Hierarchy of topological order from finite-depth unitaries, measurement and feedforward

• URL: http://arxiv.org/abs/2209.06202v2
• Date: Sun, 11 Jun 2023 08:20:36 GMT
• Title: Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Authors: Nathanan Tantivasadakarn, Ashvin Vishwanath, Ruben Verresen
• Abstract summary: Single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases. 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" This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 two (such as $D_4$ or $Q_8$). We show that after the measurement, the wavefunction 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 which 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 double for all solvable groups. Third, we argue that certain topological orders, such as non-solvable quantum doubles or Fibonacci anyons, define non-trivial 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.

Related papers

• Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits [0.0]
  We show how to prepare anyon theories that admit a gapped boundary via Adaptive Finite-Depth Local Unitary (AFDLU) Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons.
  arXiv  Detail & Related papers  (2024-11-07T18:55:09Z)
• Taming Quantum Time Complexity [45.867051459785976]
  We show how to achieve both exactness and thriftiness in the setting of time complexity. We employ a novel approach to the design of quantum algorithms based on what we call transducers.
  arXiv  Detail & Related papers  (2023-11-27T14:45:19Z)
• Intrinsic Mixed-state Quantum Topological Order [4.41737598556146]
  We show that decoherence can give rise to new types of topological order. We construct concrete examples by proliferating fermionic anyons in the toric code via local quantum channels. The resulting mixed states retain long-range entanglement, which manifests in the nonzero topological entanglement negativity.
  arXiv  Detail & Related papers  (2023-07-25T18:34:10Z)
• Sparse random Hamiltonians are quantumly easy [105.6788971265845]
  A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. This paper shows that, for most random Hamiltonians, the maximally mixed state is a sufficiently good trial state. Phase estimation efficiently prepares states with energy arbitrarily close to the ground energy.
  arXiv  Detail & Related papers  (2023-02-07T10:57:36Z)
• Non-Abelian braiding of graph vertices in a superconducting processor [144.97755321680464]
  Indistinguishability of particles is a fundamental principle of quantum mechanics. braiding of non-Abelian anyons causes rotations in a space of degenerate wavefunctions. We experimentally verify the fusion rules of the anyons and braid them to realize their statistics.
  arXiv  Detail & Related papers  (2022-10-19T02:28:44Z)
• Witnessing superpositions of causal orders before the process is completed [0.0]
  What is the most general representation of a quantum state at a single point in time? Can we adapt the current formalisms to situations where the order of quantum operations is coherently or incoherently superposed?
  arXiv  Detail & Related papers  (2022-09-19T16:47:53Z)
• Shortest Route to Non-Abelian Topological Order on a Quantum Processor [0.0]
  We show there exists a broad family of non-Abelian states -- namely those with a Lagrangian subgroup. We show how $D_4$ non-Abelian topological order can be realized on Google's quantum processors.
  arXiv  Detail & Related papers  (2022-09-08T18:00:00Z)
• Growth of entanglement of generic states under dual-unitary dynamics [77.34726150561087]
  Dual-unitary circuits are a class of locally-interacting quantum many-body systems. In particular, they admit a class of solvable" initial states for which, in the thermodynamic limit, one can access the full non-equilibrium dynamics. We show that in this case the entanglement increment during a time step is sub-maximal for finite times, however, it approaches the maximal value in the infinite-time limit.
  arXiv  Detail & Related papers  (2022-07-29T18:20:09Z)
• Adaptive constant-depth circuits for manipulating non-abelian anyons [65.62256987706128]
  Kitaev's quantum double model based on a finite group $G$. We 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.
  arXiv  Detail & Related papers  (2022-05-04T08:10:36Z)
• Efficiently preparing Schr\"odinger's cat, fractons and non-Abelian topological order in quantum devices [0.0]
  Long-range entangled quantum states -- like cat states and topological order -- are key for quantum metrology and information purposes. We propose how to scalably prepare a broad range of long-range entangled states with the use of existing experimental platforms. Remarkably, this protocol can prepare the 1D Greenberger-Horne-Zeilinger (GHZ) 'cat' state and 2D toric code with fidelity per site exceeding $0.9999$.
  arXiv  Detail & Related papers  (2021-12-02T18:58:34Z)
• Observing a Topological Transition in Weak-Measurement-Induced Geometric Phases [55.41644538483948]
  Weak measurements in particular, through their back-action on the system, may enable various levels of coherent control. We measure the geometric phases induced by sequences of weak measurements and demonstrate a topological transition in the geometric phase controlled by measurement strength. Our results open new horizons for measurement-enabled quantum control of many-body topological states.
  arXiv  Detail & Related papers  (2021-02-10T19:00:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.