Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix-product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases—for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.
Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix-product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases—for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.
Although tensor network states constitute a broad range of exotic quantum states, their realization is challenging and often requires resources whose depth scales with system size. In this work, we explore criteria on the local tensors for enabling deterministic state preparation via a single round of measurements and on-site unitary feedback. We use these criteria to construct families of measurement-preparable states in one and two dimensions, tuning between distinct symmetry-breaking, symmetry-protected, and intrinsic topological phases of matter. For instance, in one dimension we chart out a three-parameter family of preparable states which interpolate between the AKLT, cluster, GHZ and other states of interest. Our protocol even allows one to engineer preparable quantum states with a range of desired correlation lengths and entanglement properties. In addition to such constructive approaches, we present diagnostics for verifying whether a given tensor network state is preparable using measurements. We conclude by charting out generalizations, such as considering multiple rounds of measurements, implementing matrix product operators, and using incomplete basis measurements.
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.
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
In the era of noisy, intermediate-scale quantum (NISQ) devices, the efficient preparation of many-body resource states is a task of paramount importance. In this paper we focus on the deterministic preparation of matrix-product states (MPS) in constant depth by utilizing measurements and classical communication to fuse smaller states into larger ones. We place strong constraints on the MPS that can be prepared using this method, which we refer to as MPS fusion. Namely, we establish that it is necessary for the MPS to have a flat entanglement spectrum. Using the recently introduced split-index MPS (SIMPS) representation, we then introduce a family of states that belong to interesting phases of matter protected by non-onsite symmetries, including anomalous and non-invertible symmetries, and also serve as resources for long-range quantum teleportation, but which lie beyond the scope of ordinary MPS fusion. It is shown constructively that these states can be prepared in constant depth using a broader class of measurement-assisted protocols, which we dub SIMPS fusion. Even in cases when MPS fusion is possible, using SIMPS fusion can give rise to significantly reduced resource overhead. We also discuss constraints on SIMPS fusion and propose a general framework for fusion that encompasses the MPS and SIMPS protocols. Our results therefore simultaneously establish the boundaries of conventional MPS fusion and push the envelope of which states can be prepared using measurement-assisted protocols.
While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.
We give a classification of gapped quantum phases of one-dimensional systems in the framework of matrix product states (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states and in both the absence and the presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground-state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, that is, the equivalence classes of its projective representations, a result first derived by Chen, Gu, and Wen [Phys. Rev. B 83, 035107 (2011)]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labeled by both the permutation action of the former and the cohomology class of the latter. Using projected entangled pair states (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular, systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS, which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus we can focus on to the classification of isometric forms.