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arXiv:2205.01933

Adaptive constant-depth circuits for manipulating non-abelian anyons

Sergey B. Bravyi, Isaac H. Kim, Alexander Kliesch, Robert König

Published May 4, 2022• 8 citations
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Abstract

We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group $G$. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-$1$ circuit composed of generalized Pauli gates.

DOI
10.48550/arxiv.2205.01933

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We consider Kitaev's quantum double model based on a finite group GGG and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group GGG all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group GGG. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-111 circuit composed of generalized Pauli gates.
The classification of topological phases of matter is a fundamental challenge in quantum many-body physics, with applications to quantum technology. Recently, this classification has been extended to the setting of adaptive finite-depth local unitary (AFDLU) circuits, which allow global classical communication. In this setting, the trivial phase is the collection of all topological states that can be prepared via AFDLU. Here, we propose a complete classification of the trivial phase by showing how to prepare all solvable anyon theories that admit a gapped boundary via AFDLU, extending recent results on solvable groups. Our construction includes non-Abelian anyons with irrational quantum dimensions, such as Ising anyons, and more general acyclic anyons. Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase described by a solvable anyon theory with gapped boundary. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons and symmetry twist defects in solvable anyon theories. We apply our procedure to the quantum double of the group S 3 and to several examples that are beyond solvable groups, including the doubled Ising theory, the Z 3 Tambara-Yamagami string net, and doubled SU ( 2 ) 4 anyons.
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.
Adaptive quantum circuits employ unitary gates assisted by mid-circuit measurement, classical computation on the measurement outcome, and the conditional application of future unitary gates based on the result of the classical computation. In this paper, we experimentally demonstrate that even a noisy adaptive quantum circuit of constant depth can achieve a task that is impossible for any purely unitary quantum circuit of identical depth: the preparation of long-range entangled topological states with high fidelity. We prepare a particular toric code ground state with fidelity of at least 76.9±1.3%76.9\pm 1.3\%76.9±1.3% using a constant depth (d=4d=4d=4) adaptive circuit, and rigorously show that no unitary circuit of the same depth and connectivity could prepare this state with fidelity greater than 50%50\%50%.
The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback ("adaptive circuits") can evade such restrictions.We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs).The three classes are inspired by distinct physical insights, including tensor-network constructions, the multiscale entanglement renormalization ansatz, and parton constructions.A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and onedimensional CFT states and non-Abelian topological orders with both solvable and nonsolvable groups can be prepared in depth scaling logarithmically with system size.We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary Calderbank-Shor-Steane codes.Our work illustrates the practical and conceptual versatility of measurement for state preparation.
Creating and manipulating anyons and symmetry defects in topological phases, especially those with a non-Abelian character, constitutes a primitive for topological quantum computation. We provide a physical protocol for implementing the ribbon operators of non-Abelian anyons and symmetry defects. We utilize dualities, in particular, the Kramers-Wannier or gauging map, which have previously been used to construct topologically ordered ground states by relating them to simpler states. In this Letter, ribbon operators are implemented by applying a gauging procedure to a lower-dimensional region of such states. This protocol uses sequential unitary circuits or, in certain cases, constant-depth adaptive circuits. We showcase this for anyons and defects in the Z_{3} toric code and S_{3} quantum double. The general applicability of our method is demonstrated by deriving unitary expressions for ribbon operators of various (twisted) quantum doubles.
A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order of the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the lengths of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.