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Hybrid quantum-classical approach to correlated materials
Bela Bauer, Dave Wecker, Andrew J. Millis, Matthew B. Hastings, M. Troyer
TL;DR
Complex correlated materials are difficult to solve directly because classical many-body methods scale exponentially and naive quantum simulations require prohibitive resources. The paper combines classical DFT+DMFT embedding with a quantum impurity solver, enabling substantially larger simulations and targeting problems within reach of roughly one hundred logical qubits.
Problem
Directly simulating complex correlated materials remains resource-intensive, while idealized models do not quantitatively describe real materials.
Method
A hybrid DFT+DMFT algorithm uses classical computation for most orbitals and a quantum computer to solve the self-consistently determined correlated impurity problem.
Results
A small quantum computer could address impurity problems with approximately 10^2 degrees of freedom, including systems with multiple correlated atoms per unit cell and cluster-DMFT calculations.
Takeaways & Limitations
The approach could make small quantum computers useful for simulating larger strongly correlated materials and complex molecules.
Abstract
from arXiv · showhide
Recent improvements in control of quantum systems make it seem feasible to finally build a quantum computer within a decade. While it has been shown that such a quantum computer can in principle solve certain small electronic structure problems and idealized model Hamiltonians, the highly relevant problem of directly solving a complex correlated material appears to require a prohibitive amount of resources. Here, we show that by using a hybrid quantum-classical algorithm that incorporates the power of a small quantum computer into a framework of classical embedding algorithms, the electronic structure of complex correlated materials can be efficiently tackled using a quantum computer. In our approach, the quantum computer solves a small effective quantum impurity problem that is self-consistently determined via a feedback loop between the quantum and classical computation. Use of a quantum computer enables much larger and more accurate simulations than with any known classical algorithm, and will allow many open questions in quantum materials to be resolved once a small quantum computer with around one hundred logical qubits becomes available.
I. INTRODUCTION
DFT makes many weakly correlated materials tractable, but strongly correlated materials remain difficult to describe quantitatively. Direct many-body methods and naive quantum simulations face prohibitive resource requirements for complex correlated materials.
- DFT maps the interacting electronic problem onto non-interacting electrons with a density-dependent potential and self-consistency, enabling classical simulation.
- DFT approximations such as LDA yield reliable results for many weakly correlated materials, including band insulators, metals, and semiconductors.
- Full configuration interaction is limited to small systems because its classical computational effort scales exponentially with the number of orbitals.
- Naive direct quantum simulations of complex correlated materials remain impractical because relevant interaction terms and thousands of electrons must be considered.
- Idealized models such as the Hubbard model capture qualitative phenomena but do not provide quantitative descriptions of real materials.
II. A HYBRID QUANTUM-CLASSICAL APPROACH
The proposed DFT+DMFT framework assigns the large, inexpensive part of the electronic structure to classical DFT and the small correlated impurity problem to a quantum computer. This embedding reduces the quantum workload while retaining a self-consistent, frequency-dependent treatment, potentially extending simulations to larger correlated systems.
- A hybrid quantum-classical approach: DFT defines orbitals and treats most of the electronic structure, while DMFT solves a reduced model containing a smaller set of correlated orbitals.
- A hybrid quantum-classical approach: The hybrid method requires a highly accurate, full-frequency Green’s-function solution of the reduced correlated problem, while remaining an approximation to the true answer.
- A hybrid quantum-classical approach: DMFT approximates the full correlated problem through an interacting impurity cluster self-consistently coupled to a non-interacting electron bath.
- A hybrid quantum-classical approach: An impurity problem with approximately 10^2 degrees of freedom could enable multiple correlated atoms per unit cell and cluster-DMFT calculations of real materials.
- A hybrid quantum-classical approach: Bath energies and couplings are iteratively determined from a self-consistency condition involving impurity and lattice Green’s functions.
- A hybrid quantum-classical approach: Classical impurity solvers scale exponentially with total system size and typically limit practical calculations to about five or fewer correlated orbitals.
III. QUANTUM ALGORITHM FOR THE IMPURITY SOLVER
The quantum impurity solver combines ground-state preparation, real-time Green’s-function measurements, and classical transformations within a self-consistent DMFT loop. Scaling analysis indicates that impurity problems relevant to materials could fit small quantum computers of about one hundred qubits.
- Quantum resources: The impurity wavefunction requires only Nso + Nd logical qubits, while bath size increases qubit requirements without strongly affecting computation time.The method exploits the Hamiltonian’s sparsity structure.
- Quantum resources: Ground-state preparation combines adiabatic interpolation with quantum phase estimation, which projects the evolved state onto an energy eigenstate.QPE avoids separately measuring noncommuting Hamiltonian terms that would destroy the state.
- Green’s-function estimation: Real-time particle and hole Green’s functions are measured and transformed to Matsubara frequencies for enforcing DMFT self-consistency.The transformation uses a Hilbert integral evaluated on a discrete logarithmic frequency grid.
- Numerical baseline: The numerical baseline uses a single spinful impurity coupled to five spinful bath sites and performs a self-consistent DMFT simulation for two interaction strengths.The simulated impurity has Nso = 2 and Nb = 10.
- Sampling and scaling: O((1/ϵ) log(1/ϵ)) coherent sampling replaces O(1/ϵ^2) incoherent sampling, reducing the example’s measurements from 400 to 20 and yielding roughly a ten-fold improvement.The improvement becomes more significant when higher accuracy is required.
