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arXiv:2201.11381 · Physical review. B./Physical review. B

Gutzwiller wave function on a quantum computer using a discrete Hubbard-Stratonovich transformation

Kazuhiro Seki, Yuichi Otsuka, Seiji Yunoki

Published Apr 13, 2022• 14 citations
arXivHTMLPDF

Abstract

We propose a quantum-classical hybrid scheme for implementing the nonunitary Gutzwiller factor using a discrete Hubbard-Stratonovich transformation, which allows us to express the Gutzwiller factor as a linear combination of unitary operators involving only single-qubit rotations, at the cost of the sum over the auxiliary fields. To perform the sum over the auxiliary fields, we introduce two approaches that have complementary features. The first approach employs a linear-combination-of-unitaries circuit, which enables one to probabilistically prepare the Gutzwiller wave function on a quantum computer, while the second approach uses importance sampling to estimate observables stochastically, similar to a quantum Monte Carlo method in classical computation. The proposed scheme is demonstrated with numerical simulations for the half-filled Fermi-Hubbard model. Furthermore, we perform quantum simulations using a real quantum device, demonstrating that the proposed scheme can reproduce the exact ground-state energy of the two-site Fermi-Hubbard model within error bars.

DOI
10.1103/physrevb.105.155119

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We propose a quantum-classical hybrid scheme for implementing the nonunitary Gutzwiller factor using a discrete Hubbard-Stratonovich transformation, which allows us to express the Gutzwiller factor as a linear combination of unitary operators involving only single-qubit rotations, at the cost of the sum over the auxiliary fields. To perform the sum over the auxiliary fields, we introduce two approaches that have complementary features. The first approach employs a linear-combination-of-unitaries circuit, which enables one to probabilistically prepare the Gutzwiller wave function on a quantum computer, while the second approach uses importance sampling to estimate observables stochastically, similar to a quantum Monte Carlo method in classical computation. The proposed scheme is demonstrated with numerical simulations for the half-filled Fermi-Hubbard model. Furthermore, we perform quantum simulations using a real quantum device, demonstrating that the proposed scheme can reproduce the exact ground-state energy of the two-site Fermi-Hubbard model within error bars.
We implement the Gutzwiller wave function for attractive SU(3) fermion systems on a quantum computer using a quantum-classical hybrid scheme based on the discrete Hubbard-Stratonovich transformation. In this approach, the nonunitary Gutzwiller operator is decomposed into a linear combination of unitaries constructed from two-qubit fermionic Givens-rotation gates, whose rotation angles are dictated by the auxiliary fields. We develop and reformulate two complementary methods to perform the sum over these auxiliary fields. In the first method, the Gutzwiller wave function is probabilistically prepared on the register qubits by projectively postselecting the desired state via measurements of ancilla qubits. We analyze the success rate both analytically and numerically as a function of the Gutzwiller variational parameter g for the Fermi-sea and BCS-like trial states at half filling. The success rate is found to decay exponentially for small | g | , but remains finite in the | g | → ∞ limit, with increasing | g | . In the second method, we employ importance sampling to address the Gutzwiller variational problem, where the central objective is to estimate the expectation values of observables. We demonstrate the proposed scheme by calculating the energy and triple occupancy of the attractive SU(3) Hubbard model in the framework of digital quantum simulation. Moreover, we present experimental results obtained on a trapped-ion quantum computer for the two-site attractive SU(3) Hubbard model, showing good agreement with exact values within statistical errors.
The determination of the ground state of quantum many-body systems via digital quantum computers rests upon the initialization of a sufficiently educated guess. This requirement becomes more stringent the greater the system. Preparing physically motivated Ans\"atze on quantum hardware is therefore important to achieve a quantum advantage in the simulation of correlated electrons. In this spirit, we introduce the Gutzwiller wave function (GWF) within the context of the digital quantum simulation of the Fermi-Hubbard model. We present a quantum routine to initialize the GWF that comprises two parts. In the first, the noninteracting state associated with the U=0U=0U=0 limit of the model is prepared. In the second, the nonunitary Gutzwiller operator that selectively removes states with doubly occupied sites from the wave function is performed by adding to every lattice site an ancilla qubit, the measurement of which in the $|0\ensuremath{\rangle}$ state confirms the operator was applied. Due to its nondeterministic nature, we estimate the success rate of the algorithm in generating the GWF as a function of the lattice size and the interaction strength U/tU/tU/t. The scaling of the quantum circuit metrics and its integration in general quantum simulation algorithms are also discussed.
Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are also suitable for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered a generalization of the discrete HSTs by J. E. Hirsch [Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by N. Rom et al. [Chem. Phys. Lett. 270, 382 (1997)]. An improvement of the acceptance ratio is found for a real auxiliary field, while an improvement of the average sign is found for a purely imaginary auxiliary field. Efficiencies of the different HSTs are tested in the single-band Hubbard model at and away from half filling by studying the staggered magnetization and energy expectation values, respectively.
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
We report a random-sampling method for computing the expectation value of physical quantities based on the Gutzwiler variational wave function. As the first application, we calculated the double occupancy, which is a critical quantity for understanding the correlation effects in many-body systems, for single-band 1D and 2D lattices. We demonstrated that the random-sampling scheme is more efficient than an existing Metropolis Monte-Carlo algorithm. For the 1D Hubbard model with only nearest-neighbour hopping, our results are almost identical to the exact analytic solution. We have also studied systems to which analytic solutions are not available, including the 1D lattices with next-nearest-neighbour hopping and 2D lattices. In addition, constraints on real-space configurations can be easily implemented in the current scheme to further improve the Gutzwiller wave function. As an example, we calculated the double occupancy for 1D Hubbard model by applying the constraint that all double-occupied sites are paired with an empty site. With enhanced correlation between double-occupied and empty sites, the constraint results in much improved ground-state energy for 1D Hubbard model with strong on-site repulsion.
We use a variational Monte Carlo method (VMC) and a quantum Monte Carlo method to study the two-dimensional half-filled Hubbard model. We use the Gutzwiller wave function as a trial wave function in VMC and measure energy, correlation functions, and momentum distribution for lattices up to 12\ifmmode\times\else\texttimes\fi{}12 sites. We then use a Monte Carlo algorithm to study the Hubbard model with the same boundary conditions as in VMC at very low temperatures. By comparing the results from these two methods, we assess the accuracy of the Gutzwiller wave function as a representation of the ground state of the Hubbard model. Major differences are found, implying that the Gutzwiller wave function is not a useful approximation to the ground state of the Hubbard model in two dimensions.