Dynamical mean field theory algorithm and experiment on quantum computers

• Strong electron correlations can make DFT predictions incorrect, including for Mott transitions and oxygen binding to haemoglobin.Source: §I · p. 1
• DMFT treats strongly interacting local orbitals as an effective impurity coupled to a bath and neighboring orbitals.Source: §I · p. 1
• DMFT is computationally limited on conventional computers, while current quantum hardware is noisy and too small for large materials simulations.Source: §I · pp. 1–2
• The proposed VQE-based hybrid algorithm solves the effective quantum impurity problem and performs the remaining DMFT computation classically.Source: §I · p. 2
• The algorithm implements a two-site DMFT calculation on superconducting and trapped-ion quantum devices, with scalability tied to VQE scalability.Source: §I · p. 2

A. Dynamical mean field theory

• The ED impurity solver replaces the infinite bath with finitely many effective sites and maps the resulting Hamiltonian to qubits using Jordan-Wigner transformation.Source: A. Dynamical mean field theory · p. 2
• The impurity Hamiltonian contains onsite energies, electron interactions, impurity-bath hopping, and bath onsite energies, requiring 2(Nimp + Nb) qubits.Source: A. Dynamical mean field theory · p. 2
• The solver obtains eigenvalues and eigenvectors by electron-number sector, then uses particle and hole excitations to construct the impurity Green’s function.Source: A. Dynamical mean field theory · pp. 2–3
• DMFT seeks self-consistency by iteratively updating impurity and bath parameters so the impurity and lattice Green’s functions agree as closely as possible.Source: A. Dynamical mean field theory · p. 3
• Only low-energy excitations up to a specified cutoff are typically needed, although the total number of excited states grows exponentially with impurity and bath size.Source: A. Dynamical mean field theory · p. 3
• Quantum-computed Green’s functions and classical self-energy calculations are combined within the iterative loop to update the impurity Hamiltonian parameters.Source: A. Dynamical mean field theory · p. 3, Figure 1 · p. 3

B. Quantum Circuits

• Real-time Green’s-function Fourier transforms require very small time steps because finite-step errors grow with interaction strength and current hardware noise prevents accurate evaluation.Source: B. Quantum Circuits · pp. 3–4
• The proposed method obtains spectral energies and matrix elements through VQE total-energy calculations, which are generally more noise resilient than real-time evolution.Source: B. Quantum Circuits · p. 4
• A hardware-efficient ansatz prepares eigenstates, using only Ry rotations when real Hamiltonian parameters permit real expansion coefficients.Source: B. Quantum Circuits · p. 4
• The ground state is found by minimizing total energy, while modified Hamiltonians with penalty terms target the N0 ± 1 electron sectors for excitation spectra.Source: B. Quantum Circuits · p. 4
• Higher excited states are obtained by penalizing overlap with previously identified lower-energy states.Source: B. Quantum Circuits · p. 4
• The required matrix elements are evaluated from quantum circuits without increasing the qubit count.Source: B. Quantum Circuits · p. 4

C. Regularization

• Small deviations in quantum-computed quantities can leave self-energy divergences only partially cancelled, causing the DMFT loop to fail to converge.Source: C. Regularization · p. 4
• The divergences occur where the noninteracting Green’s function vanishes, corresponding to poles of the hybridization function at the bath energies.Source: C. Regularization · p. 4
• Regularization enforces sum rules by modifying spectral weights while fixing excitation energies, because the weights are expected to be less accurate and enter linearly.Source: C. Regularization · p. 4
• Restricting constrained minimization to the spectral weights makes the regularization straightforward and scalable to larger system sizes.Source: C. Regularization · p. 4

• The two-site model represents the Bethe-lattice Hubbard model with one interacting impurity site and one bath site using exact diagonalization.Source: A. 2-site DMFT model · pp. 4–5
• The four-qubit mapping assigns qubits to spin orbitals while exploiting spin symmetry to evaluate the Green’s function and self-energy for one spin only.Source: A. 2-site DMFT model · p. 5
• Only two bath parameters, ϵ2 and V = V12, are optimized in the two-site DMFT self-consistency loop.Source: A. 2-site DMFT model · p. 5
• The impurity and lattice occupations are obtained by integrating their respective densities of states, while z is introduced as the quasiparticle weight.Source: A. 2-site DMFT model · p. 5

• At particle-hole symmetry, only particle contributions need calculation because hole energies and matrix elements mirror them, reducing the required quantum evaluations.Source: B. Quantum computing implementation · p. 5
• The penalty-term circuit uses four qubits and eight optimized angles, whereas the reduced circuits exploit conserved electron number and spin sectors.Source: B. Quantum computing implementation · pp. 5–6, Figure 2 · p. 5
• The circuit-reduction approach uses separate reduced circuits, reaching two qubits for all electron sectors and requiring fewer angles, gates, and qubits than the penalty-term approach.Its drawback is that projecting a general Hamiltonian onto conserved-sector subspaces can be challenging.Source: B. Quantum computing implementation · pp. 5–6
• The two-site DMFT problem requires three energies, E0, E3,0, and E3,2, plus one independent spectral weight λ on the quantum computer.Source: B. Quantum computing implementation · p. 6
• For U = 4 and V = 0.745356, infinite-shot simulations reproduce exact energies and λ, while 5000-shot calculations show energy discrepancies within about 2% and λ discrepancies within about 10%.Source: B. Quantum computing implementation · p. 6, Table I · p. 6
• With regularization, the quantum DMFT loop converges for all U and its self-consistent z agrees well with exact analytical values; without regularization, convergence usually fails.Finite-shot noise slightly smooths the transition from finite to zero z.Source: Figure 3 · p. 6, B. Quantum computing implementation · p. 7

