Weihua Xie of The University of Texas at Dallas and colleagues demonstrate that even infinitesimally small amounts of noise can disrupt the persistent temporal oscillations expected in quantum time crystals realized with non-Hermitian dynamics. The researchers used a generalized Floquet analysis to show infinitely long-lived oscillations are generically lost with “arbitrarily weak values of common types of noise,” calculating a corresponding damping rate. Simulations using IBM’s Qiskit platform confirmed these theoretical predictions; however, experiments performed on the ibmq-lima device failed to detect any trace of the predicted oscillations, revealing a discrepancy between modeled and real-world performance.
Non-Hermitian Ising Model Drives Quantum Time Crystallinity
A two-qubit system, under ideal conditions, can theoretically maintain infinitely long-lived periodic steady states when governed by non-Hermitian dynamics, a surprising result suggesting inherent stability within these systems. This model maps the partition function of a classical 2D Ising model to a one-dimensional quantum circuit using non-unitary gates generated by non-Hermitian transverse-field Ising terms, a technique similar to that used in studies revealing steady phases with robust edge modes and spatiotemporal long-range order.
The team’s initial goal was to understand the noise tolerance of this few-qubit version of the non-Hermitian time crystal, an important step toward realizing such systems in practical quantum computing. The study’s approach involved constructing a quantum circuit to simulate the time crystal, using post-selection to approximate the non-Hermitian terms that are difficult to implement directly with unitary gates.
This method couples the system to an ancilla qubit, followed by measurement and post-selection, mirroring techniques employed in recent experiments realizing non-Hermitian dynamics and stochastic Schrödinger equation simulations. However, the researchers quickly encountered a challenge: accurately modeling the impact of noise on the system, a factor critical to bridging the gap between theoretical predictions and experimental results. The analysis computed a corresponding damping rate, quantifying the speed at which these oscillations decay under noisy conditions.
The team investigated this discrepancy by modeling the ibmq-lima device as an open Floquet system, incorporating realistic noise channels into the simulation. They considered depolarizing errors, which occur during gate application, and parameterized the noise to match the observed data.
While the open system model improved the agreement between simulation and experiment, the researchers found that the parameters provided by Qiskit did not fully capture the characteristics of the actual quantum device. “We observe that the expected oscillations are absent on an actual device,” the researchers noted, highlighting the difficulty of accurately simulating the complex noise environment of a real quantum computer.
The researchers’ work demonstrates a clear link between noise and the stability of quantum time crystals, revealing that even small amounts of disturbance can destroy the persistent oscillations that define this exotic phase of matter. The study’s focus on a two-qubit system provides a simplified yet insightful model for understanding the fundamental limits of non-Hermitian dynamics in noisy environments. The team’s exploration of noise channels and their impact on the Floquet operator provides a framework for assessing the feasibility of realizing time crystals on near-term quantum devices.
The results suggest that significant advances in noise reduction and error correction will be necessary to observe and control these delicate quantum phenomena in practice. The team concludes that further research is needed to extend these findings to larger, many-body systems and to explore alternative approaches for realizing non-Hermitian dynamics on quantum computers.
Mapping Non-Hermitian Dynamics to a Quantum Circuit
The team’s investigation focused on understanding noise tolerance in a few-qubit version of a non-Hermitian time crystal, a system exhibiting persistent temporal oscillations, a form of time crystallinity, and the challenges of implementing such dynamics on real-world hardware. The study highlights a fundamental distinction between unitary and non-unitary quantum dynamics regarding their susceptibility to noise; while late-time unitary dynamics are generally destroyed by environmental interactions, non-unitary dynamics, like those found in open quantum systems, can exhibit greater robustness.
Non-Hermitian Floquet systems occupy an intermediate ground, and their response to noise remains an open question. This circuit structure allowed for the exploration of how realistic noise impacts the post-selected system, an important step in assessing the feasibility of building such time crystals on existing quantum computers.
Initial simulations using IBM’s Qiskit platform showed little difference between the noiseless case and models incorporating specific noise parameters, a result that prompted further investigation into potential miscalibration of the ibmq-lima device or inaccuracies in the noise modeling. The team acknowledges that fully resolving this discrepancy requires additional work, but their primary focus remained on understanding the role of non-Hermitian dynamics and post-selection in the context of noisy quantum computation.
This focus led to a key clarification regarding the relationship between degenerate subspaces in non-Hermitian dynamics and decoherence-free subspaces in open quantum systems, both of which exhibit pure state dynamics within the steady state subspace but differ in their behavior outside of it. “Both of these cases involve pure state dynamics in the steady state subspace, but behave differently outside of this subspace – whether or not they maintain purity of pure states, whether or not they are trace preserving, etc,” the authors note, emphasizing the detailed interplay between these concepts. The work demonstrates that despite these differences, the superoperator spectra within the degenerate steady state manifold remain identical, offering a deeper understanding of the underlying principles governing these systems.
IBM Qiskit Simulations Reveal Noise-Induced Damping
Simulations performed using IBM’s Qiskit platform revealed a discrepancy between theoretical predictions and experimental results concerning the damping of oscillations in a non-Hermitian Ising Floquet model, a system designed to exhibit characteristics of a time crystal. The team’s approach centered on realizing non-Hermitian dynamics through a quantum circuit incorporating measurement and post-selection, a technique mirroring recent experimental efforts in the field. This method effectively simulates non-Hermitian evolution by coupling the system to an ancillary qubit, measuring its state and then selectively retaining only specific outcomes.
The model, a non-Hermitian Floquet version of the transverse-field Ising model, relies on alternating applications of bond and field terms, each with both real and imaginary components, defining the system’s evolution. Parameters were set to a specific slice through the complex plane, with the field term becoming unitary at R = 1 and h approaching -iπ/6, a configuration predicted to support the time crystal phase.
Analysis of the system’s behavior involved calculating two-time correlation functions, specifically the correlation of the z-magnetization on a single qubit, to identify the presence and decay of the predicted oscillations. The theoretical framework predicts power-law decaying correlations in the thermodynamic limit, indicative of quasi-long-range order in time, a hallmark of the time crystal phase. However, experiments on a real device (ibmq-lima) do not show remnants of these oscillations.
Noise Sensitivity of Non-Unitary Floquet Systems
The theoretical longevity of time crystalline phases in non-Hermitian Floquet systems faces a sharp limit imposed by even minimal environmental noise. Generalized Floquet analysis reveals that infinitely long-lived oscillations, predicted under ideal conditions, are generically destroyed by arbitrarily weak disturbances, establishing a surprisingly low threshold for disruption. However, experiments conducted on the ibmq-lima device yielded a stark contrast. The process involves discarding unwanted measurement outcomes, effectively projecting the system onto a specific subspace.
The superoperator describing the combined effect of unitary gates, noise, and post-selection is important for understanding the system’s behavior. For quantum gates, this includes contributions from thermal and depolarizing noise, while measurement and post-selection are represented by a projector onto the post-selected subspace.
The data from ibmq-lima, presented in accompanying figures, visually demonstrates the absence of oscillations and the deviation from expected behavior under various noise models. The researchers further explored the impact of different noise parameters, including reset errors and dephasing, on the system’s dynamics. They found that even small values of these parameters can significantly reduce the post-selection probability and suppress the oscillations.
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