Google implements the largest chemical simulation on a quantum computer

In the "Science" paper published on August 28, 2020 [1], the Google quantum artificial intelligence team successfully simulated the heterogeneous chemical reaction of diazene with 12 qubits on the "planet tree" quantum computer, which is a quantum The first chemical simulation ever performed on a computer.

On March 16, the "Nature" paper [2] published by Google's quantum artificial intelligence team showed that the team had completed a chemical simulation of 16 qubits on the "plane tree" quantum computer. This is the largest chemical simulation ever performed on a quantum computer.

16 qubits perform QMC calculations

The problem of interacting multiple electrons presents some of the greatest computational challenges in science, with important applications in many fields. Solutions to these problems will provide precise predictions of chemical reactivity and kinetics, as well as other properties of quantum systems. The Fermi Quantum Monte Carlo (QMC) method, which uses statistical sampling of ground states, is one of the most powerful approaches to these problems. Classical computation to control the Fermi sign problem by imposing constraints can ensure the efficiency of QMC, but at the cost of a potentially significant bias.

Here, the Google team proposes a way to reduce this bias by combining constrained QMC and quantum computing, and demonstrates the scheme experimentally, using up to 16 qubits to perform on chemical systems with up to 120 orbitals Unbiased Constrained QMC Calculations. This is the largest chemical simulation ever conducted with the help of a quantum computer.

The QMC method determines the exact ground-state wave function |Ψ0⟩ of the many-body Hamiltonian H^ by imaginary-time evolution of the initial state |φ0⟩, where |φ0⟩ and |Ψ0⟩ have non-zero overlap:

where τ is the imaginary time, and |ψ(τ)⟩ represents the wave function of |φ0⟩⟩ evolving with τ time (Fig. 1a).

Exact deterministic imaginary time evolution and unconstrained QMC computations are accurate on average, but the signal-to-noise ratio diverges with increasing τ due to the sign problem. Using quantum constraints helps reduce the non-negligible bias when using classical constraints (Fig. 1b).

Google's quantum-classical hybrid QMC algorithm (QC-QMC) utilizes a quantum trial wavefunction while performing most of the imaginary-time evolution on a classical computer, as shown in Figure 1c.

Essentially, on a classical computer, it is common to perform imaginary time evolution for each wave function statistical sample |φi(τ)⟩ and collect observables such as the ground state energy estimate E(i)(τ). In the process, a constraint related to the quantum tentative wave function is imposed to control the sign problem. To perform constrained time evolution, the only quantity researchers need to compute on a quantum computer is the overlap between the tentative wavefunction |ψT⟩ and the statistical sample of the wavefunction at imaginary time τ |φi(τ)⟩.

Figure 1 Imaginary time evolution, symbolic problem and the quantum-classical hybrid algorithm in this paper

The researchers say their method is generally applicable to any form of constrained QMC, but discuss here only an experimental demonstration of one algorithm, the hybrid quantum-classical AFQMC algorithm (QC-AFQMC), which uses a method called auxiliary field QMC. (AFQMC) implementation of QMC.

They first demonstrated quantum primitives for conducting experiments on the H4 molecule, which involves eight qubits, as shown in Figure 2. Here, the eight-spin-orbit quantum probe wavefunction consists of a valence bond wavefunction called a perfectly paired state and a hardware-efficient quantum circuit with offline single-particle rotation, which is classically difficult to use as a probe wavefunction for AFQMC .

The state preparation circuit in Figure 2a shows how this tentative wave function can be efficiently prepared on a quantum computer. The H in the circuit diagram represents the Hadamard gate, the G represents the Givens revolving gate (generated by XX+YY), the P represents the single-qubit Clifford gate, and |ψT⟩ represents the quantum tentative wave function. Off-line orbital rotation does not exist in actual quantum circuits because it is handled by classical postprocessing.

The convergence of H4 atomization energy as a function of the number of measurements is shown in Figures 2b and 2c. The smallest basis set with 4 orbitals (STO-3G) is from four independent experiments (2b), and the quadruple zeta basis set (cc-pVQZ) with a total of 120 orbitals is from two independent experiments (2c). In Figure 2b, the ideal (ie, noise-free) atomization energy of the quantum trial (Q.trial) is just above the exact atomization energy, and the QC-AFQMC energy is also accurate without noise. In Fig. 2c, the QC-AFQMC with this quantum heuristic produces a bias of 0.2 kcal mol-1, but is much smaller than the variational energy bias of the quantum heuristic.

Figure 2 8-qubit experiment

The researchers then experimented with nitrogen molecules (N2) using 12 qubits. Finally, 16-qubit experiments with ground-state simulations were performed on the smallest unit (two-atom) model of periodic solid-state diamond.

The 12-qubit experiment is shown in Figure 3a, showing the results of calculations performed on a three-ζ basis set (cc-pVTZ), which corresponds to a 60-orbital or 120-qubit Hilbert space. Exact in the figure is the exact result, as well as the calculated results of five other methods: QC-AFQMC (experimental), QC-AFQMC (ideal), Q.trial (experimental), CCSD(T) (classical "gold standard") and AFQMC (classic). It can be seen that the calculated results of QC-AFQMC have the smallest deviation from the exact results.

Similarly, in the 16-qubit experiment, the calculation result with the smallest deviation is still QC-AFQMC, as shown in Fig. 3b.

Figure 3 12-qubit and 16-qubit experiments

In conclusion, the Google team proposes a scalable, noise-resistant hybrid quantum-classical algorithm that seamlessly embeds special-purpose quantum primitives into an accurate many-body approach to quantum computing, known as QMC. The team's work provides a computational strategy to effectively de-bias the Fermi QMC approach by leveraging state-of-the-art quantum information tools. Its performance was finally demonstrated in a 16-qubit experiment on a NISQ processor, producing electron energies comparable to state-of-the-art classical quantum chemistry methods.

Link:

[1] https://www.science.org/doi/10.1126/science.abb9811

[2] https://www.nature.com/articles/s41586-021-04351-z