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.
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.
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.
Two prevalent approaches for preparing long-range entangled quantum states are (i) linear-depth sequential unitary (SU) circuits, which apply local unitary gates sequentially, and (ii) constant-depth measurement-feedback (MF) circuits, which employ mid-circuit measurements and conditional feedback based on measurement outcomes. Here, we establish that a broad class of SU and MF circuits are dual to each other under a spacetime rotation. We investigate this spacetime duality in the preparation of various long-range entangled states, including GHZ states, topologically ordered states, and fractal symmetry-breaking states. As an illustration, applying a spacetime rotation to a linear-depth SU circuit that implements a non-invertible Kramers-Wannier duality, originally used to prepare a 1D GHZ state, yields a constant-depth MF circuit that implements a Z2 symmetry gauging map, which equivalently prepares the GHZ state. Leveraging this duality, we further propose experimental protocols that require only a constant number of qubits to measure unconventional properties of 1D many-body states. These include (i) measurement of disorder operators, which diagnose the absence of spontaneous symmetry breaking, and (ii) postselection-free detection of measurement-induced long-range order, which emerges in certain symmetry-protected topological phases. We also show that measurement-induced long-range order provides a lower bound for strange correlators, which may be of independent interest.
Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states. Published by the American Physical Society 2024
Quantum circuits with measurements and unitary feedback (MF) can prepare long-range entangled states in constant depth, but a systematic construction of the MF preparation circuit for a given target state remains underexplored. We develop such a scheme for one-dimensional states, based on the notion of pushable defects: virtual-bond operators of a matrix product state that can be pushed through the tensor at the price of a physical feedback unitary. We show that the set of pushable defects, together with their pushing relations classifies finite-depth MF-preparable states and dictates their preparation circuits. To each class of the target state ∣A⟩, we associate a state ∣B⟩ from which ∣A⟩ can be prepared using a 1-round MF circuit; in particular, ∣A⟩ is preparable from a product state using a circuit with 1 round of MF whenever ∣B⟩ is preparable by a finite-depth local unitary (FDLU) circuit. For a general target state, the scheme is obtained by iterating this procedure until the associated state is FDLU-preparable. For open-boundary matrix product states, the scheme is complete: it constructs a preparation circuit whenever finite-depth MF preparation with left-conditioned feedback corrections is possible. Pushable defects and pushing relations thus emerge as a unifying principle for quantum state preparation via measurements and feedback. This characterization further reveals an intrinsic connection between MF circuits and non-invertible symmetries: states with certain classes of pushing relations are related to a product state by Tambara-Yamagami duality operators, or by continuous cosine symmetry operators with fusion rules LαLα′=Lα+α′+Lα−α′, together with their generalizations up to (not necessarily transversal) gates.
Symmetry-protected topological (SPT) phases extend the Landau paradigm of quantum matter by admitting distinct symmetry-preserving phases that lack any local order parameter. Demonstrating these phases at scale on programmable quantum processors is a key milestone in using quantum hardware to probe emergent many-body phenomena, yet it is impeded by the circuit depth normally required to capture non-trivial entanglement. Here we use a tensor network based approximate quantum compiling (AQC) protocol to construct shallow quantum circuits (18-39 CNOT depth), which prepare 100-site ground states of the spin-1/2 bond-alternating Heisenberg chain in both SPT phases, to 97.9-99.0% fidelity. Upon executing the circuits on IBM quantum hardware, the resulting states exhibit all defining signatures of SPT order including non-local string order for strings of up to length 20, characteristic degeneracies in the entanglement spectrum and clear evidence of symmetry-protected edge modes. The simultaneous observation of these independent diagnostics establishes current quantum computers as versatile platforms for large-scale studies of symmetry-protected quantum matter. More broadly, our results establish a practical foundation for probing non-equilibrium quench dynamics of such systems in regimes that challenge classical computational methods.
Preparing long-range entangled states poses significant challenges for near-term quantum devices. It is known that measurement and feedback (MF) can aid this task by allowing the preparation of certain paradigmatic long-range entangled states with only constant circuit depth. Here, we systematically explore the structure of states that can be prepared using constant-depth local circuits and a single MF round. Using the framework of tensor networks, the preparability under MF translates to tensor symmetries. We detail the structure of matrix-product states (MPSs) and projected entangled-pair states (PEPSs) that can be prepared using MF, revealing the coexistence of Clifford-like properties and magic. In one dimension, we show that states with Abelian-symmetry-protected topological order are a restricted class of MF-preparable states. In two dimensions, we parametrize a subset of states with Abelian topological order that are MF preparable. Finally, we discuss the analogous implementation of operators via MF, providing a structural theorem that connects to the well-known Clifford teleportation. Published by the American Physical Society 2024