Scalable Bounds for Many-Body Properties Achieved with Finite Measurements and Semidefinite Programming

Determining the limits of many-body properties represents a fundamental challenge in modern physics, offering crucial insights into complex systems and complementing traditional estimation techniques. Luke Mortimer, Leonardo Zambrano, and Antonio Acín, from ICFO , The Barcelona Institute of Science and Technology, alongside Donato Farina from the University of Naples Federico II, demonstrate a scalable method for establishing probabilistic bounds on these properties using finite measurement statistics. Their research introduces a novel framework utilising moment-matrix relaxations, significantly improving upon existing semidefinite programming approaches in terms of computational efficiency for larger quantum systems. This advancement allows for the practical certification of quantum states, even when limited by unavoidable experimental noise, and opens avenues for deeper understanding of emergent phenomena in quantum mechanics. The team’s work establishes a robust and scalable real-world scheme for verifying quantum systems with incomplete information.

Their research introduces a novel framework utilising moment-matrix relaxations, significantly improving upon existing semidefinite programming approaches in terms of computational efficiency for larger quantum systems. This advancement allows for the practical certification of quantum states, even when limited by unavoidable experimental noise, and opens avenues for deeper understanding of emergent phenomena in quantum mechanics.

Calculating bounds on properties of many-body quantum systems is of paramount importance, as these bounds guide understanding of emergent quantum phenomena and complement insights obtained from estimation methods. Recent semidefinite programming approaches enable probabilistic bounds from finite-shot measurements of easily accessible, albeit informationally incomplete, observables. This work renders these methods scalable, focusing on developing and implementing techniques to determine rigorous bounds on physical quantities within complex quantum systems, moving beyond traditional computational limitations. Specifically, the approach utilises semidefinite programming to extract maximal information from limited measurement data, offering a pathway to characterise quantum states and dynamics even with incomplete information.

Certifying Quantum States via Semidefinite Programming Relaxations

The research details a method for certifying quantum states using semidefinite programming relaxations, focusing on the use of moment matrices and reduced density matrices. Certification of quantum states is achieved through techniques like shadow tomography with n-representability conditions, which can reduce the number of measurements needed. Moment matrices are used in relaxations of state positivity, where positivity of the moment matrix serves as a relaxation of state positivity. Reduced density matrices can also be used for relaxations, particularly the reduced density matrix for the first two qubits. These methods are applied to various problems, including predicting properties from few measurements, mapping phase diagrams of quantum spin systems, and certifying steady-state properties of open quantum systems.

Moment-Matrix Bounds for Quantum System Properties

Scientists achieved a scalable method for calculating bounds on properties of quantum systems, utilising moment-matrix relaxations instead of traditional semidefinite programming. The research introduces a framework leveraging experimental estimations and semidefinite programming relaxations to define a certification scheme applicable to an increasing number of qubits. This work addresses the challenge of determining bounds on system properties from finite-shot measurements of incomplete observables, offering a pathway to understanding emergent phenomena. The team’s approach adapts to specific system knowledge, such as ground state or steady-state characteristics, to refine the bounds obtained.

Experiments revealed the ability to bound the steady-state heat current in an open quantum system and obtain a lower bound on its ground-state energy, demonstrating the method’s practical application. Researchers successfully derived lower bounds on subsystem purity for a full system in its ground state, showcasing applicability to non-linear functions. The study employed a protocol focusing on n-qubit systems, selecting moment variables associated with multi-qubit Pauli strings. Measurements confirm the framework combines information from measurement data, in the form of bounds on expectation values, with fundamental system guarantees via a semidefinite programming relaxation. The breakthrough delivers a scalable certification tool, tested on a 50-qubit Majumdar, Ghosh model, highlighting its potential for larger systems.

Rigorous Quantum State Bounds via Semidefinite Programming

This work introduces a scalable method for calculating rigorous bounds on properties of quantum states, moving beyond estimation techniques to provide guaranteed intervals within a defined confidence level. By utilising moment-matrix relaxations within a semidefinite programming framework, the researchers have developed an approach that scales polynomially with system size, a significant improvement over previous methods. This allows for the certification of bounds based on finite-shot measurements of observables, combined with prior knowledge of the system’s characteristics, such as its Hamiltonian, symmetries, or steady-state properties. The demonstrated formalism offers a versatile tool for analysing many-body quantum systems, successfully bounding quantities like steady-state heat current, ground-state energy, and subsystem purity. While acknowledging that the tightness of these bounds cannot be universally guaranteed, the numerical results confirm their informative nature and broad applicability, even extending to non-linear functions like purity, although only lower bounds can currently be derived for such cases. Future research could focus on refining the method to improve bound tightness and exploring its application to a wider range of quantum systems and observables, complementing existing estimation-based approaches.