Add support for torus QPU graph variations

docs/bibliography.rst

@@ -1,12 +1,18 @@
 Bibliography
 ============
 
-.. [NX] Aric A. Hagberg, Daniel A. Schult and Pieter J. Swart, “Exploring network structure, dynamics, and function using NetworkX”, in Proceedings of the 7th Python in Science Conference (SciPy2008), Gäel Varoquaux, Travis Vaught, and Jarrod Millman (Eds), (Pasadena, CA USA), pp. 11–15, Aug 2008
+.. [NX] A. A. Hagberg, D. A. Schult and P. J. Swart, “Exploring network structure, dynamics, and function using NetworkX”, in Proceedings of the 7th Python in Science Conference (SciPy2008), Gäel Varoquaux, Travis Vaught, and Jarrod Millman (Eds), (Pasadena, CA USA), pp. 11–15, Aug 2008
 
-.. [GD] Gogate & Dechter, "A Complete Anytime Algorithm for Treewidth", https://arxiv.org/abs/1207.4109
+.. [GD] V. Gogate and R. Dechter, "A Complete Anytime Algorithm for Treewidth", https://arxiv.org/abs/1207.4109
 
-.. [AL] Lucas, A. (2014). Ising formulations of many NP problems. Frontiers in Physics, Volume 2, Article 5.
+.. [AL] A. Lucas (2014). Ising formulations of many NP problems. Frontiers in Physics, Volume 2, Article 5.
 
-.. [FIA] Facchetti, G., Iacono G., and Altafini C. (2011). Computing global structural balance in large-scale signed social networks. PNAS, 108, no. 52, 20953-20958
+.. [FIA] G. Facchetti, G. Iacono and C. Altafini (2011). Computing global structural balance in large-scale signed social networks. PNAS, 108, no. 52, 20953-20958
 
-.. [DWMP] Dahl, E., "Programming the D-Wave: Map Coloring Problem", https://www.dwavesys.com/media/htfgw5bk/map-coloring-wp2.pdf
+.. [DWMP] E. Dahl, "Programming the D-Wave: Map Coloring Problem", https://www.dwavesys.com/media/htfgw5bk/map-coloring-wp2.pdf
+
+.. [BBRR] K. Boothby, P. Bunyk, J. Raymond and A. Roy (2019). Next-Generation Topology of D-Wave Quantum Processors. https://arxiv.org/abs/2003.00133
+
+.. [BRK] K. Boothby, J. Raymond and A. D. King (2021). Zephyr Topology of D-Wave Quantum Processors. https://dwavesys.com/media/fawfas04/14-1056a-a_zephyr_topology_of_d-wave_quantum_processors.pdf
+
+.. [RH] J. Raymond, R. Stevanovic, W. Bernoudy, K. Boothby, C. C. McGeoch, A. J. Berkley, P. Farré and A. D. King (2021). Hybrid quantum annealing for larger-than-QPU lattice-structured problems. https://arxiv.org/abs/2202.03044

