From transistor to trapped-ion computers for quantum chemistry

Abstract

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

Figure 1. Simulating quantum chemistry with trapped ions.

(a) Scheme of a trapped-ion setup for quantum simulation, which contains a linear chain of trapped ions confined by a harmonic potential, and external lasers that couple the motional and internal degrees of freedom. (b) Transitions between internal and motional degrees of freedom of the ions in the trap. (c) The normal modes of the trapped ions can simulate the vibrational degrees of freedom of molecules. (d) The internal states of two ions can simulate all four possible configurations of a molecular orbital.

Figure 2. Outline of the quantum-assisted optimization method.

(a) The key steps for quantum assisted optimization, which starts from classical solutions. For each new set of parameters λ's, determined by a classical optimization algorithm, the expectation value 〈H〉 is calculated. The potential energy surface is then obtained by quantum phase estimation. (b) Quantum measurements are performed for the individual terms in H, and the sum is obtained classically. (c) The same procedure is applied for each nuclear configuration R to probe the energy surface.

Figure 3

Digital error 1 – F (curves) along with the accumulated gate error (horizontal lines) versus time in h₁₁ energy units, for n = 1, 2, 3 Trotter steps in each plot, considering a protocol with an error per Trotter step of (a), (b) and (c). The initial state considered is |↑↑↓↓〉, in the qubit representation of the Hartree-Fock state in a molecular orbital basis with one electron on the first and second orbital. Vertical lines and arrows define the time domain in which the dominant part of the error is due to the digital approximation. d) Energy of the system, in h₁₁ units, for the initial state | ↑↑↓↓〉 for the exact dynamics, versus the digitized one. For a protocol with three Trotter steps the energy is recovered up to a negligible error.