Jacek Dziarmaga of Jagiellonian University is simulating quantum annealing on a three-dimensional cubic lattice, moving from a paramagnetic to a spin-glass phase using a tensor network approach. The work tests the Kibble-Zurek power law, a mechanism originally proposed to describe cosmological phase transitions, by evaluating residual energy across a range of annealing times.

Researchers introduce a new Monte Carlo sampling approach for finite lattices, complementing a deterministic method used for infinite lattices, as the residual energy consistently approaches the KZ power law with increased annealing time. This simulation offers new opportunities to explore the quantum version of the Kibble-Zurek mechanism in higher dimensions.

Monte Carlo Sampling Simulates 3D Quantum Annealing

A new Monte Carlo sampling approach accelerates the simulation of quantum annealing on three-dimensional lattices, offering a significant efficiency gain over deterministic methods when evaluating energy in finite systems. This advancement addresses a key bottleneck in tensor-network simulations, allowing for the exploration of longer annealing times for testing the Kibble-Zurek mechanism. The technique was applied to a cubic lattice modeling the random Ising model, a common framework for studying quantum annealing as implemented in systems like the D-Wave Advantage computer.

The computational challenge stems from the need to accurately assess expectation values within the three-dimensional tensor network representing the quantum system; researchers adapted the Monte Carlo Metropolis-Hastings algorithm to achieve this. Unlike previous methods, this approach prioritizes speed by employing a simplified proposal function, accepting a slight trade-off in accuracy for a substantial reduction in computational overhead.

Even with this simplification, the resulting autocorrelation times remained short, indicating the efficiency of the sampling process. This efficiency is particularly important for extending simulations to longer annealing ramps, a requirement for verifying theoretical predictions. Results demonstrate that the residual energy, as measured through Monte Carlo sampling, increasingly aligns with the KZ power law as annealing time increases.

Deterministic methods, while accurate, become increasingly time-consuming as the system size and annealing time grow due to the long-range correlations that develop. The Monte Carlo approach circumvents this limitation, allowing researchers to extend their simulations into the KZ scaling regime, where the power law behavior is expected to be most pronounced.

For example, evaluating data at 5.12 nanoseconds proved particularly demanding for the deterministic method, while the Monte Carlo data readily extended into this region. This capability is critical for accurately characterizing the behavior of quantum annealers and validating theoretical models.

Projected Entangled-Pair States Model Random Ising Systems

Simulating quantum annealing in three dimensions presents a significant computational challenge, and researchers are now using Monte Carlo sampling to overcome limitations inherent in traditional deterministic methods. The work focuses on the random Ising model, a common testbed for these systems, and uses a three-dimensional tensor network representation, extending beyond the more frequently employed two-dimensional projected entangled-pair states (PEPS). The team at Jagiellonian University developed a method to address the computational bottleneck previously identified in evaluating expectation values within three-dimensional tensor networks.

Their solution involves a dual approach: a deterministic method for infinite lattices possessing periodic boundaries, and a newly introduced Monte Carlo sampling technique tailored for finite lattices with open boundaries. This flexibility is important, as real-world quantum annealers are necessarily finite in size, demanding accurate modeling of boundary effects.

The Monte Carlo scheme allows for a more efficient estimation of system behavior than previously possible with solely deterministic methods, particularly as system size increases. The KZ mechanism predicts how systems transition through phase transitions, and its application to quantum annealing offers a means of predicting the final energy state after the annealing process. The Kibble-Zurek mechanism, originally proposed to describe the formation of topological defects during cosmological phase transitions, was later adapted to continuous phase transitions in condensed matter systems.

The work builds upon previous research that identified expectation value estimation as the primary obstacle in three-dimensional tensor network simulations. In this study, the team demonstrated that these values can be computed using either deterministic methods or, more efficiently, via the newly developed 3D Monte Carlo sampling approach. The methodology employed involved driving the Hamiltonian across a quantum phase transition, moving from a paramagnetic phase to a spin-glass phase.

This transition is simulated using a tensor product state, a representation known as a projected entangled-pair state in two dimensions but extended to three dimensions for this research. The team used a neighborhood tensor update (NTU) technique, previously applied to simulations of many-body localization, Kibble-Zurek ramps and the preparation of ground states using shallow quantum circuits. The NTU method was also used in simulations of thermal states obtained through imaginary-time evolution in the fermionic Hubbard model.

