Quantum Simulation of the Schrodinger Equation using IBM's Quantum Computers

Madita Willsch

Tino Gramß

Minjae Jo, Myungshik Kim

Digital quantum computers are potentially an ideal platform for simulating open quantum many-body systems beyond the digital classical computers. Many studies have focused on obtaining the ground state by simulating time dynamics or variational approaches of closed quantum systems. However, dynamics of open quantum systems has not been given much attention with a reason being their non-unitary dynamics not natural to simulate on a set of unitary gate operations in quantum computing. Here we study prototypical models in open quantum systems with Trotterisations for the modified stochastic Schr{ö}dinger equation (MSSE). Minimising the leading error in MSSE enables to optimise the quantum circuits, and we run the optimised circuits with the noiseless \textit{quantum assembly language (QASM) simulator} and the noisy IBM Quantum devices. The \textit{QASM simulator} enables to study the reachable system size that is comparable to the limits of classical computers. The results show that the nonequilibrium critical phenomena in open quantum systems are successfully obtained with high precision. Furthermore, we run the algorithm on the IBM Quantum devices, showing that the current machine is challenging, to give quantitatively accurate time dynamics due to the noise. Despite errors, the results by IBM devices qualitatively follow the trend of critical behaviour and include a possibility to demonstrate quantum advantage when the noise is reduced. We discuss how much noise should be reduced for a certain fidelity using the noise model, which will be crucial to demonstrate quantum advantage from future quantum devices.

Malte Schade, Cyrill Bösch, Václav Hapla +1 more

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

LEVI, Benjamin, Bertrand Georgeot

Manik Kapil, Bikash K. Behera, Prasanta Kumar Panigrahi

The Klein Gordon equation was the first attempt at unifying special relativity and quantum mechanics. While initially discarded this equation of "many fathers" can be used in understanding spinless particles that consequently led to the discovery of pions and other subatomic particles. The equation leads to the development of Dirac equation and hence quantum field theory. It shows interesting quantum relativistic phenomena like Klein Paradox and "Zitterbewegung", a rapid vibrating movement of quantum relativistic particles. The simulation of such quantum equations initially motivated Feynman to propose the idea of quantum computation. While many such simulations have been done till date in various physical setups, this is the first time a digital quantum simulation of Klein Gordon equation is proposed on IBM's quantum computer. Here we simulate the time-dependent Klein Gordon equation in a barrier potential and clearly observe the tunnelling of the particle and anti-particle through a strong potential claiming Klein Paradox. The simulation technique used here inspires the quantum computing community for further studying Klein Gordon equation and applying it to more complicated quantum mechanical systems.