Instances of pygsti.objects.StateSpaceLabels
describe the structure of a model's state space and associate labels with the parts of that structure. This is particularly useful when dealing with multiple qubits or a qubit and its environment, as it can be useful to reference subspaces or subsystems of the entire quantum state space.
In general, a state space is the direct sum of one or more tensor product blocks, each of which is the tensor product of one or more factors:
$$ \mbox{State space} = (\mathcal{H}_1^A \otimes \mathcal{H}_2^A \otimes \cdots) \oplus (\mathcal{H}_1^B \otimes \mathcal{H}_2^B \otimes \cdots) \oplus \cdots$$In the above expression the tensor product blocks are in parenthesis and labelled by $A$, $B$, etc., and the $\mathcal{H}_i^X$ are the factors. We can initialize a StateSpaceLabels
object using a list of tuples containing labels and dimensions which mirror this structure, i.e.
StateSpaceLabels( [(H1A_label, H2A_label, ...), ((H1B_label, H2B_label, ...), ... ],
[(H1A_dim , H2A_dim, ...), ((H1B_dim , H2B_dim, ...), ... ])
where _label
variables are either strings or integers (it can be convenient to label qubits, for instance, by integers) and _dim
variables are always integers. Here are some examples:
import pygsti
from pygsti.objects import StateSpaceLabels
lbls = StateSpaceLabels([('H0','H1')], [(2,3)])
print(lbls) # label(dim) notation, '*' means 'otimes', '+' means 'oplus'
lbls2 = StateSpaceLabels(('H0','H1'), (2,3)) # same as above - a *single* tensor product block
print(lbls2)
lbls3 = StateSpaceLabels([('H0',), ('H1',)], [(2,),(3,)]) # direct sum
print(lbls3)
lbls4 = StateSpaceLabels([('H1a','H2a'), ('H1b','H2b')], [(2,1),(3,4)])
print(lbls4)
Since we're often dealing with qubits (dimension = 2 factors), the labels beginning with 'Q' or that are integers default to dimension 2. Similarly, labels beginning with 'L' default dimension 1 (an additional "Level"). If all the labels in the first argument passed to the StateSpaceLabels
constructor have defaults, then the second argument (the dimensions) may be omitted. For example:
lbls5 = StateSpaceLabels(['Q0','Q1']) # 2 qubits
print(lbls5)
lbls6 = StateSpaceLabels(list(range(3))) # 3 qubits
print(lbls6)
lbls7 = StateSpaceLabels([('Q0','Q1'),('Leakage',)])
print(lbls7)
The raw data within a StateSpaceLabels
object is stored in the .labels
and .labeldims
members (but be careful, labeldims
is a dictionary):
lbls7.labels
lbls7.labeldims
You can access the total dimension of the state space using the .dim
member ($d^2$ if the density-matrix dimension is $d$, i.e. when density matrices are $d \times d$), and the tensor-product block dimensions $k_i^2$ (there is a kite structure of are $k_i\times k_i$ nonzero blocks that form the the $d \times d$ density matrix, so $\sum_i k_i = d$) via the .tpb_dims
property. You can also access the per-label dimension via the .labeldims
dictionary.
print("Total dim = ",lbls7.dim)
#print("Separately: ",lbls7.dim, lbls7.dim.blockDims, lbls7.dim.opDim, lbls7.dim.embedDim)
print("Dimensions of tensor product blocks = ", lbls7.tpb_dims)
print("Dimension of the space associated with each label: ", lbls7.labeldims)
There are also few convenience functions that make it easier to access the raw data:
print("Number of tensor product blocks = ",lbls7.num_tensor_prod_blocks())
print("The labels in the 0th tensor product block are: ",lbls7.tensor_product_block_labels(0))
print("The dimensions corresponding to those labels are: ",lbls7.tensor_product_block_dims(0))
print("The 'Q0' labels exists in the tensor product block w/index=",lbls7.tpb_index['Q0'])
print("The product of the dimensions associated with 'Q0' and 'Leakage' = ",lbls7.product_dim( ('Q0','Leakage') ))
That's it! You know all there is to know about the StateSpaceLabels
object. Remember you can pass a StateSpaceLabels
object to pygsti.construction.build_explicit_model
to create a model which operates on the given state space.