Scalable Quantum Simulation of Molecular Energies

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.

Popular Summary

Universal simulation of physical systems is among the most compelling applications of quantum computing. In particular, quantum simulation of molecular energies promises significant advances in our understanding of chemistry, including precise predictions of chemical reaction rates. These advances, which may require relatively few qubits to achieve, may have broad industrial impact given the wide importance of chemistry. Here, we report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation.

We use a programmable array of superconducting qubits held at a base temperature of 20 mK in a dilution refrigerator; each qubit is composed of a superconducting quantum interference device (SQUID) and a capacitor, and is coupled to its nearest neighbors. We compute the energy surface of molecular hydrogen (i.e., the potential energy curve) using two distinct quantum algorithms: the variational quantum eigensolver and the phase estimation algorithm with Trotterization. We compare the experimental performance of these approaches and show that the variational quantum eigensolver yields chemically accurate results. Our experiments also reveal that the variational quantum eigensolver is robust to certain systematic errors.

Our findings suggest that the quantum simulation of classically intractable molecules may be possible without the overhead of quantum error correction.

Figure 1

Hardware and software schematic of the variational quantum eigensolver. (Hardware) micrograph shows two Xmon transmon qubits and microwave pulse sequences to perform single-qubit rotations (thick lines), dc pulses for two-qubit entangling gates (dashed lines), and microwave spectroscopy tones for qubit measurements (thin lines). (Software) quantum circuit diagram shows preparation of the Hartree-Fock state, followed by application of the unitary coupled cluster ansatz in Eq. (3) and efficient partial tomography ($R_t$) to measure the expectation values in Eq. (1). Finally, the total energy is computed according to Eq. (4) and provided to a classical optimizer which suggests new parameters.

Figure 2

Variational quantum eigensolver: raw data and computed energy surface. (a) Data showing the expectation values of terms in Eq. (1) as a function of $\theta$, as in Eq. (3). Black lines nearest to the data show the theoretical values. While such systematic phase errors would prove disastrous for PEA, our VQE experiment is robust to this effect. (b) Experimentally measured energies (in hartree) as a function of $\theta$ and $R$. This surface is computed from (a) according to Eq. (4). The white curve traces the theoretical minimum energy; the values of theoretical and experimental minima at each $R$ are plotted in Fig. 3. Errors in this surface are given in Fig. 6.

Figure 3

Computed ${\mathrm{H}}_2$ energy curve and errors. (a) Energy surface of molecular hydrogen as determined by both VQE and PEA. VQE approach shows dissociation energy error of $(8\pm 5)\times {10}^{-4}$ hartree (error bars on VQE data are smaller than markers). PEA approach shows dissociation energy error of $(1\pm 1)\times {10}^{-2}$ hartree. (b) Errors in VQE energy surface. Red dots show error in the experimentally determined energies. Green diamonds show the error in the energies that would have been obtained experimentally by running the circuit at the theoretically optimal $\theta$ instead of the experimentally optimal $\theta$. The discrepancy between blue and red dots provides experimental evidence for the robustness of VQE, which could not have been anticipated via numerical simulations. The gray band encloses the chemically accurate region relative to the experimental energy of the atomized molecule. The dissociation energy is relative to the equilibrium geometry, which falls within this envelope.

Figure 4

Hardware and software schematic of the Trotterized phase estimation algorithm. (Hardware) micrograph shows three Xmon transmon qubits and microwave pulse sequences, including (i) the variable amplitude ${\mathrm{CZ}}_{\varphi }$ (not used in Fig. 1) and (ii) dynamical decoupling pulses not shown in logical circuit. (Software) state preparation includes putting the ancilla in a superposition state and compensating for previously measured bits of the phase using the gate $Z_{{\mathrm{\Phi }}_k}$ (see text). The bulk of the circuit is the evolution of the system under a Trotterized Hamiltonian controlled by the ancilla. Bit $j_k$ is determined by a majority vote of the ancilla state over 1000 repetitions.

Figure 5

A flow chart describing steps required to quantum compute molecular energies using both PEA and VQE.

Figure 6

Errors in the VQE energy surface (in hartree) as a function of bond length and rotation angle. This plot looks somewhat like the derivative of Fig. 2 with respect to $R$ and $\theta$ because errors are greatest where the energy is most sensitive to changes in system parameters. As in Fig. 2, the white curve traces the theoretical minimum energy, which is seen to be in good agreement with the data. Note that while errors in the energy surface are sometimes negative, all energies are bounded from below by the variational minimum.

Figure 7

Example data for a single PEA experiment, run at $R=1.55\AA$. The results are shown without phase kickback from the measurements of the previous bit. The line at $P_{|1⟩}=0.5$ discriminates a measurement of 1 from 0.