Lecture 13 - Adiabatic Quantum Computing

Adiabatic Quantum Computing

• Overview: Adiabatic Quantum Computing (AQC) is a paradigm of quantum computing that utilizes the principles of quantum mechanics to solve optimization problems by adiabatically evolving a quantum system from an initial Hamiltonian to a problem Hamiltonian.

Key Concepts

• Subset Sum Problem (SSP):

  • Definition: Given a set of integers, find a subset whose sum equals a target integer, denoted as T.

  • Characteristics: NP-complete; requires exponential time for exact algorithms, but polynomial time for verification.

• Quantum Advantage:

  • Quantum computers can solve SSP with a better complexity of $O(2^\frac{1}{2^n}) = O(\sqrt{2^n})$ compared to classical algorithms.

  • Practical implementations of adiabatic quantum computers, like those from D-Wave Systems, leverage this advantage.

• Quadratic Unconstrained Binary Optimization (QUBO):

  • SSP can be rewritten as a QUBO form, allowing AQC to target specific solutions in optimization problems.

Adiabatic Theorem

• Definition: A quantum system remains in its instantaneous eigenstate if perturbed slowly enough, and if there is a gap between the instantaneous eigenvalue and the rest of the spectrum of the Hamiltonian $\hat{H} = H$.

• Mathematical Expression:

  • If a quantum system begins in an eigenstate $| \Psi(0) \rangle$ of $H(t=0)$, it evolves to the corresponding eigenstate $| \Psi(t_{max}) \rangle \text{ of } H(t=t_{max})$ under slow perturbations.

AQC Process Steps

1. Initialization:

   • Prepare the system in the ground state of the initial Hamiltonian

     $H_B = -\sum_{j=1}^n X_j$

     where $X_j$ represents qubit interactions.

2. Evolution:

   • Evolve the state through a Hamiltonian that transitions from the initial state to the problem Hamiltonian $H_P$. The evolution follows:

     $\frac{d}{dt}| \Psi(t) \rangle = -\frac{i}{\hbar} H(t)| \Psi(t) \rangle$

   • The Hamiltonian transitions from $H_B$ to $H_P$ gradually over time.

3. Measurement:

   • Measure the state at the end of the evolution period to obtain a solution. If the QUBO has many solutions, the result will be in a superposition of the ground states.

Problem Solving with AQC

• Example:

  • Given a QUBO formulated from the SSP, the AQC procedure reveals solutions progressively as the time parameter goes to $t_{max}$.

  • With more qubits, the computation grows significantly, but different instances may have varying difficulties.

• Amplitudes of Solutions:

  • The amplitude squared provides probabilities for each final state, illustrating likelihoods of obtaining specific solutions.

Advantages of AQC

• Simultaneous Exploration: Works on all $2^n$ potential solutions at once, increasing the chances of finding satisfactory solutions when measurements are taken.

• Requires a process duration of $O(\sqrt{2^n})$ to yield optimal results if energy gap parameters are satisfied.

Challenges & Future Directions

• Questions: Formation of Hamiltonians ($H_B$ and $H_P$) and dependency on quantum mechanics principles.

• Future Study: Exploration of quantum algorithms and deeper implications for optimization and computational theory.

Summary

• Adiabatic Quantum Computing provides a framework for solving complex problems like the SSP efficiently through the adiabatic theorem and QUBO formulations, setting the stage for future advancements in quantum algorithm development.