Unbiasing fermionic quantum Monte Carlo with a quantum computer

Abstract

Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems^1-4. Fermionic quantum Monte Carlo (QMC) methods^5,6, which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver^7,8, our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction.

Fig. 1. Imaginary-time evolution, sign problem and our quantum-classical hybrid algorithm.

a, Depiction of the imaginary-time evolution, which shows an exponential convergence to the ground state as a function of imaginary time, τ. b, Illustration of the fermionic sign problem. Exact deterministic imaginary-time evolution and an unconstrained QMC calculation, which is exact on average but has a signal-to-noise ratio that diverges with increasing τ due to the sign problem (top). Constrained QMC calculations with classical and quantum constraints. The use of quantum constraint helps to reduce the bias that is non-negligible when using the classical constraint (bottom). c, Overview of the QC-QMC algorithm. Stochastic wavefunction samples, represented as ${\left\{|{\phi }_i⟩\right\}}_{\tau }$, are evolved in time along with associated weights {w_i}_τ. Throughout the time evolution, queries to the quantum processor about the overlap value between the quantum trial wavefunction $|{\mathrm{\Psi }}_{\text{T}}⟩$ and a stochastic wavefunction sample ${\left\{|{\phi }_i⟩\right\}}_{\tau }$ are made while updating the gate parameters to describe ${\left\{|{\phi }_i⟩\right\}}_{\tau }$. Our quantum processor uses N qubits to efficiently estimate the overlap, which is then used to evolve w_i and to discard stochastic wavefunction samples with w_i < 0. Finally, observables, such as $⟨E\left(\tau \right)⟩$, are computed on the classical computer using overlap queries to the quantum processor (Supplementary Section C).

Fig. 2. 8-qubit experiment.

a, Circuit used for the 8-qubit H₄ experiment over a 2 × 4 qubit grid (from Q_1,1 to Q_2,1) on the Sycamore quantum processor. In the circuit diagram, H denotes the Hadamard gate, G denotes a Givens rotation gate (generated by XX + YY), P denotes a single-qubit Clifford gate and $|{\mathrm{\Psi }}_{\text{T}}⟩$ denotes the quantum trial wavefunction. Note that the ‘offline’ orbital rotation is not present in the actual quantum circuit because it is handled via classical post-processing, as discussed in Supplementary Section C. b, c, Convergence of the atomization energy of H₄ as a function of the number of measurements. A minimal basis set (STO-3G) with four orbitals total from four independent experiments (exp.) (b) and a quadruple-zeta basis set (cc-pVQZ) with 120 orbitals total from two independent experiments (c). The different symbols in b and c show independent experimental results. Note that the ideal (that is, noiseless) atomization energy of the quantum trial (Q. trial) in b is precisely on top of the exact one and that the QC-AFQMC energy would likewise be exact in the absence of noise. For the system in c, QC-AFQMC with this quantum trial would yield an error of 0.2 kcal mol⁻¹ despite a much larger error in the variational energy of the quantum trial. Further note that the quantum resource used in c is 8 qubit, but, as shown in Supplementary Section C, our algorithm enables the addition of ‘virtual’ electron correlation in a much larger Hilbert space. The top panels of b and c magnify the energy range near the exact answer. See Extended Data Tables 1–8 for the raw data for b, c, as well as other relevant data. Source data

Fig. 3. 12-qubit and 16-qubit experiments.

a, Circuit layout showing spin-up and spin-down qubits for the 12-qubit experiment (top). Potential energy surface of N₂ in a triple-zeta basis set (cc-pVTZ; 60 orbital) (bottom). The relative energies are shifted to zero at 2.25 Å. Inset shows the error in total energy relative to the exact results in kcal mol⁻¹. The shaded region in the inset shows the bounds for chemical accuracy (1 kcal mol⁻¹). Neither the variational energy of the quantum trial nor the statistical error bars of the AFQMC methods are visible on this scale. b, Circuit layout showing spin-up and spin-down qubits for the 16-qubit experiment (top). Error in total energy as a function of lattice constant of diamond in a double-zeta basis (DZVP-GTH; 26 orbitals) (bottom). The shaded region shows the bounds for chemical accuracy. Our quantum trial results are not visible on this scale. Inset shows a supercell structure of diamond in which two highlighted atoms form the minimal unit cell. See Extended Data Tables 9, 10 for the raw data for a, b, respectively. Source data