Quantum computing with Nonlinear oscillator networks 

Overview:

As opposed to classical computing, quantum computing (QC) methods employ quantum mechanical phenomena such as interference, superposition, entanglement, etc. In QC, quantum superpositions of two coherent states representing opposing conditions at the same time can be encoded within a single electronic circuit called a cat qubit. In adiabatic quantum computing (AQC), the system is prepared with an initial Hamiltonian having a simple ground state and computation proceeds from here to a more complicated Hamiltonian whose ground state gives the solution to the computational problem. The Hamiltonian is varied slowly throughout the procedure so that, by the adiabatic theorem, the system remains always in its instantaneous ground state. Thus it is a formulation designed to overcome the problem of irreversible dissipation or loss of energy. AQC has been shown in the literature to be equivalent to quantum computing in the circuit model where a computation is encoded in a series of unitary quantum logic gates,  qubits initialised to known values, etc., potentially involving the entire Hilbert space.  

In AQC, as the system slowly evolves with the slowly varying Hamiltonian, several qubits could be in the proximity of points involving change of stability. At these points, the ground state could get close to the nearest first energy state. External or quantum fluctuations could then push the system away from the ground state, damaging the computation process; therefore quantum fluctuation terms are decreased slowly.  

On the other hand, recently, a quantum computer made up of continuous degrees of freedom of quantum nonlinear oscillators was proposed [1], which employs the superposition of an exponentially large number of the oscillator network states to solve computational problems. In this formulation, called bifurcation-based adiabatic quantum computation (B-AQC), non-dissipative quantum mechanical oscillators having a suitable nonlinearity evolve adiabatically,  yielding entangled Schroedinger cat states at the bifurcation point. In B-AQC, the nonlinear terms are increased very slowly and the network employs quantum adiabatic evolution to find the optimal solution. B-AQC has been so far discussed as being composed of Kerr parametric oscillators and Ising machines (designed for finding ground states of the Ising model) based on this have been termed as quantum bifurcation machines (QbMs) in the literature [2] .  

Significant interest has been shown recently in the use of mechanical systems in quantum buses and qubits. Since any force applied to mechanical oscillators and resonators can yield mechanical displacements, mechanical qubits would be ideal for quantum sensing [3]. That mechanical oscillators can exhibit very high quality factors and long coherence times would imply that multiple such units could be used in quantum circuits with long quantum decoherence time, a feature absent in superconducting qubits that have been integrated into circuits with very many elements. An energy-dependent spacing in a quantum oscillator’s energy spectrum can be brought about by a controlled anharmonicity, and a corresponding selective excitation of these energy states, leaving others untouched, a feature usually absent in quantum harmonic systems.  One example of high-Q mechanical oscillators are carbon nanotubes. We have shown  earlier [4,5] that we can model the energy states of multi-walled  carbon nanotubes through mechanical analogues with nonlinear elastic elements. We propose to extend earlier work in the literature that investigates B-AQC in the context of Kerr nonlinear parametric oscillators (KPOs) to other  mechanical systems that can be modelled through the introduction of nonlinear coupling and non-Hookean elastic elements, in a theoretical  investigation of their application as quantum computing elements.                                                                                         

References

• Hayato Goto. “Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network”, Scientific Reports 6, 21686 (2016).
• Hayato Goto, Zhirong Lin  & Yasunobu Nakamura. “Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators”, Scientific Reports 8, 7154 (2018).
• F. Pistolesi, A. N. Cleland, & A. Bachtold. “Proposal for a Nanomechanical Qubit”, Phys. Rev. X 11, 031027 (2021).
• Brijesh Kumar Mishra & Balakrishnan Ashok. “Coaxial carbon nanotubes: from springs to ratchet wheels and nanobearings”, Mater. Res. Express 5, 075023 (2018).
• B. Ashok. “Tubes and containers at the nano and microscales: Statics and dynamics”. Indian Academy of Sciences Conference Series 2, 19 (2019)