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In the context of quantum physics, the term "duality" refers to transformations that link apparently distinct physical theories, often unveiling hidden symmetries. Some recent studies have been aimed at understanding and implementing duality transformations, as this could aid the study of quantum states and symmetry-protected phenomena.

Researchers at the University of Cambridge, Ghent University, Institut des Hautes Études Scientifiques and the University of Sydney recently demonstrated the implementation of dualities in symmetric 1-dimensional (1D) quantum lattice models, outlining a method to turn duality operators into unitary linear-depth quantum circuits.

Their paper, in Â鶹ÒùÔºical Review Letters, is part of a larger research effort aimed at better understanding symmetries and dualities in quantum lattice models.

"In recent years, there has been a lot of progress in our understanding of how symmetries can interact with the locality structure inherently present in a quantum lattice model," Laurens Lootens, co-first author of the paper, told Â鶹ÒùÔº.

"In particular, studies showed that symmetries do not necessarily need to act in an on-site manner, but that they can also create correlations between neighboring degrees of freedom and entangle them. Such operations are very naturally described as matrix product operators, a type of tensor network that explicitly encodes the non-trivial entanglement structure in these more general symmetry operators," Lootens added.

As part of their earlier studies, Lootens and his colleagues showed that one can systematically construct matrix product operator symmetries, using mathematical objects called fusion categories and their corresponding representation theories. This led them to the realization that duality transformations can also be written as matrix product operators.

Ultimately, they uncovered the entanglement structure of duality transformations. This inspired their recent studies, which were aimed at writing these transformations as quantum circuits that could be implemented on quantum hardware.

"The conventional wisdom has been that dualities are not reversible; some information is lost when performing the duality transformation," explained Lootens. "In specific cases where these dualities leave the theory unchanged, this has earned them the name of 'non-invertible symmetries.' Mathematically, this implies that these transformations cannot be represented as unitary operators, which prohibits them from being written as a quantum circuit."

In their earlier studies, Lootens and his colleagues showed that when they allowed a duality transformation to act on the boundary condition of its corresponding theory, no information was lost and when they inverted the transformation they ended up with the original theory. This essentially entails that by including the boundary condition as a degree of freedom, which essentially enlarged the space in which the transformation acted, the transformation became unitary.

"Together with the matrix product operator representation of the duality, this allowed us to write this seemingly non-invertible transformation as a unitary quantum circuit," said Lootens. "Since dualities can turn local operators into non-local string operators, we knew the depth of the circuit—the number of consecutive steps needed to implement it—had to scale at least linearly with the system size."

In the explicit construction produced by the researchers, the depth of the circuit was found to be equal to the system's size. This suggests that the circuit they realized is optimal.

"The dualities we consider essentially amount to generalized gauging procedures of (possibly non-invertible) symmetries," Clement Delcamp, co-first author of the paper, told Â鶹ÒùÔº. "As such, these are typically non-invertible operations in closed boundary conditions, and a fortiori non-unitary. So, what kind of operations do these correspond to?

"Using our previous results, we demonstrated that the duality operators can be realized as unitary linear-depth quantum circuits when supplementing the Hilbert space with ancillary degrees of freedom that keep track of the various sectors, and subsequently measuring the ancillary degrees of freedom so as to project input and output states into definite sectors."

Lootens, Delcamp and their colleagues showed that the non-unitarity of an operator is explained by the necessity to measure ancillary degrees of freedom. This implies that dualities are typically encoded into quantum channels.

"What makes this result especially noteworthy is that we were able to show it for arbitrary dualities between arbitrary (1+1)d models possessing arbitrary internal symmetries," explained Delcamp.

This recent study could soon open new opportunities for research focusing on symmetries and dualities in quantum lattice models. Lootens and Delcamp now plan to continue conducting research on this topic, building on their recent efforts and results.

"For the one-dimensional (1D) case, there is still an open question concerning what exactly characterizes those dualities that can be performed in constant depth, independent of the system size," said Lootens. "We showed that for a certain class of dualities that we refer to as 'nilpotent,' this can be done through an intricate sequence of quantum gates and measurements, but we know from examples that this is not the most general classification."

In their future work, the researchers also hope to demonstrate that their results can be generalized across higher dimensions. This could prove challenging, as it will entail the use of higher category theory and its associated mathematical tools have only very recently been developed.

"For specific examples, however, quantum circuits have already been devised and used to great success in the preparation of exotic topological states of matter on real quantum devices," added Lootens. "We are thus hopeful that the ideas in our paper can be used to obtain a similarly general theory in these higher-dimensional settings as well."

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