docs/reference/generators.rst

@@ -17,6 +17,8 @@ D-Wave Systems
    pegasus_graph
    zephyr_graph
 
+
+
 Example
 ~~~~~~~
 
@@ -48,6 +50,16 @@ the `find_chimera()` function to determine the Chimera indices.
 
 	Indices of a Chimera unit cell found by creating a lattice of size (1, 1, 4).
 
+Toruses
+-------
+
+.. autosummary::
+   :toctree: generated/
+
+   chimera_torus
+   pegasus_torus
+   zephyr_torus
+
 Other Graphs
 ------------
 

dwave_networkx/algorithms/coloring.py

@@ -171,7 +171,8 @@ def _chromatic_number_upper_bound(G):
             # be the largest eigenvalue of A. Then chi <= theta_1 + 1 with
             # equality iff G is complete or an odd cycle.
             # this is strictly better than brooks theorem
-            bound = math.ceil(max(np.linalg.eigvals(nx.to_numpy_array(G))))
+            # G is real symmetric, use eigvalsh for real valued output.
+            bound = math.ceil(max(np.linalg.eigvalsh(nx.to_numpy_array(G))))
     else:
         # we know it's connected
         bound = n_nodes

dwave_networkx/algorithms/elimination_ordering.py

@@ -497,9 +497,7 @@ def treewidth_branch_and_bound(G, elimination_order=None, treewidth_upperbound=N
 
     References
     ----------
-    .. [GD] Gogate & Dechter, "A Complete Anytime Algorithm for Treewidth",
-       https://arxiv.org/abs/1207.4109
-
+    Based on the algorithm presented in [GD]_
     """
     # empty graphs have treewidth 0 and the nodes can be eliminated in
     # any order
@@ -875,7 +873,7 @@ def pegasus_elimination_order(n, coordinates=False):
     """Provides a variable elimination order for the Pegasus graph.
 
     The treewidth of a Pegasus graph ``pegasus_graph(n)`` is lower-bounded by 
-    :math:`12n-11` and upper bounded by :math:`12n-4` [bbrr]_ .
+    :math:`12n-11` and upper bounded by :math:`12n-4` [BBRR]_ .
 
     Simple pegasus variable elimination order rules:
 
@@ -897,11 +895,6 @@ def pegasus_elimination_order(n, coordinates=False):
     order : list
         An elimination order that provides an upper bound on the treewidth.
 
-
-    .. [bbrr] Boothby, K., P. Bunky, J. Raymond, A. Roy. Next-Generation Topology
-       of D-Wave Quantum Processors. Technical Report, Februrary 2019.
-       https://www.dwavesys.com/resources/publications?type=white
-
     """
     m = n
     l = 12
@@ -932,7 +925,7 @@ def zephyr_elimination_order(m, t=4, coordinates=False):
     """Provides a variable elimination order for the zephyr graph.
 
     The treewidth of a Zephyr graph ``zephyr_graph(m,t)`` is upper-bounded by 
-    :math:`4tm+2t` and lower-bounded by :math:`4tm` [brk]_ .
+    :math:`4tm+2t` and lower-bounded by :math:`4tm` [BRK]_ .
 
     Simple zephyr variable elimination rules:
        - eliminate vertical qubits, one column at a time
@@ -953,13 +946,6 @@ def zephyr_elimination_order(m, t=4, coordinates=False):
     order : list
         An elimination order that achieves an upper bound on the treewidth.
 
-
-    References
-    ----------
-    .. [brk] Boothby, Raymond, King, Zephyr Topology of D-Wave Quantum
-        Processors, October 2021.
-        https://dwavesys.com/media/fawfas04/14-1056a-a_zephyr_topology_of_d-wave_quantum_processors.pdf
-
     """
     order = ([(0,w,k,j,z) for w in range(2*m+1) for k in range(t) for z in range(m) for j in range(2)]
              + [(1,w,k,j,z) for z in range(m) for j in range(2)  for w in range(2*m+1) for k in range(t)])

dwave_networkx/generators/chimera.py

@@ -26,14 +26,15 @@
 
 from itertools import product
 
-from .common import _add_compatible_edges
+from .common import _add_compatible_nodes, _add_compatible_edges, _add_compatible_terms
 
 __all__ = ['chimera_graph',
            'chimera_coordinates',
            'find_chimera_indices',
            'chimera_to_linear',
            'linear_to_chimera',
            'chimera_sublattice_mappings',
+           'chimera_torus',
            ]
 
 
@@ -78,7 +79,8 @@ def chimera_graph(m, n=None, t=None, create_using=None, node_list=None, edge_lis
         the graph topology and node labeling conventions, and an error is thrown
         if any node is incompatible or duplicates exist. 
         In other words, the ``node_list`` must specify a subgraph of the 
-        full-yield graph described below.
+        full-yield graph described below. An exception is allowed if 
+        ``check_edge_list=False``, in which case any node in ``edge_list`` is treated as valid.
     check_edge_list : bool (optional, default :code:`False`)
         If :code:`True`, the ``edge_list`` elements are checked for compatibility with
         the graph topology and node labeling conventions, an error is thrown
@@ -95,14 +97,14 @@ def chimera_graph(m, n=None, t=None, create_using=None, node_list=None, edge_lis
     A Chimera lattice is an m-by-n grid of Chimera tiles. Each Chimera
     tile is itself a bipartite graph with shores of size t. The
