Abstract
Many recent tensor network algorithms apply unitary operators to parts of a tensor network in order to reduce entanglement. However, many of the previously used iterative algorithms to minimize entanglement can be slow. We introduce an approximate, fast, and simple algorithm to optimize disentangling unitary tensors. Our algorithm is asymptotically faster than previous iterative algorithms and often results in a residual entanglement entropy that is within 10 to 40% of the minimum. For certain input tensors, our algorithm returns an optimal solution. When disentangling order-4 tensors with equal bond dimensions, our algorithm achieves an entanglement spectrum where nearly half of the singular values are zero. We further validate our algorithm by showing that it can efficiently disentangle random 1D states of qubits.
Highlights
Many recent tensor network algorithms [1,2,3,4,5,6,7] rely on the application of unitary tensors in order too reduce the short-ranged entanglement and correlations within the tensor network
The fast algorithm typically results in an entanglement Sfast within 10 to 40% of the global minimum Smin
In the 5th column of Tab. 1, we show how much longer it takes the gradient descent algorithm to optimize down to the entanglement Sfast reached by our fast algorithm; we find speedups ranging from 20 to 20,000 times as the bond dimension is increased from 2 to 16
Summary
Many recent tensor network algorithms [1,2,3,4,5,6,7] rely on the application of unitary (or isometry) tensors in order too reduce the short-ranged entanglement and correlations within the tensor network. The CPU time of our algorithm scales as O(χ13χ32 + χ16) when χ1 = χ2 and χ3 = χ4.4 This CPU complexity is as fast or faster (when χ3 χ1) than the complexity O(χ13χ33) for computing the singular values of A across the dotted red line, which are needed to calculate the entanglement entropy across the dotted red line. This makes our algorithm asymptotically faster than just a single step of any iterative algorithm that attempts to minimize the entanglement entropy.
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