Fig 1.
An example of a tensor network.
The number of edges (or wires) connecting to a tensor is equal to that tensor’s rank. When an index (edge) is contracted by combining two tensors according to Eq 1, the two tensor are replaced by a new one. The number of scalar entries in the tensor scales exponentially in the number of edges to which it connects. In general it is not trivial to choose an efficient contraction ordering that minimizes the total number of floating point operations.
Fig 2.
A tetrahedral graph illustrates why it is often unavoidable to form tensors of higher rank while contracting a tensor network.
In this example, contracting any of the six edges produces a tensor of rank 4, even though all of the original tensors were of rank 3.
Fig 3.
Results from implementing 10,000 trials of Eq 6.
We use n = 6 qubits, m = 10, and p = 2. (a) The histogram plots how close the method’s output string is to the actual most likely string. qTorch’s procedure for the “estimate” is given in the text. The horizontal axis Ranking is the number of computational basis states in |Ψ〉 with higher probability than the estimated string—a lower number ranking indicates a better estimate. (b) Distribution of the 1-norm distance between the approximate distribution p′ arising from the product state approximation |Ψ′〉 in and the distribution p arising from the exact state |Ψ〉.
Fig 4.
Time results for simulating quantum circuits of the Hubbard model.
LG, Stoch, and LIQUi|> denote linegraph-based tensor contraction, stochastic tensor contraction, and LIQUi|>, respectively. LIQUi|>’s full Hilbert simulation method is substantially faster than either tensor contraction method. Missing data points resulted from exceeding memory capacity.
Fig 5.
Simulation time plotted against number of qubits for Max-Cut/QAOA circuits.
LG, Stoch, and LIQUi|> denote linegraph-based tensor contraction, stochastic tensor contraction, and the LIQUi|> software package, respectively. Tree decompositions for the LG method were determined by running the QuickBB simulation for 3000 seconds. For lower regularities, the tensor contraction methods outperform LIQUi|>, since LIQUi|> simulates the full Hilbert space. However, as the regularity of the Max-Cut graphs (and hence the treewidth of the quantum circuits’ line graphs) increase, full Hilbert space simulation using LIQUi|> becomes more efficient.
Fig 6.
Simulation time plotted against number of qubits for 3-regular Max-Cut/QAOA circuits.
LG and Stoch denote linegraph-based tensor contraction and stochastic tensor contraction respectively. For 3-regular Max-Cut/QAOA circuits, we were able to simulate a small subset of the 100-qubit circuits we created, not shown here.
Fig 7.
Simulation time plotted against the regularity of the underlying Max-Cut graph, for Max-Cut/QAOA circuits.
LG, Stoch, and LIQUi|> denote linegraph-based tensor contraction, stochastic tensor contraction, and LIQUi|>, respectively. As regularity increases, full Hilbert space simulation (using LIQUi| >) becomes a more competitive simulation method. Missing data points resulted from running out of memory.
Fig 8.
Simulation time plotted against approximate treewidth, for all simulated Max-Cut/QAOA quantum circuits of 18 qubits.
The plot demonstrates the general trend of increased simulation time with the quantum circuit’s line graphs’s treewidth, despite a constant number of qubits.