NAE3SAT is an NP-complete
Boolean satisfiability (SAT) problem that has clauses of three literals each and
requires that for every clause the literals are not all equal to each other; i.e.,
all configurations are valid except (-1, -1, -1) and (+1, +1, +1) (for
spin-valued variables, or (0,0,0) and (1, 1, 1) for binary-valued variables).
SAT Terminology:
The SAT problem problem is to decide whether the literals in its clauses can be assigned values that satisfy all the clauses. In CNF, the SAT is satisfied only if all its clauses are satisfied.
- Literal is a Boolean variable such as
xand its negation. - Clause is a disjunction of literals such as
x1 OR x0. - conjunctive normal form (CNF) conjoins clauses by the AND operator; i.e., (clause 1) AND (clause 2) AND (clause 3).
NAE3SAT problems are known to transition from the satisfiable to the unsatisfiable
regime at a clause-to-variable ratio ρ=2.1. This code example solves problems
with ρ=2.1 (critical point) as well as with ρ=3.0 (max-SAT regime).
This example explores solving this satisfiability problem with two different generations of D-Wave quantum computers:
- A Pegasus-topology Advantage system.
- An experimental prototype of a Zephyr-topology quantum processing unit (QPU) for the Advantage2 next-generation system currently under development.
You can run this example without installation in cloud-based IDEs that support the Development Containers specification (aka "devcontainers").
For development environments that do not support devcontainers, install
requirements:
pip install -r requirements.txt
If you are cloning the repo to your local system, working in a virtual environment is recommended.
Your development environment should be configured to access Leap’s Solvers. You can see information about supported IDEs and authorizing access to your Leap account here.
To run the example:
python nae3sat_example.pyThe code saves the resulting graphics under a plots folder.
The code reformulates NAE3SAT instances as Ising problems, embeds these onto the QPU graphs, and analyzes the solution quality obtained from both QPUs.
NAE3SAT problems can be mapped to the
Ising model by
anti-ferromagnetically coupling the variables of a clause. For example, a clause
for variables (s0, s1, s2) is represented by the Ising Hamiltonian,
H(s0, s1, s2) = s0*s1 + s1*s2 + s0*s2.
The table below shows the energy for all configurations of the variables. The valid not-all-equal configurations have a lower energy than the all-equal ones. Therefore, the NAE3SAT clause is solved when minimizing the Hamiltonian objective.
| s0 | s1 | s2 | E |
|---|---|---|---|
| -1 | -1 | -1 | 3 |
| 1 | -1 | -1 | -1 |
| 1 | 1 | -1 | -1 |
| -1 | 1 | -1 | -1 |
| -1 | 1 | 1 | -1 |
| 1 | 1 | 1 | 3 |
| 1 | -1 | 1 | -1 |
| -1 | -1 | 1 | -1 |
Negated variables are represented by a
spin-reverse transform
of such variables. For example, a clause with -s1, the negation of s1,
is represented by the Ising Hamiltonian,
H(s0, -s1, s2) = -s0*s1 - s1*s2 + s0*s2.
A NAE3SAT problem is generated by adding all its clauses together; for example:
H(s0,..., sN) = H(s0, s1, s2) + H(s6, -s1, s4) + H(-s3, s5, -s9) + ...
The minimization of this objective solves the NAE3SAT problem.
For more information on reformulating SAT problems in Ising format, see the system documentation.
Both the Pegasus and Zephyr graphs natively support triangles, enabling the embedding of three-variable clauses without chains. Nevertheless, large NAE3SAT problems typically include interactions between variables that are not natively connected and do require chains.
The graphic below shows the distribution of chain lengths for a ρ=3.0 NAE3SAT
problem of 75 variables minor-embedded into the two topologies using Ocean software's
minorminer
heuristic. Smaller chains typically improve the success of quantum-annealing computations.
The Zephyr topology of the Advantage2 prototype has a greater connectivity than the Pegasus topology of Advantage. This enables more-compact embedding (minor embeddings with shorter chains) of problems in the Advantage2 prototype, which in turn improves the performance of the QPU on embedded problems.
The graphic below shows the solution quality of 100 samples from a ρ=3.0 problem.
The distribution of samples obtained by the Advantage2 prototype tends to have lower energy than that of the Advantage QPU. Lower energies mean better solutions.
Note: The data displayed in the graphics above was generated for a particular execution of this code example; results will slightly vary from run to run. If you wish to experiment further to get more consistent results, you can increase the number of samples or the number of calls (keep in mind these runs will consume your solver access time at the standard rate).
Released under the Apache License 2.0. See LICENSE file.