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Semiconductor adiabatic qubits

Malcolm S. Carroll

Published Jan 23, 2023• 0 citations
OpenAlex

Abstract

A quantum computing device that includes a plurality of semiconductor adiabatic qubits is described herein. The qubits are programmed with local biases and coupling terms between qubits that represent a problem of interest. The qubits are initialized by way of a tuneable parameter, a local tunnel coupling within each qubit, such that the qubits remain in a ground energy state, and that initial state is represented by the qubits being in a superposition of |0> and |1> states. The parameter is altered over time adiabatically or such that relaxation mechanisms maintain a large fraction of ground state occupation through decreasing the tunnel coupling barrier within each qubit with the appropriate schedule. The final state when tunnel coupling is effectively zero represents the solution state to the problem represented in the |0> and |1> basis, which can be accurately read at each qubit location.

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We apply a quantum adiabatic evolution algorithm to a combinatorial optimization problem where the cost function depends entirely on the of the number of unit bits in a n-bit string (Hamming weight). The solution of the optimization problem is encoded as a ground state of the problem Hamiltonian H_p for the z-projection of a total spin-n/2. We show that tunneling barriers for the total spin can be completely suppressed during the algorithm if the initial Hamiltonian has its ground state extended in the space of the z-projections of the spin. This suppression takes place even if the cost function has deep and well separated local minima. We provide an intuitive picture for this effect and show that it guarantees the polynomial complexity of the algorithm in a very broad class of cost functions. We suggest a simple example of the Hamiltonian for the adiabatic evolution: H(tau) = (1-tau) hat S_{x}^{2} + tau H_p, with parameter tau slowly varying in time between 0 and 1. We use WKB analysis for the large spin to estimate the minimum energy gap between the two lowest adiabatic eigenvalues of H(tau).
We present a quantum algorithm for implementing ${\ensuremath{\phi}}^{4}$ lattice scalar field theory on qubit computers. The field is represented in the discretized field amplitude basis. The number of qubits and elementary gates required by the implementation of the evolution operator is proportional to the lattice size. The algorithm allows efficient ${\ensuremath{\phi}}^{4}$ state preparation for a large range of input parameters in both the normal and broken-symmetry phases. The states are prepared using a combination of variational and adiabatic evolution methods. First, the ground state of a local Hamiltonian, which includes the ${\ensuremath{\phi}}^{4}$ self-interaction, is prepared using short variational circuits. Next, this state is evolved by switching on the coupling between the lattice sites adiabatically. The parameters defining the local Hamiltonian are adjustable and constitute the input of our algorithm. We present a method to optimize these parameters in order to reduce the adiabatic time required for state preparation. For preparing broken-symmetry states, the adiabatic evolution problems caused by crossing the phase transition critical line and by the degeneracy of the broken-symmetry ground state can be addressed using an auxiliary external field which gradually turns off during the adiabatic process. We show that the time dependence of the external field during the adiabatic evolution is important for addressing the broken-symmetry ground state degeneracy. The adiabatic time dependence on the inverse error tolerance can be reduced from quadratic to linear by using a field strength that decreases exponentially in time relative to one that decreases linearly.
Quantum computing seeks to use the powers of quantum mechanics to accomplish tasks that classical computers cannot easily accomplish. Adiabatic quantum computing is one flavor of quantum computing that slowly changes a system with time in such a way that the state remains close to the ground state for the entire evolution. An easy way to implement and study adiabatic computing is using stoquastic Hamiltonians with no-sign problem. These Hamiltonians can be readily simulated on classical computers using quantum Monte Carlo schemes; though, it is unknown whether such simulation captures the speed and power of the quantum computation.In this work, I study stoquastic Hamiltonians, focusing on quantum mechanical tunneling through classically inaccessible regions and barriers. I study multi-qubit tunneling problems to determine what properties of the barrier make this problem quantum mechanically or classically hard or inefficient. I compare quantum adiabatic computing with a path-integral Quantum Monte Carlo algorithm, which is a classical algorithm designed to simulate the quantum mechanical dynamics of the adiabatic evolution. Limited numerical data shows strong correlation between adiabatic and Monte Carlo runtimes, but due to computational limits, only small system sizes could be sampled.Additionally, I study the properties of the quantum adiabatic algorithm in both the asymptotic limit of a large number of qubits and long runtimes and more realistic finite system sizes and non-adiabatic conditions. I develop a modification of the existing Villain transformation that allows it to find the asymptotic running time of the quantum evolution for different barrier sizes. Furthermore, I compare this to finite system sizes and discover the extremely large numbers of qubits are needed for these asymptotics to be useful.When the quantum dynamics are run faster than adiabatically, new behavior arises that could potentially lead to enhancements in the quantum runtimes. I explore these enhancements, discovering when they do and do not occur. I also develop a broad framework for determining the nature of the near-adiabatic behavior of any quantum evolution, using only information about the energy gap between the ground state and the first excited state.