Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
AI Summary
The paper asks whether entanglement generated by known many-body Hamiltonian dynamics can be certified using fewer Pauli measurements than tomography, without accurately tracking the full evolving state. It constructs witnesses $W=\beta I-\sum_i\tilde S_i(t)$ by approximately evolving the initial state’s generalized stabilizers. Evolution uses segmented second-order Taylor expansions, with small Pauli coefficients discarded after each time interval. A semidefinite program maximizes the retained observable over PPT states to determine a valid witness threshold $\beta$.
Numerical tests cover power-law and nearest-neighbor XY models, random-coupling Heisenberg chains, an XY chain with an $X$ field, and the isotropic Heisenberg chain, using 6–12 qubits. They address bipartite entanglement and genuine tripartite entanglement between three spatial blocks. With heuristically chosen truncation thresholds and imposed polynomial witness-size budgets, examples at 11–12 qubits retain roughly 200–11,000 Pauli terms. Maximum detection times are model-dependent: approximately $1$–$1.5$ inverse-coupling units for several XY examples and $0.3$–$0.5$ for random or isotropic chains. A special nearest-neighbor XY case admits evolution within a closed Pauli-operator span, illustrated by 72 strings for six qubits.
The principal limitation is progressive spreading of Pauli coefficients: fixed-threshold truncation eventually removes information needed for detection; retaining more terms extends detection but can approach the full Pauli basis. Resource scaling is assessed on small systems rather than established asymptotically. The implemented search also uses exact numerical target-state evolution and a full-state-space PPT optimization. Experimental noise tolerance and finite-shot measurement costs are not quantified.
Some signs of AI-generated textOverclaimsSmall numerics onlyMissing comparisonStrong assumptions
Central claim may not hold: The witness-design algorithm retains exponential-size ideal-state calculations and full-system PPT optimization, so symbolic stabilizer propagation alone does not establish computational scalability.
Polynomial measurement size is imposed heuristically and tested only on small systems, without establishing that effective detection survives that constraint at larger sizes.
The claimed detection-time advantage over naive approaches is not quantitatively benchmarked, and several reported models exhibit appreciably decreasing detection times with system size.
Appendix A uses the wrong sign for the quadratic Taylor term, contradicting the main text and introducing a second-order error in the printed observable-update prescription.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
The paper asks whether entanglement generated by known many-body Hamiltonian dynamics can be certified using fewer Pauli measurements than tomography, without accurately tracking the full evolving state. It constructs witnesses W=βI−∑iS~i(t) by approximately evolving the initial state’s generalized stabilizers. Evolution uses segmented second-order Taylor expansions, with small Pauli coefficients discarded after each time interval. A semidefinite program maximizes the retained observable over PPT states to determine a valid witness threshold β.
Numerical tests cover power-law and nearest-neighbor XY models, random-coupling Heisenberg chains, an XY chain with an X field, and the isotropic Heisenberg chain, using 6–12 qubits. They address bipartite entanglement and genuine tripartite entanglement between three spatial blocks. With heuristically chosen truncation thresholds and imposed polynomial witness-size budgets, examples at 11–12 qubits retain roughly 200–11,000 Pauli terms. Maximum detection times are model-dependent: approximately 1–1.5 inverse-coupling units for several XY examples and 0.3–0.5 for random or isotropic chains. A special nearest-neighbor XY case admits evolution within a closed Pauli-operator span, illustrated by 72 strings for six qubits.
The principal limitation is progressive spreading of Pauli coefficients: fixed-threshold truncation eventually removes information needed for detection; retaining more terms extends detection but can approach the full Pauli basis. Resource scaling is assessed on small systems rather than established asymptotically. The implemented search also uses exact numerical target-state evolution and a full-state-space PPT optimization. Experimental noise tolerance and finite-shot measurement costs are not quantified.
The generation and verification of large-scale entanglement are essential to the development of quantum technologies. In this paper, we present an efficient scheme to generate genuine multipartite entanglement of a large number of qubits by using the Heisenberg interaction. This method can be conveniently implemented in various physical platforms, including superconducting, trapped-ion, and cold-atom systems. In order to characterize the entanglement of the output quantum state, we generalize the stabilizer formalism and develop an entanglement witness method. In particular, we design a generic searching algorithm to optimize entanglement witness with a minimal number of measurement settings under a given noise level. From the perspective of practical applications, we numerically study the trade-off between the experiment efficiency and the detection robustness.
