Hybrid Quantum-Classical Approach to Correlated Materials

Recent improvements in the 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 100 logical qubits becomes available.

Popular Summary

Quantum computing holds promise for solving problems that seem far beyond the reach of even the largest current supercomputers: the simulation of quantum systems, cryptography, and machine learning. Recently, advances in the control of quantum systems have made it appear feasible to build the first devices that can potentially reap these benefits within a decade. These advances have prompted the question of which real-world problems would most benefit from such small, first-generation quantum computers. One example that has been widely studied is the simulation of molecular energies in quantum chemistry. Meanwhile, the highly relevant problem of simulating strongly correlated materials appears to require a prohibitive amount of resources when solved using conventional, direct approaches. Here, we show how this limitation can be overcome using a hybrid quantum-classical approach.

We make use of a well-established framework of embedding methods to reduce the description of a complex material to a simple but highly entangled impurity model. As a first step, we use a computationally inexpensive density-functional-theory calculation to approximately determine the electronic structure of the material. We then focus on the most relevant orbitals and use dynamical mean-field theory, which is more computationally expensive, to study the effects of strong interactions on these orbits. A key part of this second step is that a highly entangled impurity model must be solved. While solving this impurity model on a classical computer remains an enormous challenge and can be achieved only in special cases, we demonstrate that this task is ideally suited for a small quantum computer with roughly 100 logical qubits.

We expect that our results will motivate future studies focusing on the experimental feasibility of this hybrid quantum-classical approach.

Figure 1

Overview of the $\mathrm{DFT}+\mathrm{DMFT}$ approach. In our proposal, the solution of the impurity problem (highlighted in red), which is the computationally limiting step in computations using classical computers, is performed by a quantum computer.

Figure 2

Spectral function of a Hubbard model on the Bethe lattice (with hopping $t_{ij}=t/\sqrt{2z}$ in the limit $z\to \infty$), with $U/t=8$ (upper panel) and $U/t=2$ (lower panel) and $N_b=10$ bath sites. For this simulation, both the imaginary-time Green’s function required in the self-consistency as well as the spectral function are extracted from real-time Green’s function data. For the self-consistency, a time grid of 1200 points from $T_{\min }={10}^{-5}$ to $T_{\max }=40$ is used. Other parameters of the self-consistency are $\beta =20$, $N_{\omega }=400$. For extracting the spectral function, a time grid of 1000 points with the same limits is used. We emphasize that no analytic continuation is required to obtain $A(\omega )$ in our approach. $N_{\mathrm{meas}}$ in the legend indicates how many good samples of the real-time Green’s function are obtained at each time slice. The results for $N_{\mathrm{meas}}=100$ are very similar to those for larger $N_{\mathrm{meas}}$, and those for $N_{\mathrm{meas}}=400$ are almost indistinguishable from those for $N_{\mathrm{meas}}=1600$.

Figure 3

Overview of the incoherent estimation of the Green’s function.

Figure 4

Incoherent measurement circuit for $U_{\mathrm{meas}}$. Here, $S=\sqrt{Z}$ is applied only if the imaginary part of the expectation value is desired.

Figure 5

(a) High-level overview of one step of the Trotterized time evolution. (b) Circuit $h_1=\exp (-iTϵc_i^{†}{c}_i)$. (c) Circuit $h_U=\exp (-iTU{n}_i{n}_j)$, where $\theta =TU/2$. (d) Circuit $h_2=\exp (-iTt[c_i^{†}{c}_j+c_j^{†}{c}_i])$. Here, we show the simplest case $j=i+1$, which does not require a Jordan-Wigner transformation.