The researchers constructed upper and lower boundary states for the PEPS network, a process that previously presented a computational bottleneck. By carefully controlling the bond dimension of these boundary states, they were able to significantly reduce computational demands. These boundary states were then used to evaluate the probability amplitude by sandwiching a layer of the projected PEPS between them, allowing for a more efficient calculation of system properties.

Kibble-Zurek Mechanism Tested with Annealing Ramps

The efficiency of evaluating observable quantities in three-dimensional tensor networks has been significantly improved through the implementation of a Monte Carlo sampling approach, allowing for more extensive simulations of quantum annealing. Researchers found that both deterministic methods and this new Monte Carlo technique converge on results consistent with the predicted power law governing residual energy during the annealing process. The Monte Carlo method proved particularly effective for finite lattices with open boundaries, offering a computational advantage over deterministic approaches in these scenarios.

The Monte Carlo sampling technique involved a Metropolis-Hastings algorithm adapted for evaluation of expectation values on an eight-by-eight-by-eight lattice. To maximize efficiency, the algorithm sweeps the lattice layer by layer, and within each layer, row by row, using six different orientations sequentially. While more accurate proposal functions exist, their computational cost was deemed prohibitive, as even the simpler function employed yielded acceptably short autocorrelation times.

This pragmatic approach allowed the team to extend their simulations into the Kibble-Zurek scaling regime, a region previously inaccessible due to computational limitations. The deterministic evaluation of data at 5.12 nanoseconds proved particularly time-consuming due to long-range correlations and slow convergence of the upper and lower boundaries of the tensor network. In contrast, the Monte Carlo data extended into the scaling regime, demonstrating its efficiency in capturing the relevant physics. The combination of deterministic and Monte Carlo methods offers a versatile toolkit for tackling complex quantum simulations.

Monte Carlo Approach Extends Accessible Quench Times

The adaptation of the Monte Carlo Metropolis-Hastings algorithm allows researchers to evaluate expectation values on a cubic lattice of 8x8x8 sites, a scale previously difficult to achieve with comparable accuracy. This is accomplished by sampling over the network layer by layer, using two-dimensional projected entangled-pair states and one-dimensional matrix product states for the boundaries, creating a single-layer structure that reduces computational cost. The resulting data, used to generate figures detailing excitation energy and correlation functions, are openly available for further analysis.

This shift in the computational bottleneck is particularly beneficial for simulating imaginary-time evolution towards the ground state, where the Monte Carlo approach offers a distinct advantage. By varying the annealing time, researchers can predict the resulting energy levels and compare them to the theoretical predictions of the KZ power law. The observed approach to this law validates the application of the KZ mechanism to quantum annealing, providing insights into the dynamics of these systems.

The work builds on previous benchmarks of coherent annealers against classical simulation methods, demonstrating an advantage of quantum devices in specific scenarios. The researchers note that the computational gain from the single-layer structure outweighs the cost of multiple Monte Carlo sweeps needed to achieve desired error bars, making the method a viable alternative to deterministic approaches. This is especially important for finite systems, where deterministic methods struggle with the increased complexity of open boundaries.

The technique allows for a more comprehensive exploration of the system’s behavior, extending the accessible range of quench times and providing a more complete picture of the underlying physics. The development of this efficient Monte Carlo scheme not only addresses a computational bottleneck but also opens new avenues for investigating ground state properties and furthering the development of quantum annealing techniques. The success of this approach suggests that Monte Carlo sampling can be a powerful tool for tackling complex quantum systems, particularly those involving finite lattices and open boundaries.

By providing an efficient means of evaluating expectation values, it allows researchers to focus on the more challenging aspects of the simulation, such as the time-evolution itself. In the study, the researchers revisit this problem using both deterministic and Monte Carlo approaches, highlighting the complementary nature of these methods and their combined power in tackling challenging computational problems.

👉 More information
🗞 Monte Carlo sampling from a projected entangled-pair state in simulations of quantum annealing in the three-dimensional random Ising model
✍️ Jacek Dziarmaga
🧠 DOI: http://link.aps.org/doi/10.1103/xb18-52hq

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