrasts IBMQ Santiago and Bogota hardware with and without error mitigation at 5,000 shots.The plotted quantity is the converged effective on-site Matsubara self-energy.Source: Figure 7 · p. 11
V. CONCLUSIONS
The paper presents RDM sampling as a unified route to self-consistent quantum-classical embedding in quantum chemistry and condensed matter, with encouraging robustness on current hardware.
Contributions
- The approach self-consistently couples correlated quantum subspaces to classical environments through faithful, efficient sampling of reduced density matrices.The same framework targets molecular dipoles and condensed-matter self-energy and mass-renormalization quantities.Source: §V · p. 11
- Compact tensor-product bases reduce the number of sampled RDM terms, while simple error mitigation controls noise in the sampled density matrices.These strategies are used across the quantum chemistry and condensed-matter settings.Source: §V · p. 12
Conclusions
- Self-consistent optimization is particularly robust to sampling noise on current-generation quantum hardware and can be more reliable than single-iteration VQE energy uncertainties.The authors connect this robustness to prospects for improved hardware and state-preparation algorithms.Source: §V · p. 12
Limitations
- The conclusions are restricted to relatively small correlated subspaces, while quantitative accuracy requires higher-body density matrices and larger-qubit studies.Sampling higher-body RDMs and understanding scalability remain future research directions.Source: §V · p. 12
A. LIGHT TOUCH ERROR MITIGATION
The error-mitigation scheme estimates and removes measurement bias using a simple gate-error model, with electron-number deviations providing the required bias estimate.
Assumptions
- The mitigation method assumes a probability Perr that at least one gate error occurs during circuit propagation and measurement.The erroneous expectation value and its measurement distribution are treated as unknown.Source: A. LIGHT TOUCH ERROR MITIGATION · p. 15
Estimator
- Using large-sample normality, the measured expectation is modeled as a mixture of the true and erroneous expectations.The resulting expression is ⟨Ômes⟩=(1−Perr)⟨Ô⟩+Perr⟨Ôerr⟩.Source: A. LIGHT TOUCH ERROR MITIGATION · p. 15
- Assuming errors yield either ±1 eigenvalue with equal probability removes the erroneous-expectation term and supplies a lower-bound mitigation estimate without extra computation.The approximation is intended for the experiments rather than as a rigorous restoration of state purity.Source: A. LIGHT TOUCH ERROR MITIGATION · pp. 15–16
Bias estimation
- Perr is estimated from the electron-number operator by comparing the measured electron count with the target number of electrons.The one-body RDM terms needed for this estimate are already computed during VQE.Source: A. LIGHT TOUCH ERROR MITIGATION · p. 16
Comparison
- Calibration-based Perr estimates are generally less reliable because they treat circuit-error bias classically and miss effects associated with reduced quantum-state purity.The extrapolation approach instead captures this component of the bias.Source: A. LIGHT TOUCH ERROR MITIGATION · p. 16
B. OPERATOR GROUPINGS FOR RDM ELEMENT SAMPLING
The section groups commuting Pauli strings into tensor product bases to reduce measurements for reduced density matrices, while documenting higher-body extensions and experimental QPU layouts.
B. OPERATOR GROUPINGS FOR RDM ELEMENT SAMPLING
- The LDFC graph-colouring heuristic groups Pauli strings by commuting relationships, then assigns joint-measurement bases and circuits for each group.The procedure constructs a commutation graph, ranks vertices by degree, colors connected terms, identifies a multiplicative basis, and maps basis operators to single-qubit Z measurements.Source: B. OPERATOR GROUPINGS FOR RDM ELEMENT… · p. 16
- The TPB reduces the number of unique Pauli strings required to sample one- and two-body RDMs as active-space size increases.Table I compares direct Jordan–Wigner terms with grouped JW-TPB terms, while the reported results describe a substantial reduction.Source: B. OPERATOR GROUPINGS FOR RDM ELEMENT… · p. 16, Table I · p. 17
- Tables I and II distinguish grouped measurement requirements for one- and two-body versus three- and four-body RDMs across active-space sizes.The higher-body grouping study extends to six orbitals, but larger cases became computationally demanding for the available algorithm and resources.Source: B. OPERATOR GROUPINGS FOR RDM ELEMENT… · p. 16, Table I · p. 17, Table II · p. 17
- The greedy LDFC method does not guarantee optimal groups, so more efficient Pauli-string groupings may exist.Joint measurement follows the method presented in Ref. 36.Source: B. OPERATOR GROUPINGS FOR RDM ELEMENT… · p. 17
A. Lattice structures
- The experiments used IBMQ Bogota for error-mitigated CASSCF and EwDMET, Santiago without mitigation, and Athens for RDM sampling with and without mitigation.Bogota, Santiago, and Athens are five-qubit IBM Canary r3 QPUs with reported quantum volume 32.Source: Figure 8 · p. 17, Figure 9 · p. 17, A. Lattice structures · p. 17
B. Calibration information
The calibration section reports IBMQ hardware metrics averaged across qubits and connections near the experiment time, with values subject to later recalibration.
B. Calibration information
- The calibration values may change over time as IBM recalibrates the processors.The appendix notes that the machine information is sourced from the IBMQ portal and is time-dependent.Source: B. Calibration information · p. 18
- Table III summarizes selected IBMQ calibration metrics averaged over all qubits and connections near the experiment time.The reported values describe the hardware at a particular calibration point.Source: Table III · p. 18, B. Calibration information · p. 18