Recent advances in the field of quantum computing hardware motivate the search for applications which demonstrate so-called quantum advantage. One promising use case that has been identified is the simulation of quantum many-body systems. The computational resources required for performing such a simulation on a classical computer generally grow exponentially with the size of the system being modeled, which is ultimately due to the exponential growth of the Hilbert space in which the dynamics of such a system take place. While digital quantum computers natively evolve quantum states directly in such a Hilbert space, thus naively avoiding this problem, the result of such a computation is typically not obtained as a deterministic output. Rather, it requires projective measurements which are fundamentally affected by shot noise. Any desired expectation values must therefore be reconstructed from repeated measurements, making the number of those measurements a relevant computational resource, and thus an important consideration for any potential claims of quantum advantage. In this work, we study how this resource scales with system size (a scaling which itself depends on the desired accuracy of the final result), using the simulation of nuclear magnetic resonance (NMR) spectra as a test case. We study this scaling for both real-world molecules, as well as a class of model NMR Hamiltonians which allow for efficient large-scale simulations with one-dimensional, two-dimensional, and all-to-all interactions. We find that the required resources increase only weakly with molecular size, far below the exponential growth of the underlying Hilbert space. This result suggests that shot noise should not pose a fundamental barrier to achieving quantum advantage in the simulation of NMR systems, and perhaps for many-body systems more broadly.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Quantum simulators enable the exploration of complex quantum phenomena in condensed-matter systems by reproducing their dynamics on controllable quantum devices. However, experimental constraints often restrict the class of Hamiltonians that can be realized natively. Hamiltonian engineering addresses this limitation by expanding the set of accessible Hamiltonians beyond the fixed system Hamiltonian provided by the hardware. We introduce a framework for fermionic Hamiltonian engineering based on interleaving free evolution under the system Hamiltonian with a sequence of experimentally feasible local fermionic unitaries. The required sequences and free-evolution times are obtained efficiently through a linear program, enabling effective time evolution under a broad class of target Hamiltonians. In particular, our method realizes arbitrary complex tunnelling coefficients, constrained only by the connectivity of the underlying system Hamiltonian. We further show that symmetries of the Hamiltonian engineering task can be exploited to drastically reduce the cost of computing the control parameters, and that errors arising from finite pulse times can be compensated within the same framework through a simple modification of the linear program. To our knowledge, this is the first automated method to realize locally tunable artificial gauge fields in interacting fermionic lattice models using only site-resolved energy shifts as control. We demonstrate the method by engineering the dynamics of interacting Fermi-Hubbard chains with complex tunnelling coefficients and of the Harper-Hofstadter model on a 1088-mode lattice. For the latter, we exploit the translation symmetry of the model to determine a provably optimal six-pulse sequence that produces the desired dynamics independent of the lattice size.
Entanglement is a central concept in quantum information and a key resource for many quantum protocols. In this work we propose and analyze a class of entanglement witnesses that detect the presence of entanglement in subsystems of experimental multiqubit stabilizer states. The witnesses we propose can be decomposed into sums of Pauli operators and can be efficiently evaluated by either two measurement settings only or at most a number of measurements that only depends on the size of the subsystem of interest. We provide two constructive methods to design the local witness operators, the first one based on the local unitary equivalence between graph and stabilizer states, and the second one based on sufficient and necessary conditions that the respective set of constituent Pauli operators needs to fulfill. We theoretically establish the noise tolerance of the proposed witnesses and benchmark their practical performance by analyzing the local entanglement structure of an experimental seven-qubit quantum error correction code.
Quantum observables in the form of few-point correlators are the key to characterizing the dynamics of quantum many-body systems. In dynamics with fast entanglement generation, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. In experimental systems, repeated time-reversal protocols have been successfully implemented to restore sensitivities of quantum observables. Using a 103-qubit superconducting quantum processor, we characterize ergodic dynamics using the second-order out-of-time-order correlators, OTOC(2). In contrast to dynamics without time reversal, OTOC(2) are observed to remain sensitive to the underlying dynamics at long time scales. Furthermore, by inserting Pauli operators during quantum evolution and randomizing the phases of Pauli strings in the Heisenberg picture, we observe substantial changes in OTOC(2) values. This indicates that OTOC(2) is dominated by constructive interference between Pauli strings that form large loops in configuration space. The observed interference mechanism endows OTOC(2) with a high degree of classical simulation complexity, which culminates in a set of large-scale OTOC(2) measurements exceeding the simulation capacity of known classical algorithms. Further supported by an example of Hamiltonian learning through OTOC(2), our results indicate a viable path to practical quantum advantage.