- Sampling and scaling: A 10-orbital problem with 60–100 bath sites appears within reach of a small quantum computer of about one hundred qubits.The estimate involves about 10^8 measurements, each requiring a coherent run of about 10^8 gates, for a total of 10^16 gates across independent runs.
IV. OUTLOOK AND DISCUSSION
The approach embeds a quantum impurity solver within classical embedding methods and can generalize beyond DFT+DMFT. The authors argue that small quantum computers could address larger, strongly correlated materials, while contrasting their method with variational and finite-temperature alternatives.
- Embedding framework: The hybrid method selects orbitals for higher-accuracy treatment and feeds the quantum impurity solution back into the DFT calculation.This resembles complete active space methods but adds feedback into the DFT problem.
- Embedding framework: The framework can generalize from DFT+DMFT to other embedding approaches, including density-matrix embedding theory.DMET also seeks self-consistency between an extended non-interacting lattice model and an interacting impurity problem.
- Outlook: Small quantum computers could support simulations of larger systems, especially strongly correlated crystalline materials and complex molecules.The paper presents these systems as targets with broad physical phenomena and applications.
- Comparison with alternatives: Variational quantum eigensolvers are well suited to simple local model Hamiltonians because they rely on few non-commuting terms, limiting their practicality for more complex systems.The paper positions its embedding strategy as a route beyond this restriction.
- Comparison with alternatives: Compared with a related impurity-solver approach, this work uses the more broadly applicable DMFT method and a zero-temperature formulation.The related method operates at finite temperature and does not estimate algorithmic scaling or baseline gate counts.
Appendix A: Introduction to Quantum circuits
The appendix introduces controlled-gate circuits for measuring unitary expectation values and implementing the Hamiltonian terms used in the simulation.
- An ancilla prepared in (|0⟩+|1⟩)/√2 becomes entangled through a controlled unitary, making the unitary expectation value accessible from ancilla measurements.
- The Hamiltonian terms are represented by chemical-potential, hopping, and interaction circuits, with a control qubit included for subsequent algorithmic steps.
- General hopping circuits require a Jordan-Wigner transformation to account for fermionic signs, although the simplest nearest-neighbor case does not.
1. Quantum simulation of time evolution
Quantum time evolution maps spin-orbital occupations onto qubits and approximates exp(−iHt) with a Trotter-Suzuki decomposition assembled from simple term-wise circuits.
- The simulation allocates one qubit per spin-orbital in a second-quantized occupation-number basis and applies the time-evolution unitary exp(−iHt).
- The Hamiltonian is decomposed into non-commuting one-body and two-body terms, whose exponentials are applied sequentially in a Trotter-Suzuki approximation.
- The Trotter approximation becomes exact as the number of time subdivisions N approaches infinity, and higher-order decompositions may improve performance.
- A single Trotter step is implemented with circuits for chemical-potential, interaction, and hopping terms.
- For adiabatic state preparation, the Hamiltonian parameters are updated at each of the N time steps.
2. Quantum phase estimation
Quantum phase estimation extracts energies from controlled time evolution, while Green’s-function measurements use controlled unitaries and require additional terms when superconductivity breaks particle-number conservation.
- Quantum phase estimation controls exp(−iHt) with an ancillary qubit and infers energy from the resulting phase difference.
- Larger evolution times make the phase difference sensitive to smaller energy differences, enabling precise energy measurement from multiple times.
- Real-time Green’s-function expectation values are converted into expectations of unitary operators measured with a controlled-unitary ancilla protocol.
- Without superconductivity, particle-number-nonconserving terms vanish in the ground state, so obtaining the real and imaginary parts of Gp(t) and Gh(t) requires 4 measurements.
- With superconductivity, cross-terms contribute and the required measurements increase from 4 to 8 at each time point.
Appendix C: Self-consistency and bath fitting
The appendix describes reconstructing real- and imaginary-frequency Green’s functions and iteratively fitting discrete bath parameters to enforce DMFT self-consistency.
- The self-consistency loop is most conveniently executed in Matsubara frequencies, requiring the impurity solver’s Green’s function in imaginary time.
- The real-time Green’s function is Fourier transformed, numerically integrated over selected times, and optionally broadened by η when extracting spectral information.
- The spectral function has particle contributions at positive frequencies, hole contributions at negative frequencies, and satisfies A(ω) ≥ 0 in equilibrium.
- A Hilbert transformation converts the spectral function or real-frequency Green’s function into the imaginary-frequency Green’s function.
- DMFT self-consistency relates impurity and lattice Green’s functions through the self-energy and a hybridization function determined by discrete bath parameters.
- Each iteration computes the self-energy, updates the non-interacting Green’s function, and fits bath energies ϵ_i and couplings V_i by nonlinear optimization.
- For the infinite-coordination Bethe lattice, the semicircular density of states yields a concise self-consistency condition used with the same bath-fitting procedure.
1. Incoherent approach
The incoherent workflow repeatedly prepares a ground state, measures an expectation value, and uses energy-based projection to reuse the state when successful. Its preparation and measurement costs remain substantial, with re-preparation frequency depending on the model.
- The workflow prepares the ground state by adiabatic evolution from an easily prepared initial state.
- It then measures the expectation value of the unitary operator U αβ meas.
- Quantum phase estimation on U = e−itH can project the system back into the known ground state for reuse after a successful measurement.If a different energy is obtained, the state is orthogonal to the ground state and must be re-prepared.
- The probability of returning to the ground state depends on model details and must be estimated case by case.
- 1.7 · 105 ground-state preparations and 6.8 · 106 QPE executions were required in numerical simulations of the example.
- The coherent measurement procedure reduces the number of measurements required at each time step quadratically in accuracy.