Wave-function positivization via automatic differentiation

Abstract

We introduce a procedure to systematically search for a local unitary transformation that maps a wave function with a nontrivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one-and two-qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign for a two-leg triangular Heisenberg ladder with next-to-nearestneighbor and ring-exchange interactions. This model exhibits phases where the sign structure can be removed by simple local one-qubit unitaries, but also an exotic Bose-metal phase whose sign structure induces "Bose surfaces" with a fermionic character and a higher entanglement that requires deeper circuits.

Intrinsic sign problem in fermionic and bosonic chiral topological matter

“…It may therefore be possible to obtain fine-tuned models and QMC methods that lead to a small enough to be practically useful. More generally, it may be possible to search for such models and methods algorithmically, thus easing the intrinsic sign problem [10,[122][123][124]. We also note that the results presented in this paper do not exclude approaches to the sign problem based on a modified or constrained Monte Carlo sampling [13,125,126], as well as machine-learning-aided QMC [127], and infinite-volume diagrammatic QMC [128].…”

Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians

“…Other methods for generating positive-valued decompositions of the partition function include, e.g., resummation techniques wherein negativevalued weights in the decomposition are grouped together with positive ones to form positive ``superweights"" that can in turn be treated as probabilities in a quantum Monte Carlo algorithm [35,13,19]. Other methods also include applying a constantdepth quantum circuit [32]. These other methods are beyond the scope of this paper.…”

Symmetry-protected sign problem and magic in quantum phases of matter

“…We assume that these low-energy states are similar to the ground state of the un-truncated Hamiltonian in regions far from the boundary. 5 Hence, the low-energy states correspond to excitations localized near the boundary or degenerate ground states.…”

“…While the magic in a many-body state and the notion of a sign problem are promising metrics for the quantum complexity of states, they are notoriously challenging to study analytically and numerically, although substantial progress has been made [1,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. We therefore propose a simplification by imposing symmetry constraints.…”