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 preparation certain cases. Here we show how this observation imposes complexity hierarchy on long-range entangled states based the minimal number measurement layers required create state, which call "shots"....

In the field of monitored quantum circuits, it has remained an open question whether finite-time protocols for preparing long-range entangled states lead to phases matter that are stable gate imperfections, can convert projective into weak measurements. Here, we show in certain cases, entanglement persists presence measurements, and gives rise novel forms criticality. We demonstrate this explicitly two-dimensional Greenberger-Horne-Zeilinger cat state three-dimensional toric code as minimal...

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 devices provide new hope for scalably preparing such long-range entangled states, existing protocols using the experimentally established ingredients of a finite-depth circuit single round measurement produce only Abelian states. Surprisingly, we show there exists broad family states-namely those with Lagrangian...

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, criticality. Remarkably, examples known where LRE is obtained performing single-site measurements on SRE, such as toric code from measuring a sublattice 2D cluster state. However, systematic understanding when how SRE give rise to still...

Abstract Quantum systems evolve in time one of two ways: through the Schrödinger equation or wavefunction collapse. So far, deterministic control quantum many-body lab has focused on former, due to probabilistic nature measurements. This imposes serious limitations: preparing long-range entangled states, for example, requires extensive circuit depth if restricted unitary dynamics. In this work, we use mid-circuit measurement and feed-forward implement non-unitary dynamics Quantinuum’s H1...

Finite-depth quantum circuits preserve the long-range entanglement structure in states and map between within a gapped phase. To of different phases, we can use sequential circuits, which apply unitary transformations to local patches, strips, or other subregions system way. The circuit, on one hand, preserves area law hence gappedness states. On circuit has generically linear depth, hence, it is capable changing correlation phase they belong to. In this paper, systematically discuss...

We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits gapped boundary. Our primary example is on four-dimensional qudits belongs to the double semion (DS) phase of matter. The DS Hamiltonian constructed by condensing an emergent boson in Z4 toric code, where condensation implemented at level ground states two-body measurements. rigorously verify identifying explicit finite-depth quantum circuit (with ancillary qubits) maps its ground-state...

(3+1)D topological phases of matter can host a broad class non-trivial defects codimension-1, 2, and 3, which the well-known point charges flux loops are special cases. The complete algebraic structure these defines higher category, be viewed as an emergent symmetry. This plays crucial role both in classification possible fault-tolerant logical operations quantum error correcting codes. In this paper, we study several examples such codimension from distinct perspectives. We mainly invertible...

We propose a new class of error-correcting dynamic codes in two and three dimensions that has no explicit connection to any parent subsystem code. The two-dimensional code, which we call the CSS (Calderbank-Shor-Steane) honeycomb is geometrically similar code by Hastings Haah also dynamically embeds an instantaneous toric However, unlike it possesses structure its gauge checks do not form Nevertheless, show our protocol conserves logical information threshold for error correction. generalize...

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends classification of systems composite-dimensional qudits establishes that is at least as rich theories. exemplify construction defined on four-dimensional based <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow...

It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth unitary operator U <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi></mml:math> . Here, we consider obtaining entangler local ‘pivot’ Hamiltonian H_\text{pivot} display="inline"><mml:msub><mml:mi>H</mml:mi><mml:mtext mathvariant="normal">pivot</mml:mtext></mml:msub></mml:math> such = e^{i\pi H_\text{pivot}}...

Floquet codes are a novel class of quantum error-correcting with dynamically generated logical qubits arising from periodic schedule noncommuting measurements. We utilize the interpretation measurements in terms condensation topological excitations and rewinding measurement sequences to engineer new examples codes. In particular, is advantageous for obtaining desired set instantaneous stabilizer groups on both toric planar layouts. Our first example code that have same order as...

We propose a new model of quantum computation comprised low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply gates. These constitute class error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. construct an explicit example, the DA color code, is assembled from short can realize all 72 automorphisms 2D code. On stack <mml:math...

We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, choice an Abelian normal subgroup $N$, foliation structure. These hybrid examples were introduced in previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, therefore give rise...

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schr\"odinger cat states, topological order, criticality. Remarkably, examples known where LRE is obtained performing single-site measurements on SRE, such as toric code from measuring a sublattice 2D cluster state. However, systematic understanding when how SRE give rise to still...

We construct a novel three-dimensional quantum cellular automaton (QCA) based on system with short-range entangled bulk and chiral semion boundary topological order. argue that either the QCA is nontrivial, i.e. not finite-depth circuit of local gates, or there exists two-dimensional commuting projector Hamiltonian realizing order (characterized by $U(1)_2$ Chern-Simons theory). Our obtained first constructing Walker-Wang certain premodular tensor category four, then condensing deconfined...

Progress in understanding symmetry-protected topological (SPT) phases has been greatly aided by our ability to construct lattice models realizing these states. In contrast, a systematic approach constructing that realize quantum critical points between SPT is lacking, particularly dimension d&amp;gt;1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . Here, we show how the...

Fracton order describes novel quantum phases of matter that host quasiparticles with restricted mobility and, thus, lies beyond the existing paradigm topological order. In particular, excitations cannot move without creating multiple are called fractons. Here, we address a fundamental open question—can notion self-exchange statistics be naturally defined for fractons, given their complete immobility as isolated excitations? Surprisingly, demonstrate how fractons can exchanged and show...

Despite growing interest in beyond-group symmetries quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>G</a:mi><a:mo>×</a:mo><a:mi>Rep</a:mi><a:mo...

The author generalizes the Jordan-Wigner Transformation to any translation-invariant fermion system, including cases where parity is conserved on individual subdimensional manifolds, such as planes or fractals. paper finds that in three dimensions, dual spin model an exotic gauge theory point-like excitations have fundamentally restricted mobility called fractons, yet simultaneously are emergent fermions.

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional topological orders and orders. Hybrid host (i) mobile quasiparticles loop excitations, as well (ii) pointlike excitations with restricted mobility, nontrivial fusion rules mutual braiding statistics between two sets excitations. Furthermore, can realize either or after undergoing a phase transition driven by condensation certain Therefore they serve parent for...

Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations promising building blocks fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its have remained elusive, unlike simpler or defects Abelian TO. In this work, we present first unambiguous realization demonstrate control anyons. Using an adaptive circuit on...

Elementary point charge excitations in three-plus-one-dimensional $(3+1\mathrm{D})$ topological phases can condense along a line and form descendant excitation called the Cheshire string. Unlike elementary flux loop system, strings do not have to appear as boundary of 2D disk exist on open segments. On other hand, are different from trivial that be created with local unitaries zero dimensions finite depth quantum circuits one dimension higher. In this paper, we show create string, needs...

Long-range entangled quantum states -- like cat and topological order are key for metrology information purposes, but they cannot be prepared by any scalable unitary process. Intriguingly, using measurements as an additional ingredient could circumvent such no-go theorems. However, efficient schemes known only a limited class of long-range states, their implementation on existing devices via sequence gates is hampered high overheads. Here we resolve these problems, proposing how to scalably...