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arXiv:1602.00063

Towards Quantum Simulation of Chemical Dynamics with Prethreshold Superconducting Qubits

Phillip C. Stancil, Hao You, Cook, A., Andrew T. Sornborger, Michael R. Geller

Published Jan 30, 2016• 6 citations
arXivHTMLPDF

Abstract

The single excitation subspace (SES) method for universal quantum simulation is investigated for a number of diatomic molecular collision complexes. Assuming a system of $n$ tunably-coupled, and fully-connected superconducting qubits, computations are performed in the $n$-dimensional SES which maps directly to an $n$-channel collision problem within a diabatic molecular wave function representation. Here we outline the approach on a classical computer to solve the time-dependent Schrödinger equation in an $n$-dimensional molecular basis - the so-called semiclassical molecular-orbital close-coupling (SCMOCC) method - and extend the treatment beyond the straight-line, constant-velocity approximation which is restricted to large kinetic energies ($\gtrsim 0.1$ keV/u). We explore various multichannel potential averaging schemes and an Ehrenfest symmetrization approach to allow for the application of the SCMOCC method to much lower collision energies (approaching 1 eV/u). In addition, a computational efficiency study for various propagators is performed to speed-up the calculations on classical computers. These computations are repeated for the simulation of the SES approach assuming typical parameters for realistic pretheshold superconducting quantum computing hardware. The feasibility of applying future SES processors to the quantum dynamics of large molecular collision systems is briefly discussed.

DOI
10.48550/arxiv.1602.00063

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The single excitation subspace (SES) method for universal quantum simulation is investigated for a number of diatomic molecular collision complexes. Assuming a system of nnn tunably-coupled, and fully-connected superconducting qubits, computations are performed in the nnn-dimensional SES which maps directly to an nnn-channel collision problem within a diabatic molecular wave function representation. Here we outline the approach on a classical computer to solve the time-dependent Schrödinger equation in an nnn-dimensional molecular basis - the so-called semiclassical molecular-orbital close-coupling (SCMOCC) method - and extend the treatment beyond the straight-line, constant-velocity approximation which is restricted to large kinetic energies (≳0.1\gtrsim 0.1≳0.1 keV/u). We explore various multichannel potential averaging schemes and an Ehrenfest symmetrization approach to allow for the application of the SCMOCC method to much lower collision energies (approaching 1 eV/u). In addition, a computational efficiency study for various propagators is performed to speed-up the calculations on classical computers. These computations are repeated for the simulation of the SES approach assuming typical parameters for realistic pretheshold superconducting quantum computing hardware. The feasibility of applying future SES processors to the quantum dynamics of large molecular collision systems is briefly discussed.
Current quantum computing architectures lack the size and fidelity required for universal fault-tolerant operation, limiting the practical implementation of key quantum algorithms to all but the smallest problem sizes. In this work we propose an alternative method for general-purpose quantum computation that is ideally suited for such ``prethreshold'' superconducting hardware. Computations are performed in the nnn-dimensional single-excitation subspace (SES) of a system of nnn tunably coupled superconducting qubits. The approach is not scalable, but allows many operations in the unitary group SU(nnn) to be implemented by a single application of the Hamiltonian, bypassing the need to decompose a desired unitary into elementary gates. This feature makes large, nontrivial quantum computations possible within the available coherence time. We show how to use a programmable SES chip to perform fast amplitude amplification and phase estimation, two versatile quantum subalgorithms. We also show that an SES processor is well suited for Hamiltonian simulation, specifically simulation of the Schr\"odinger equation with a real but otherwise arbitrary $n\ifmmode\times\else\texttimes\fi{}n$ Hamiltonian matrix. We discuss the utility and practicality of such a universal quantum simulator, and propose its application to the study of realistic atomic and molecular collisions.
We introduce a protocol for the fast simulation of nnn-dimensional quantum systems on nnn-qubit quantum computers with tunable couplings. A mapping is given between the control parameters of the quantum computer and the matrix elements of Hs(t)H_{\rm s}(t)Hs​(t), an arbitrary, real, time-dependent n×nn\times nn×n dimensional Hamiltonian that is simulated in the nnn-dimensional `single excitation' subspace of the quantum computer. A time-dependent energy/time rescaling minimizes the simulation time on hardware having a fixed coherence time. We demonstrate how three tunably coupled phase qubits simulate a three-channel molecular collision using this protocol, then study the simulation's fidelity as a function of total simulation time.
We describe an alternative approach to quantum computation that is ideally suited for today's sub-threshold-fidelity qubits, and which can be applied to a family of hardware models that includes superconducting qubits with tunable coupling. In this approach, the computation on an n-qubit processor is carried out in the n-dimensional single-excitation subspace (SES) of the full 2^n-dimensional Hilbert space. Because any real Hamiltonian can be directly generated in the SES [E. J. Pritchett et al., arXiv:1008.0701], high-dimensional unitary operations can be carried out in a single step, bypassing the need to decompose into single- and two-qubit gates. Although technically nonscalable and unsuitable for applications (including Shor's) requiring enormous Hilbert spaces, this approach would make practical a first-generation quantum computer capable of achieving significant quantum speedup.
Quantum particle simulations have largely been based on time-independent, split-operator schemes in which kinetic and potential operators are interwoven to provide accurate approximations to system dynamics. These simulations can be very expensive in terms of the number of gates required, although individual cases, such as tunneling, have been found where prethreshold simulations are possible. By prethreshold, we mean a quantum computation or simulation with an existing architecture and not requiring error correction. In the case of molecular collisions, switching to a time-dependent formulation can reduce the system dimensionality significantly and provide an opportunity for pre-threshold simulation. Here, we study the efficiency of gate-based quantum simulation of a set of molecular collisions of increasing complexity. We show that prethreshold quantum simulation of such systems is feasible up to Hilbert space dimension 8, but beyond that error correction would be required.
A quantum computer will potentially solve far-reaching problems which are currently intractable on any classical computer. Many technological obstacles have prevented the realization of a quantum computer, the main obstacle being decoherence, which is the loss of quantum information. Decoherence arises from the undesired interaction between qubits and their environment. Isolated qubits have better coherence but are more difficult to control. Superconducting qubits are a promising platform since their macroscopic size allows for easy control and coupling to other qubits. While the coherence of superconducting qubits has substantially improved over the past two decades, further improvements in coherence are required.We have repeatedly and reliably increased the coherence times of superconducting qubits. Currently decoherence in these devices is dominated by coupling to material defects. These defects are present in the dielectrics used to fabricate these devices or introduced during fabrication. Using simpler resonators as a testbed, we individually isolate, characterize, and then improve each step of the more complicated fabrication of superconducting qubits. We increased the quality factor of resonators by a factor of four by first identifying the surfaces and interfaces as a major source of loss and then by optimizing the substrate preparation. Furthermore, we measure and subsequently mitigate additional defect loss, which is dependent on the position of ground plane holes used to limit the loss from magnetic vortices. Implementing these improvements led to an increase of our qubit coherence times by more than an order of magnitude.The progress made in coherence while maintaining a high degree of connectivity and controllability has been directly used in more complex circuits. One such device is a fully connected three qubit ring with both tunable qubit frequencies and adjustable qubit-qubit couplings. The considerable level of control allows us to generate the quantum dynamics resulting from a collision between Na and He atoms by encoding the problem into a time-dependent multi-qubit interaction. We compute a scattering cross section by measuring the probability of exciting the Na atom over a wide range of collision parameters. The scattering cross-section shows that there is a velocity where the excitation probability is maximized, in agreement with scattering experiments.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.