IBM superconducting qubits

• The IBM 2-qubit CR measurements produced energies within about 5% of exact values, while the 4-qubit PT energies were within about 10–20%.λ deviations were about 20% for CR and 60% for PT, but DMFT obtains λ from an exact sum rule, leaving final accuracy determined by the energies.Source: C. Experiments on quantum computers · p. 7, Table II · p. 7

Trapped-ion qubits

• On the trapped-ion device, energy errors were about 1–3% for 2 qubits and 6–8% for 4 qubits, while CR λ essentially matched the exact value.The reported measurements use the particle-hole-symmetric case, with a general non-particle-hole-symmetric example provided separately.Source: Table III · p. 7, C. Experiments on quantum computers · p. 8

VQE optimization

• Device-side VQE optimization converged within 2% of exact energies and produced λ accurate to about 5% on the IBM quantum computer.The results were better than measurements using simulator-optimized angles, indicating partial compensation of hardware calibration errors.Source: Table II · p. 7, C. Experiments on quantum computers · p. 8

Full DMFT loop

• After 7 iterations, the IBM 2-qubit CR DMFT loop reached V = 0.755, close to the exact value, with convergence tolerance |V − √z| < 0.01.The resulting energies and λ were also close to their exact values after regularization.Source: C. Experiments on quantum computers · p. 8

Electronic structure

• The DMFT+VQE experiment reproduced the impurity and Bethe-lattice density of states well using the IBM self-consistent V = 0.755, compared with the exact V = 0.745356.The experiment used the 2-qubit reduced circuit approach and included many-electron interactions.Source: C. Experiments on quantum computers · p. 8, Figure 4 · p. 8
• The interacting DMFT density of states showed the correct three-peak structure, unlike the non-interacting Bethe-lattice result with a single broad peak.Small deviations from exact results were attributed to differences between experimental and exact VQE energies.Source: C. Experiments on quantum computers · p. 8

IV. CONCLUSIONS

• The algorithm performs DMFT calculations on currently available quantum computers, with 2-site benchmarks on superconducting and trapped-ion qubits showing low error.The approach relies on VQE total-energy calculations, which are generally more noise-resilient than real-time state evolution.Source: §IV · p. 8

1. General relations

• The Hamiltonian commutes with total electron number and spin, allowing eigenstates to be evaluated separately in fixed-N and fixed-Sz sectors.These conserved quantities enable reduced-qubit Hamiltonians for the relevant sectors.Source: §1 · pp. 9–10

2. N =0

• For N = 0 and N = 4, the constraints leave one possible state, whose energies are given analytically by the reduced-sector Hamiltonian.The appendix gives EN=0,Sz=0 = µ − U/4 − ϵ2 and EN=4,Sz=0 = −µ + 3U/4 + ϵ2.Source: §2 · p. 10, §6 · p. 11

3. N =1

• For N = 1, transformed operators determine spin imbalance and electron-number imbalance across two sites; spin flips and spin-preserving electron hops generate their ladder operations, yielding a two-qubit representation.Source: §3 · p. 10

4. N =2

• For N = 2 and Sz = 0, transformed Pauli operators enforce the sector constraints and represent the Hamiltonian on 2 qubits.The Sz = ±2 sectors contain single states with analytically specified energies.Source: §4 · pp. 10–11

7. Matrix elements with circuit reduction

• The reduced circuits can evaluate selected particle and hole Green’s-function matrix elements entirely on 2 qubits by exploiting sector selection rules and appropriate Ry rotations.The appendix explains when matrix elements vanish and how circuit ordering or an added Ry(π) rotation accesses finite elements.Source: §7 · pp. 11–12

Appendix C: Additional 4-qubit experimental data

• SPAM correction generally improves the measured data, but the improvement is not systematic and does not improve the λ3,1,n matrix elements.The exact particle-hole relations are fulfilled to a good approximation overall.Source: Appendix C · p. 12
• Table IV measures E3,n and λp,1,n for n ∈ {0, 1, 2, 3} on IBM and UMD quantum computers, comparing raw and SPAM-corrected results.The measurements extend the 4-qubit results for the optimized ansatz parameters.Source: Table IV · p. 12

Appendix D: Non-particle-hole symmetric system

• Away from particle-hole symmetry, one- and three-electron quantities must be computed separately, which mainly doubles the quantum computations.The 2-qubit circuit-reduction approach still fully describes all states, whereas the original 4-qubit circuit is only approximate.Source: Appendix D · p. 12
• The extended 4-qubit circuit in Fig. 5 is the smallest extension that fully reproduces exact results for all parameters of the general 2-site DMFT Hamiltonian.The extension adds one Ry rotation, so the expected result quality is similar to the particle-hole-symmetric case.Source: Figure 5 · p. 12, Appendix D · p. 13
• For the ion-trap experiments, 2-qubit results compute only states with finite λp/h,1,n; the remaining states can be obtained by adding an Ry(π) rotation.The table compares optimized-ansatz measurements with numerically exact values.Source: Table V · p. 13