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arXiv:2209.06202 · PRX Quantum

Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward

Nathanan Tantivasadakarn, Ashvin Vishwanath, Ruben Verresen

Published Jun 9, 2023• 123 citations
arXivHTMLPDF

Abstract

Long-range entanglement-the backbone of topologically ordered states-cannot be created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation.Recently, it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases.Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots."First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class 2 (such as D 4 or Q 8 ).We show that after the measurement, the wave function always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome.Moreover, the clean quantum double ground state can be deterministically prepared via feedforward-gates that depend on the measurement outcomes.Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group.As a special case, this gives an explicit protocol to prepare twisted quantum doubles for all solvable groups.Third, we argue that certain topological orders, such as nonsolvable quantum doubles or Fibonacci anyons, define nontrivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots.Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones.This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.

DOI
10.1103/prxquantum.4.020339

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Long-range entanglement-the backbone of topologically ordered states-cannot be created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation.Recently, it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases.Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots."First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class 2 (such as D 4 or Q 8 ).We show that after the measurement, the wave function always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome.Moreover, the clean quantum double ground state can be deterministically prepared via feedforward-gates that depend on the measurement outcomes.Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group.As a special case, this gives an explicit protocol to prepare twisted quantum doubles for all solvable groups.Third, we argue that certain topological orders, such as nonsolvable quantum doubles or Fibonacci anyons, define nontrivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots.Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones.This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.
A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z 2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G × H symmetry gives rise to anomalous LRE upon measuring G -charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024
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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.
A highly coveted goal is to realize emergent non-Abelian gauge theories and their anyonic excitations, which encode decoherence-free quantum information. While measurements in quantum devices provide new hope for scalably preparing such long-range entangled states, existing protocols using the experimentally established ingredients of a finite-depth circuit and a single round of measurement produce only Abelian states. Surprisingly, we show there exists a broad family of non-Abelian states-namely those with a Lagrangian subgroup-which can be created using these same minimal ingredients, bypassing the need for new resources such as feed forward. To illustrate that this provides realistic protocols, we show how D_{4} non-Abelian topological order can be realized, e.g., on Google's quantum processors using a depth-11 circuit and a single layer of measurements. Our work opens the way toward the realization and manipulation of non-Abelian topological orders, and highlights counterintuitive features of the complexity of non-Abelian phases.
Exactly solvable models of topologically ordered phases with non-abelian anyons typically require complicated many-body interactions which do not naturally appear in nature. This motivates the "inverse problem" of quantum many-body physics: given microscopic systems with experimentally realistic two-body interactions, how to design a Hamiltonian that realizes a desired topological phase? Here we solve this problem on a platform motivated by Rydberg atoms, where elementary two-level systems couple via simple blockade interactions. Within this framework, we construct Hamiltonians that realize topological orders described by non-abelian quantum double models. We analytically prove the existence of topological order in the ground state, and present efficient schemes to prepare these states. We also introduce protocols for the controlled adiabatic braiding of anyonic excitations to probe their non-abelian statistics. Our construction is generic and applies to quantum doubles D(G)\mathcal{D}(G)D(G) for arbitrary finite groups GGG. We illustrate braiding for the simplest non-abelian quantum double D(S3)\mathcal{D}(S_3)D(S3​).
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