TensorNetCanvas is a web-based visualizer and simulator for tensor networks.
Tensors can be created manually by entering a nested array in the text field in the upper left part of the user interface. For example, entering this string
[[1,0],[0,-1]]
and hitting the "Create Tensor(s)" button will create a 2×2 tensor equivalent to a Pauli Z matrix. The user can also enter a set of tensors, separated by spaces or commas or semicolons or newlines, such as
[[0,1],[1,0]]; [[0,-i],[i,0]]; [[1,0],[0,-1]]; [[1,0],[0,i]]; [[1,0],[0,0.7071+0.7071i]]
to create the 3 Pauli matrices X, Y, Z, as well as an S and T matrix. Alternatively, the user can enter tuples (with parentheses) containing a nested array followed by a name for the tensor, and optional names for each of the indices, such as
([[1,0],[0,i]],'S','out','in')
for a Pauli S matrix. Multiple tensors can be entered, separated by spaces or commas or semicolons or newlines, such as
([[0,1],[1,0]],'X','out','in')
([[0,-i],[i,0]],'Y','out','in')
([[1,0],[0,-1]],'Z','out','in')
([[1,0],[0,i]],'S','out','in')
([[1,0],[0,0.707106781+0.707106781i]],'T','out','in')
([[0.707106781,0.707106781],[0.707106781,-0.707106781]],'H','out','in')
Each tensor is shown as a rectangle with tabs below it corresponding to the indices. Edges can be created between tensors by dragging from one tab to another. For example, if the user creates a Hadamard (H) matrix and a Pauli Z matrix, and drags from index 1 of the former to index 0 of the latter, the result looks like this:
The user can then select both tensors (for example, by placing the cursor above and to the left of both tensors, holding down the Ctrl key and the left mouse button, and dragging out a rectangle to select both), and then press the "Contract selected" button. This should result in a tensor equal to the matrix product of H and Z, i.e., [[0.707,-0.707],[0.707,0.707]].
TensorNetCanvas runs inside a web browser, and the URL in the browser contains information encoding the structure of the tensors and the edges connecting them. This means that tensor networks can be bookmarked, saved in a text file, and shared in an email. As an example, opening this link should show TensorNetCanvas with the below tensor network:
The above network contains several examples of operations. Contracting the top two tensors is equivalent to performing a dot product of two vectors. Contracting the next two is equivalent to a product of a matrix with a column vector. Next, we have the outer product of two vectors, a product of three 2×2 matrices (yielding H Z H = X), and the trace of a matrix.
Contractions, as implemented in TensorNetCanvas, are a generalization of the dot product of pairs of vectors, the outer product of pairs of vectors, the matrix product, the trace, partial trace, and tensor product. Contractions can be performed on a single tensor (causing self-edges to be contracted), or on two tensors, or on many tensors. In general, contracting a set of selected tensors will first cause each tensor to have any self-edges contracted, followed by binary contractions on any connected tensors, followed by tensor products on any remaining disconnected tensors.
The lower left corner of the user interface contains a list of functions associated with mouse buttons and keyboard keys:
Mouse controls:
Left drag on unselected node: move node
Left drag on selected node: move all selected nodes
Ctrl+click on node: toggle selection
Ctrl+drag on empty space: rectangle or lasso select
Left drag from index to index: create edge
Right click on index: menu
Left drag on empty space: pan
Scroll wheel: zoom in/out
Other controls:
Ctrl+A: select all nodes
Delete or Backspace: delete selected nodes
Browser Back/Forward: undo/redo
C: contract selected node(s)
F: frame all nodes
This video demonstrates TensorNetCanvas's functionality with several examples:
The description of the video on youtube contains a table of contents with timestamps.
A companion paper is available (not yet published - email the author for a copy) that explains the design and features of TensorNetCanvas and gives more example tensors, including order-3 tensors.
A quantum circuit can be created inside Quirk or MuqcsCraft, and then the URL (which contains a description of the circuit) can be pasted into another text field in TensorNetCanvas to import the circuit and convert it to a tensor network.
Above: a circuit in MuqcsCraft.
Below: the same circuit, after importing it into TensorNetCanvas. Notice that each control or anticontrol qubit in the original circuit is converted to an order-3 tensor. Contracting the entire network results in a single tensor that can be vectorized (flattened into a vector) to yield the output state vector of the original circuit. This shows how a tensor network can be used to simulate a quantum circuit.
When TensorNetCanvas is running inside a web browser, the user can open a JavaScript console and call subroutines inside the code programmatically. The following lines of code can be pasted into the console, and illustrate how to create a few tensors and perform operations with them.
// define two vectors and one matrix
let v1 = CTensor.create([1,2,3]);
let v2 = CTensor.create([4,5,6]);
let m1 = CTensor.create([ [1,2,3], [4,5,6], [7,8,9] ]);
// dot product of two vectors
console.log( CTensor.einsum("i,i->",v1,v2).toString() );
// outer product of two vectors
console.log( CTensor.einsum("i,j->ij",v1,v2).toString() );
// trace of matrix
console.log( CTensor.einsum("ii->",m1).toString() );
// product of matrix with column vector
console.log( CTensor.einsum("ij,j->i",m1,v1).toString() );