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 or ). 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.
- Received 18 November 2022
- Accepted 12 April 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.020339
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Studying ground states of many-body quantum systems has led to a rich plethora of quantum phases of matter. A key organizational principle is to regard two quantum states as being in the same phase of matter if they can be related by a shallow circuit (dis)entangling operation. Nontrivial disordered states (that is, those that are not in the same phase as an unentangled state) exhibit so-called topological order, with great relevance to condensed matter, quantum simulators, and quantum computation. Recently, studies have shown that measurements can be a valuable resource for efficiently preparing certain topologically ordered states of matter that are out of the scope of shallow (dis)entangling operations. This raises the question of whether a coarser equivalence can be established for phases of matter based on the number of measurements required for state preparation.
In this work, we introduce a new hierarchy of quantum states based on the number of shots (rounds of single-site measurements interspersed with finite-depth unitaries) needed to prepare them. We explore the classification of states and show examples of different measurement-equivalent phases. We find that various non-Abelian topological orders can be created with a single-shot protocol, similar to their simpler Abelian counterparts. We also argue that certain phases of matter (such as Fibonacci topological order) are not reachable by a finite number of shots.
This work contributes to the understanding of emergent properties of quantum states, provides road maps for preparing long-range entangled states in quantum devices, and inspires questions about the relationship between the difficulty of preparing quantum states and their quantum computational power.
