Faster identification of optimal contraction sequences for tensor networks

Abstract

The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, including quantum many-body physics, loop quantum gravity, and quantum chemistry. The computational cost of evaluating an expression may depend strongly on the order in which the index sums are evaluated, and determination of the operation-minimizing contraction sequence for a single tensor network (single term, in quantum chemistry) is known to be NP-hard. The current preferred solution is an exhaustive search, using either an iterative depth-first approach with pruning or dynamic programming and memoization, but these approaches are impractical for many of the larger tensor network ansätze encountered in quantum many-body physics. We present a modified search algorithm with enhanced pruning which exhibits a performance increase of several orders of magnitude while still guaranteeing identification of an optimal operation-minimizing contraction sequence for a single tensor network. A reference implementation for MATLAB, compatible with the ncon() and multienv() network contractors of arXiv:1402.0939 and Evenbly and Pfeifer, Phys. Rev. B 89, 245118 (2014), respectively, is supplied.

Hand-waving and interpretive dance: an introductory course on tensor networks

“…Whilst algorithms which can speed up tensor network contractions by optimising the bubbling used [3][4][5], as discusssed above, the underlying computational problem is NP-complete [6,7] Even ignoring the specific bubbling used, the complexity of the overall contraction procedure can also be shown to be prohibitive in general. Consider a network made from the binary tensors e and n. The value of e is 1 if and only if all indices are identical, and zero otherwise, whilst n has value 1 if and only if all legs differ and 0 otherwise.…”

“…Pfeifer et al [3] provides code which allows for finding optimal bubbling order for networks of up to 30-40 tensors. This code interfaces with that provided in [4] and [5], providing a complete tensor network package.…”

Adaptive algorithm for quantum circuit simulation

“…The graph-based notation became standard in tensor network literature a decade ago. [15] Following Markov and Shi, several groups developed highly efficient algorithms for quantum circuit simulation based on this representation, see 6,11,16, and 17 for more details. The previous margin of 50 qubits was lifted, as is demonstrated by multiple authors.…”

Tensor renormalization group with randomized singular value decomposition

“…As we mentioned, the cost of contraction of the tensor network without Ω and Q † is O(χ 6 ). The order of contractions is important to reduce the computational cost [23]. For example, the computational cost of Y = S [1] (S [2] (S [3] (S [4] Ω))) scales as O(χ 5 ), but Y = (((S [1] S [2] )S [3] )S [4] )Ω scales as O(χ 6 ).…”