Chapter 6: D-Wave

Quantum Annealing and Ising Models

What is D-Wave Technology?

D-Wave builds quantum annealers designed to solve combinatorial optimization problems. Unlike gate-based quantum computers, D-Wave's systems use quantum annealing, a process based on the principle of energy minimization.

Quantum Annealing vs Gate-based QC

• Quantum Annealing (QA): Uses adiabatic evolution of a system's Hamiltonian.
• Gate-Based QC: Uses logic gates to construct quantum circuits for universal computation.

Why Do Errors in Qubits Not Greatly Affect D-Wave Results?

In QA, the final result is not a quantum state but a classical state corresponding to the minimum energy configuration. Errors during the anneal may shift the path slightly but do not necessarily prevent reaching a good approximation of the global minimum.

Why is it Called "Annealing"?

Inspired by classical annealing, which heats a system and then slowly cools it to remove defects.

In quantum annealing, we do: - Start in the ground state of a known initial Hamiltonian: $$ H(t=0) = H_{\text{driver}} = -\sum_{i} \sigma^x_i $$ - Slowly evolve to the problem Hamiltonian: $$ H(t=T) = H_{\text{problem}} = \sum_{i} h_i \sigma^z_i + \sum_{i<j} J_{ij} \sigma^z_i \sigma^z_j $$ - System ideally remains in the ground state (via adiabatic theorem).

Mapping Optimization Problems to Ising Model

A general quadratic unconstrained binary optimization (QUBO) problem: $$ \text{minimize} \quad x^T Q x \quad \text{where} \quad x_i \in {0, 1} $$

Can be transformed to an Ising form: $$ \text{minimize} \quad \sum_i h_i s_i + \sum_{i<j} J_{ij} s_i s_j \quad \text{where} \quad s_i \in {-1, +1} $$

Example: Financial Portfolio Optimization

Minimize risk and maximize return: $$ \text{maximize} \quad \mu^T x - \lambda x^T \Sigma x $$

Subject to: - Budget constraint: \(\sum x_i = k\) - Binary decision: \(x_i \in \{0, 1\}\)

This can be formulated as a QUBO and solved using D-Wave or other Ising-based methods.

Coupled Oscillator-Based Ising Model (COBI)

COBI uses coupled oscillators to represent spins. The dynamics naturally evolve toward energy-minimizing configurations: - Oscillators represent binary spins. - Couplings mimic \(J_{ij}\) terms. - Stable phase-locking corresponds to a low-energy Ising state.

COBI vs QUBO Optimization Results

Studies have shown that COBI implementations can match or approximate QUBO solutions well for small problem sizes, such as: - MAX-CUT problems - Graph coloring - Portfolio selection (toy models)

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