SBU-led Paper in Nature Explores the Stability of Quantum Systems

Stability against perturbations is a defining property of quantum many-body phases of matter. However, most stabilities are only established for static perturbations; the question of whether any system can remain stable against generic time-dependent perturbations has remained elusive. Specifically, scientists want to know whether certain quantum systems can stay stable even when they are constantly being disturbed or changed over time.

In a recent paper published in Nature Communications, "Universal energy-space localization and stable quantum phases against time-dependent perturbations," Stony Brook researchers identify a universal phenomenon, where the evolving state can be exponentially localized in an energy window of instantaneous spectrum, and prove its survival under generic time-dependent perturbations. Instantaneous spectrum is a time-varying representation that shows the frequency or energy content and distribution of a non-stationary signal at any precise moment in time.

The paper was co-authored by Tzu-Chieh Wei, a professor specializing in quantum information science at the C.N. Yang Institute for Theoretical Physics at Stony Brook University, and Hongye Yu, who recently completed his PhD degree from Stony Brook's Department of Physics and Astronomy.

Researchers can usually prove that a system is stable when the disturbance is fixed and doesn't change. But it's much harder to prove stability when the disturbance changes with time. This research shows that some quantum systems can remain stable under changing disturbances.

"Applying such energy-space localization to classical and quantum LDPC codes whose codewords (original state) are separated by extensive energy barriers, we show that the system remains localized near the original codeword for an exponentially long time under generic time-dependent perturbations," said Wei. "For classical optimization problems with clustered solution spaces, the stability becomes an obstacle for quantum Hamiltonian-based algorithms to escape local minima. Our work provides a new lens for analyzing quantum non-equilibrium dynamics and tools for establishing stability and designing quantum algorithms."

Quantum codewords are specific multi-qubit quantum states inside a protected code subspace that store logical information safely against noise and decoherence. The paper shows that certain quantum systems can resist even constantly changing disturbances and remain near their original state for an incredibly long time. This is a very useful property for error detection and correction, crucial for the survival of quantum computers against noise and errors. However, that same stability could make some quantum optimization algorithms get stuck.

"Our result is a double-edged sword," said Wei. The researchers found that a quantum system can become localized in energy, meaning its state stays within a relatively narrow range of possible energies. This localization can survive even when the system is subjected to general disturbances that change over time. For certain types of error-correcting codes, called low-density parity-check (LDPC) codes, different codewords are separated by large energy barriers. Because of these barriers, the system can stay close to its codeword for an extremely long time.

To change states and move towards optimal solutions, quantum algorithms must overcome these huge energy barriers. But even with disturbances that change over time, the system has a hard time crossing the barrier. However, this stability can actually be bad for some quantum algorithms. Some optimization problems have many groups of possible solutions, with poor solutions trapped by the barriers. A quantum algorithm might need to move from one codeword to another to find a better solution. But if the system is too stable, it can get stuck near its starting point and have difficulty escaping.

"In this work, we have introduced and proved the energy-space localization as a universal property of time-dependent q-local Hamiltonians," said Wei. Determining whether the lowest energy eigenvalue (the ground state energy) of a q-local Hamiltonian is low or high is a fundamental problem in quantum complexity.

"Our theorems, serving as both fundamental theoretical results and versatile mathematical tools, have enabled us to demonstrate wide applications in proving the stability of systems with extensive energy barriers, such as in certain LDPC codes and hard optimization problems," added Wei.

The energy-space localization allows researchers to study generic quantum evolution in the lens of energy changes.

"Using energy-space localization, we have shown that systems with extensive energy barriers exhibit extremely stable phases, even under time-dependent perturbations," said Wei.

Among the various stabilities demonstrated for LDPC codes, the exponentially long dynamical localization against generic time-dependent perturbations stands out for its theoretical novelty and practical implications.

"This result not only establishes proof of stability in a previously considered intractable time-dependent setting but also provides a theoretical foundation on how error-correcting codes protect information against realistic time-varying noise," said Wei. "We have also extended other previously known stabilities of LDPC codes to quasi-q-local perturbations. For hard optimization problems, we proved that the evolving state under an algorithmic driving Hamiltonian becomes trapped inside local minima for an exponentially long time if the total variation of the driving is not sufficiently large. In summary, our theoretical framework rigorously connects the intrinsic property of quantum evolution with the stability of error-correcting codes and the algorithmic hardness of optimization problems from the perspective of the energy space."

- Robert Emproto