     connection in a Chimera lattice can be expressed using a node-indexing
-    notation (i,j,u,k) for each node.
+    notation (i, j, u, k) for each node.
 
-    * (i,j) indexes the (row, column) of the Chimera tile. i must be
-      between 0 and m-1, inclusive, and j must be between 0 and
-      n-1, inclusive.
+    * (i, j) indexes the (row, column) of the Chimera tile. i must be
+      between 0 and m - 1, inclusive, and j must be between 0 and
+      n - 1, inclusive.
     * u=0 indicates the left-hand nodes in the tile, and u=1 indicates
       the right-hand nodes.
-    * k=0,1,...,t-1 indexes nodes within either the left- or
+    * k=0, 1, ..., t - 1 indexes nodes within either the left- or
       right-hand shores of a tile.
 
     In this notation, two nodes (i, j, u, k) and (i', j', u', k') are
@@ -159,6 +161,11 @@ def chimera_graph(m, n=None, t=None, create_using=None, node_list=None, edge_lis
 
     max_size = m * n * 2 * t  # max number of nodes G can have
 
+    if edge_list is None:
+        check_edge_list = False
+    if node_list is None:
+        check_node_list = False
+
     if edge_list is None or check_edge_list is True:
         if coordinates:
             # tile edges
@@ -169,13 +176,13 @@ def chimera_graph(m, n=None, t=None, create_using=None, node_list=None, edge_lis
                              for k1 in range(t))
 
             # horizontal edges
-            G.add_edges_from(((i, j, 1, k), (i, j+1, 1, k))
+            G.add_edges_from(((i, j, 1, k), (i, j + 1, 1, k))
                              for i in range(m)
-                             for j in range(n-1)
+                             for j in range(n - 1)
                              for k in range(t))
             # vertical edges
-            G.add_edges_from(((i, j, 0, k), (i+1, j, 0, k))
-                             for i in range(m-1)
+            G.add_edges_from(((i, j, 0, k), (i + 1, j, 0, k))
+                             for i in range(m - 1)
                              for j in range(n)
                              for k in range(t))
         else:
@@ -204,20 +211,25 @@ def chimera_graph(m, n=None, t=None, create_using=None, node_list=None, edge_lis
             _add_compatible_edges(G, edge_list)
 
     else:
+        if check_node_list or node_list is None:
+            if coordinates:
+                G.add_nodes_from((i, j, u, k) for i in range(m)
+                                  for j in range(n)
+                                  for u in range(2)
+                                  for k in range(t))
+            else:
+                G.add_nodes_from(i for i in range(m*n*t*2))
+
         G.add_edges_from(edge_list)
 
     if node_list is not None:
-        nodes = set(node_list)
-        G.remove_nodes_from(set(G) - nodes)
         if check_node_list:
-            if G.number_of_nodes() != len(node_list):
-                raise ValueError("node_list contains nodes incompatible with "
-                                 "the specified topology and node-labeling "
-                                 "convention.")
-
+            _add_compatible_nodes(G, node_list)
         else:
+            nodes = set(node_list)
+            G.remove_nodes_from(set(G) - nodes)
             G.add_nodes_from(nodes)  # for singleton nodes
-
+        
     if data:
         if coordinates:
             def checkadd(v, q):
@@ -634,7 +646,7 @@ def mapping(q):
         y, x, u, k = source_to_chimera(q)
         return chimera_to_target((y + y_offset, x + x_offset, u, k))
 
-    #store the offset in the mapping, so the user can reconstruct it
+    # store the offset in the mapping, so the user can reconstruct it
     mapping.offset = offset
 
     return mapping
@@ -656,7 +668,7 @@ def chimera_sublattice_mappings(source, target, offset_list=None):
     tile parameters, and the mappings produced are not exhaustive.  The mappings
     take the form
 
-        ``(y, x, u, k) -> (y+y_offset, x+x_offset, u, k)``
+        ``(y, x, u, k) -> (y + y_offset, x + x_offset, u, k)``
 
     preserving the orientation and tile index of nodes.  We use the notation of
     Chimera coordinates above, but either or both of the target graph may have
@@ -726,3 +738,94 @@ def chimera_to_target(q):
     for offset in offset_list:
         yield _chimera_sublattice_mapping(source_to_chimera, chimera_to_target, offset)
 
+def chimera_torus(m, n=None, t=None, node_list=None, edge_list=None):
+    """Creates a defect-free Chimera lattice of size :math:`(m, n, t)` subject to periodic boundary conditions.
+
+
+    Parameters
+    ----------
+    m : int
+        Number of rows in the Chimera torus lattice.
+        If :math:`m<3` translational invariance already applies in the rows. If 
+        :math:`m>=3` additional external couplers are added, reestablishing 
+        translational invariance.
+        Connectivity of all horizontal qubits is :math:`min(m - 1, 2) + 2t`.
