core.ising_qubo

Classes

IsingModel

Functions

calc_qubo_energy(→ float)	Calculates the energy of the state.
qubo_to_ising(→ IsingModel)	Converts a Quadratic Unconstrained Binary Optimization (QUBO) problem to an equivalent Ising model.

Module Contents

class IsingModel

quad: dict[tuple[int, int], float]

linear: dict[int, float]

constant: float

index_map: dict[int, int] | None = None

num_bits() → int

calc_energy(state: list[int]) → float

Calculates the energy of the state.

Examples

>>> ising = IsingModel({(0, 1): 2.0}, {0: 4.0, 1: 5.0}, 6.0)
>>> ising.calc_energy([1, -1])
3.0

ising2qubo_index(index: int) → int

normalize_by_abs_max()

Normalize coefficients by the absolute maximum value.

The coefficients for normalized is defined as:

\[W = \max(|J_{ij}|, |h_i|)\]

We normalize the Ising Hamiltonian as

\[\tilde{H} = \frac{1}{W}\sum_{ij}J_{ij}Z_iZ_j + \frac{1}{W}\sum_ih_iZ_i + \frac{1}{W}C\]

normalize_by_rms()

Normalize coefficients by the root mean square.

The coefficients for normalized is defined as:

\[W = \sqrt{ \frac{1}{E_2}\sum(w_ij^2) + \frac{1}{E_1}\sum(w_i^2) }\]

where w_ij are quadratic coefficients and w_i are linear coefficients. E_2 and E_1 are the number of quadratic and linear terms respectively. We normalize the Ising Hamiltonian as

\[\tilde{H} = \frac{1}{W}\sum_{ij}J_{ij}Z_iZ_j + \frac{1}{W}\sum_ih_iZ_i + \frac{1}{W}C\]

This method is proposed in [Sureshbabu et al., 2024]

[SHS+24]

Shree Hari Sureshbabu, Dylan Herman, Ruslan Shaydulin, Joao Basso, Shouvanik Chakrabarti, Yue Sun, and Marco Pistoia. Parameter Setting in Quantum Approximate Optimization of Weighted Problems. Quantum, 8:1231, January 2024. URL: https://doi.org/10.22331/q-2024-01-18-1231, doi:10.22331/q-2024-01-18-1231.

calc_qubo_energy(qubo: dict[tuple[int, int], float], state: list[int]) → float

Calculates the energy of the state.

Examples

>>> calc_qubo_energy({(0, 0): 1.0, (0, 1): 2.0, (1, 1): 3.0}, [1, 1])
6.0

qubo_to_ising(qubo: dict[tuple[int, int], float], constant: float = 0.0, simplify=False) → IsingModel

Converts a Quadratic Unconstrained Binary Optimization (QUBO) problem to an equivalent Ising model.

QUBO:

\[\sum_{ij} Q_{ij} x_i x_j,~\text{s.t.}~x_i \in \{0, 1\}\]

Ising model:

\[\sum_{ij} J_{ij} z_i z_j + \sum_i h_i z_i, ~\text{s.t.}~z_i \in \{-1, 1\}\]

Correspondence:

\[x_i = \frac{1 - z_i}{2}\]

where \((x_i \in \{0, 1\})\) and \((z_i \in \{-1, 1\})\).

This transformation is derived from the conventions used to describe the eigenstates and eigenvalues of the Pauli Z operator in quantum computing. Specifically, the eigenstates |0⟩ and |1⟩ of the Pauli Z operator correspond to the eigenvalues +1 and -1, respectively:

\[Z|0\rangle = |0\rangle, \quad Z|1\rangle = -|1\rangle\]

This relationship is leveraged to map the binary variables (x_i) in QUBO to the spin variables (z_i) in the Ising model.

Examples

>>> qubo = {(0, 0): 1.0, (0, 1): 2.0, (1, 1): 3.0}
>>> ising = qubo_to_ising(qubo)
>>> binary = [1, 0]
>>> spin = [-1, 1]
>>> qubo_energy = calc_qubo_energy(qubo, binary)
>>> assert qubo_energy == ising.calc_energy(spin)

>>> qubo = {(0, 1): 2, (0, 0): -1, (1, 1): -1}
>>> ising = qubo_to_ising(qubo)
>>> assert ising.constant == -0.5
>>> assert ising.linear == {}
>>> assert ising.quad == {(0, 1): 0.5}