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arXiv:2410.11602 · Physical review. B./Physical review. B

Preparing topological states with finite depth simultaneous commuting gates

Yarden Sheffer, Erez Berg, Ady Stern

Published Mar 28, 2025• 2 citations
arXivHTMLPDF

Abstract

We present protocols for preparing two-dimensional Abelian and non-Abelian topologically ordered states by employing finite depth unitary circuits composed of long-ranged, simultaneous, and mutually commuting two-qubit gates. Our protocols are motivated by recent proposals for circuits in trapped ion systems, which allow each qubit to participate in multiple gates simultaneously. Our circuits are shown to be optimal, in the sense that the number of two-qubit gates and ancilla qubits scales as $O({L}^{2})$, where $L$ is the linear size of the system. Examples include the ground states of the toric code, certain Kitaev quantum-double models, and string-net models. Going beyond two dimensions, we extend our scheme to more general Calderbank-Shor-Steane codes. As an application, we present protocols for realizing the three-dimensional Haah's code and X-cube fracton models.

DOI
10.1103/physrevb.111.115160

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We present protocols for preparing two-dimensional Abelian and non-Abelian topologically ordered states by employing finite depth unitary circuits composed of long-ranged, simultaneous, and mutually commuting two-qubit gates. Our protocols are motivated by recent proposals for circuits in trapped ion systems, which allow each qubit to participate in multiple gates simultaneously. Our circuits are shown to be optimal, in the sense that the number of two-qubit gates and ancilla qubits scales as O(L2)O({L}^{2})O(L2), where LLL is the linear size of the system. Examples include the ground states of the toric code, certain Kitaev quantum-double models, and string-net models. Going beyond two dimensions, we extend our scheme to more general Calderbank-Shor-Steane codes. As an application, we present protocols for realizing the three-dimensional Haah's code and X-cube fracton models.
The finding of physical realizations of topologically ordered states in experimental settings, from condensed matter to artificial quantum systems, has been the main challenge en route to utilizing their unconventional properties.We show how to realize a large class of topologically ordered states and simulate their quasiparticle excitations on a digital quantum computer.To achieve this, we design a set of linear-depth quantum circuits to generate ground states of general string-net models together with unitary open-string operators to simulate the creation and braiding of Abelian and non-Abelian anyons.We show that the Abelian (non-Abelian) unitary string operators can be implemented with a constant-(linear-) depth quantum circuit.Our scheme allows us to directly probe characteristic topological properties, including topological entanglement entropy, braiding statistics, and fusion channels of anyons.Moreover, this set of efficiently prepared topologically ordered states has potential applications in the development of fault-tolerant quantum computers.
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We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and nondiagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of S 3 to generate a controlled-charge-conjugation ( C C ) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault tolerance of our protocol, presenting insights into how to construct a heralded decoder for the quantum double of S 3 . We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into D ( G ) , where G is a semidirect product of Abelian groups. This work serves as a step toward classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.
We provide explicit constant-depth local unitary circuits that realize general anyon permutations in Kitaev's quantum double models. This construction can be naturally understood through a correspondence between anyon permutation symmetries of two-dimensional topological orders and self-dualities in one-dimensional systems, where local gates implement self-duality transformations on the boundaries of microscopic regions. From this holographic perspective, general anyon permutations in the D(G)D(G)D(G) quantum double correspond to compositions of three classes of one-dimensional self-dualities, including gauging of certain subgroups of GGG, stacking with GGG symmetry-protected topological phases, and outer automorphisms of the group GGG. We construct circuits realizing the first class by employing self-dual unitary gauging maps, and present transversal circuits for the latter two classes.
Realizing non-Abelian topological orders and their anyon excitations is an esteemed objective. In this work, we propose a novel approach towards this goal: quantum simulating topological orders in the doubled Hilbert space---the space of density matrices. We show that ground states of all quantum double models (toric code being the simplest example) can be efficiently prepared in the doubled Hilbert space; only finite-depth local operations are needed. In contrast, this is not the case in the conventional Hilbert space: ground states of only some of these models are known to be efficiently preparable. Additionally, we find that nontrivial anyon braiding effects, both Abelian and non-Abelian, can be realized in the doubled Hilbert space, although the intrinsic nature of density matrices restricts possible excitations.