New Technique Swiftly Predicts Stable States of Complex Quantum Systems

Scientists are increasingly focused on understanding the steady states of complex quantum many-body systems, and Miguel Frías Pérez and Antonio Acín, both from ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, have now developed a novel method for certifying these states locally. Their research introduces a relaxation-based approach to effectively bound expectation values within the steady state of dissipative systems governed by Lindblad master equations. Rather than attempting to fully characterise the entire quantum state, the authors cleverly focus on reduced density matrices and enforce consistency with a global steady state, formulating the problem as a semidefinite program. This allows for efficient and rapid convergence of bounds on observable values, offering highly competitive predictions for the steady state behaviour of both one- and two-dimensional models with any number of particles, and representing a significant advance in our ability to verify quantum simulations and characterise complex quantum phenomena.

This work circumvents the need to fully represent the quantum state by instead focusing on reduced density matrices as primary variables, enforcing constraints consistent with a global steady state.

The resulting mathematical framework takes the form of a semidefinite program, enabling efficient computation of bounds on expectation values for a given system. Results demonstrate rapid convergence of these bounds as the size of the reduced density matrices increases, yielding highly competitive predictions for the steady state behaviour of both one- and two-dimensional models with an arbitrary number of particles.
This research addresses a fundamental challenge in quantum mechanics: describing the behaviour of interacting many-body systems, particularly those coupled to an environment. The study introduces a technique that avoids the exponential growth in computational complexity typically associated with representing the full density matrix of such systems.

By promoting reduced density matrices to central variables, the method establishes constraints derived from the requirement of a consistent global steady state. These constraints are then formulated as a semidefinite program, a type of optimisation problem well-suited for efficient computation. The core innovation lies in the ability to obtain rigorous bounds on expectation values without requiring a complete description of the quantum state.

Benchmarking against several one- and two-dimensional models reveals that the accuracy of these bounds converges quickly with increasing size of the reduced density matrices. This fast convergence signifies a substantial improvement in predictive power for simulating the steady state of dissipative quantum systems.

The technique is applicable to systems of arbitrary dimension and particle number, offering a versatile tool for analysing complex quantum phenomena. Specifically, the method applies to systems governed by a Lindblad master equation, a standard framework for describing open quantum systems. The research focuses on translation-invariant systems, where the reduced density matrix of a subsystem remains consistent across all contiguous particles.

By combining positivity, normalisation, and translation invariance constraints with the steady-state condition, scientists derive a set of equations that can be efficiently solved using semidefinite programming. This approach provides a powerful means of certifying and benchmarking the performance of quantum devices operating under realistic conditions with unavoidable decoherence and dissipation.

Constraining expectation values via steady-state semidefinite programming of reduced density matrices

A relaxation-based method forms the core of this work, bounding expectation values within the steady state of dissipative many-body quantum systems. Rather than attempting to fully represent the quantum state, the research promotes reduced density matrices to primary variables, subsequently enforcing constraints derived from global steady-state consistency.

These resulting constraints are formulated as a semidefinite program, enabling efficient computation of bounds on expectation values. The study leverages the Lindblad form of master equations to describe the open quantum systems under investigation. Reduced density matrices, representing subsystems of the larger system, are central to the methodology, allowing the researchers to focus on relevant correlations without tracking the entire state.

By enforcing consistency with the global steady state, the work establishes a set of constraints that define the permissible range of expectation values. Implementation of the semidefinite program relies on numerical solvers designed to handle the resulting optimisation problem. The bounds obtained through this process demonstrate fast convergence as the size of the reduced density matrices increases, indicating the method’s efficiency in approximating the steady state.

Performance was benchmarked against several one- and two-dimensional models with an arbitrary number of particles, validating the approach across diverse physical scenarios. This technique offers rigorous results, providing guaranteed bounds on expectation values, unlike heuristic methods commonly employed in many-body quantum system analysis.

Convergence of semidefinite programming bounds on steady-state expectation values

Relaxation-based methods are presented to bound expectation values on the steady state of dissipative many-body quantum systems described by master equations of the Lindblad form. The research promotes reduced density matrices as variables and enforces constraints consistent with a global steady state, formulating these constraints as a semidefinite program.

Results demonstrate fast convergence of the bounds with the size of the reduced density matrices, yielding competitive predictions for the steady state of both one- and two-dimensional models with an arbitrary number of particles. The study focuses on systems governed by a Markovian equation of the Lindblad form, where the dynamics are described by a Lindbladian superoperator.

Steady states, fixed points of the evolution, satisfy the condition L(ρs) = 0, representing positive operators within the kernel of the Lindbladian. Instead of directly solving for the full density matrix, the work promotes reduced density matrices to primary variables, circumventing the exponential complexity associated with many-body systems.

This approach allows for the efficient bounding of local expectation values in the steady state, applicable to systems of arbitrary dimension and number of constituents. Benchmarking with several models in one- and two-dimensional systems reveals fast convergence of the bounds as the size of the reduced density matrices increases.

The technique provides a means to characterise open many-body quantum systems classically, offering potential benefits for certifying and benchmarking quantum devices under realistic conditions. Furthermore, the method contributes to understanding the boundary between quantum and classical computational power and may aid in the development of engineered dissipative processes for quantum computation.

Semidefinite programming constrains steady-state expectation values via reduced density matrices

Researchers have developed a relaxation-based method for determining bounds on expectation values within the steady state of dissipative quantum systems. This approach diverges from traditional methods by focusing on reduced density matrices, representing local subsystems, rather than attempting to characterise the complete many-body state.

By promoting these reduced density matrices as variables and applying constraints consistent with a global steady state, the problem is transformed into a semidefinite program, enabling efficient computation of bounds on observable values. The resulting bounds demonstrate rapid convergence as the size of the reduced density matrices increases, yielding highly competitive predictions for the steady state of both one- and two-dimensional models with an arbitrary number of particles.

Specifically, for certain single-qubit Pauli operators, the absolute values of the bounds obtained were below 0.018 for σy and 0.013 for σz, demonstrating a high degree of precision. The algorithm’s rigorous nature allows it to serve as a benchmark for evaluating other classical and quantum techniques designed to solve for steady states.

The authors acknowledge a limitation in the current implementation regarding the size of clusters that can be simulated, which restricts the complexity of the systems that can be fully analysed. Future research directions include investigating the algorithm’s performance near phase transitions and exploring the relationship between bound convergence and the spectral properties of the Lindbladian. Combining this approach with other relaxation techniques may also offer further improvements in accuracy and enable the study of dynamical relaxation processes towards the steady state, potentially broadening the scope of applicable quantum systems.