dwave.embedding.chain_strength.uniform_torque_compensation

uniform_torque_compensation(bqm, embedding=None, prefactor=1.414)

Chain strength that attempts to compensate for chain-breaking torque.

The problem’s connectivity[1] and quadratic biases are used to calculate a value of chain strength that preforms reasonably well on many problems.

As the quantum annealing progresses, and the amplitude of the transverse field (\(A(s)\) in the QA Implementation section) decreases, the wavefunction representing the QPU’s quantum state develops long-range order. A chain strength that increases the correlation of the chains’ qubits together with the development of this long-range order produces efficient quantum dynamics [Ray2020]. For many hard, frustrated problems (such as spin glasses) with typical chain topology (path or tree-like chains), the optimal chain strength is proportional to the root of typical variable connectivity and the root mean square (RMS) of coupling strength (\(\sqrt{\text{connectivity}} \times \text{coupling}_{RMS}\)).

This chain strength, chosen for its dynamically-efficient scaling, also meets another requirement, even in challenging models: it must be large enough so that, toward the end of the quantum annealing, intact chains of the embedded problem (the programmed Hamiltonian) have lower energy than broken chains for the ground state. This allows the embedded problem to recover the ground state in the adiabatic limit. Consider the following observation: in a frustrated problem, such as a spin glass, qubits in chains are subject to random energy signals from neighbors. If you split the chain in two with equal numbers of neighboring chains, the central limit theorem dictates that the signal in each half has zero mean and variance proportional to the connectivity and the RMS of the coupling values. In combination, these can create a random torque on the chain, favouring misalignment (a broken chain). For the central coupling to maintain alignment of the two halves, it needs an energy penalty larger than the torque signal, and so scales as \(\sqrt{\text{connectivity}} \times \text{coupling}_{RMS}\).

[1]

A measure of the density of interactions between variables; for example the average degree (number of edges per node in the graph representing the problem).

Parameters:

• bqm (BinaryQuadraticModel) – A binary quadratic model.

• embedding (dict/EmbeddedStructure, default=None) – Included to satisfy the chain_strength callable specifications for embed_bqm.

• prefactor (float, optional, default=1.414) – Prefactor used for scaling. For non-pathological problems, the recommended range of prefactors to try is [0.5, 2].

Returns:

The chain strength, or 1 if chain strength is not applicable.

Return type:

float