+    n : int (optional, default m)
+        Number of columns in the Chimera torus lattice.
+        If :math:`n<3` translational invariance already applies in the columns. If 
+        :math:`n>=3` additional external couplers are added, reestablishing 
+        translational invariance.
+        Connectivity of all vertical qubits is :math:`min(n - 1, 2) + 2t`.
+    t : int (optional, default 4)
+        Size of the shore within each Chimera tile.
+    node_list : iterable (optional, default None)
+        Iterable of nodes in the graph. If None, nodes are generated 
+        for an undiluted torus calculated from ``m``, ``n`` and ``t``
+        as described below. The node list must describe a subset
+        of the torus nodes to be maintained in the graph 
+        using the coordinate node labeling scheme.
+    edge_list : iterable (optional, default None)
+        Iterable of edges in the graph. If None, edges are generated
+        for an undiluted torus calculated from ``m``, ``n`` and ``t``
+        as described below. The edge list must describe 
+        a subgraph of the torus, using the coordinate node labeling scheme.
+
+    Returns
+    -------
+    G : NetworkX Graph
+        A Chimera torus with shape (m, n, t), with Chimera coordinate node labels.
+
+
+    A Chimera torus is a generalization of the standard chimera graph
+    whereby degree-six connectivity is maintained, but the boundary
+    condition is modified to enforce an additional translational-invariance 
+    symmetry [RH]_. Local connectivity in the Chimera torus
+    is identical to connectivity for chimera graph nodes away from the boundary.
+    The graph has :code:`V=8*m*n` nodes, and :code:`min(6, 4 + m)V//2 + 
+    min(6, 4 + n)V/2` edges. With the standard :math:`K_{t, t}` Chimera tile definition,
+    any tile displacement :math:`(x, y)`  modulo :math:`(m, n)`, rows and columns respectively,
+    that is, :code:`(i, j, u, k)` -> :code:`((i + x)%m, (i + y)%n, u, k)`,
+    defines an automorphism.
+
+    See :func:`.chimera_graph` for additional information.
+
+    Examples
+    ========
+    >>> G = dnx.chimera_torus(3, 3, 4)  # a 3x3 tile chimera graph (connectivity 6)
+    >>> len(G)
+    72
+    >>> any([len(list(G.neighbors(n))) != 6 for n in G.nodes])
+    False
+
+    """
+    # Graph properties are by and large inherited from chimera_graph
+    G = chimera_graph(m=m, n=n, t=t, node_list=None, edge_list=None, data=True, coordinates=True)
+    if n is None:
+        n = G.graph['columns']         
+    if t is None:
+        t = G.graph['tile']
+
+
+    # With modification of the boundary condition
+    if m>2:
+        # Wrapped around row external-coupler edges:
+        additional_edges = [((m - 1, j, 0, k), (0, j, 0, k))
+                            for j in range(n)
+                            for k in range(t)]
+    else:
+        additional_edges = []
+
+    if n>2:
+        # Wrapped around columns external-coupler edges:
+        additional_edges += [((i, n - 1, 1, k), (i, 0, 1, k))
+                             for i in range(m)
+                             for k in range(t)]
+
+    if len(additional_edges)>0:
+        G.add_edges_from(additional_edges)
+
+    _add_compatible_terms(G, node_list, edge_list)
+
+    G.graph['boundary_condition'] = 'torus'
+
+    return G

dwave_networkx/generators/common.py

@@ -4,10 +4,26 @@ def _add_compatible_edges(G, edge_list):
     # Slow when edge_list is large, but clear (non-defaulted behaviour, so fine):
     if edge_list is not None:
         if not all(G.has_edge(*e) for e in edge_list):
-            raise ValueError("edge_list contains edges incompatible with a "
-                             "fully yielded graph of the requested topology")
+            raise ValueError("edge_list contains edges incompatible with G")
         # Hard to check edge_list consistency owing to directedness, etc. Brute force
         G.remove_edges_from(list(G.edges))
         G.add_edges_from(edge_list)
         if G.number_of_edges() < len(edge_list):
             raise ValueError('edge_list contains duplicates.')
+
+def _add_compatible_nodes(G, node_list):
+    if node_list is not None:
+        if not all(G.has_node(n) for n in node_list):
+            raise ValueError("node_list contains nodes incompatible with G")
+        nodes = set(node_list)
+        remove_nodes = set(G) - nodes
+        G.remove_nodes_from(remove_nodes)
+        if G.number_of_nodes() < len(node_list):
+            raise ValueError('node_list contains duplicates.')
+
+def _add_compatible_terms(G, node_list, edge_list):
+    _add_compatible_edges(G, edge_list)
+    _add_compatible_nodes(G, node_list)
+    #Check node deletion hasn't caused edge deletion:
+    if edge_list is not None and len(edge_list) != G.number_of_edges():
+        raise ValueError('The edge_list contains nodes absent from the node_list')