Improving Quantum Imaginary Time Evolution with SA-QITE

Quantum imaginary time evolution is an important technique used in various fields such as natural sciences and optimization. It allows scientists to prepare important quantum states, which can help in understanding complex systems. Quantum computers are becoming useful tools for this purpose, as they can handle the large computations needed for imaginary time evolution.

However, traditional methods can be slow and require too many measurements, making them impractical for real applications. To address these challenges, we propose a new approach designed to reduce the amount of time and resources needed for these calculations.

#What is Quantum Imaginary Time Evolution?

In quantum mechanics, time evolution refers to how a system changes over time. For imaginary time evolution, we use a mathematical approach that is different from real-time evolution. This method is particularly powerful for preparing thermal states, which are essential in physics and machine learning.

Thermal states can help calculate various physical properties and optimize systems in different fields. Preparing Ground States is also crucial, as it has applications in physics, chemistry, classical optimization, and finance.

To simulate this process, one typically has to deal with very large wave functions. This is where quantum computers show promise, as they can handle these complex computations more efficiently than classical computers.

#The Challenges of Traditional Methods

While traditional methods, such as the Suzuki-Trotter approximation, work well for real-time evolution, they face obstacles in imaginary time evolution. They often require complex circuits that are difficult to implement on current quantum devices. As quantum computers develop, they have limitations, such as short coherence times and noise in operations, making these traditional methods less suitable.

Instead of directly evolving the quantum state, variational methods project the evolution onto adjustable parameters in an ansatz circuit. This approach is more flexible and works better with the current generation of quantum devices.

#Variational Quantum Imaginary Time Evolution (VarQITE)

Variational quantum imaginary time evolution is a framework for handling the imaginary time evolution of quantum states using variational methods. This approach involves defining a circuit with parameters that can be tuned to approximate the desired state.

The parameters are updated using a mathematical principle called the McLachlan variational principle. Essentially, the idea is to optimize the parameters iteratively by evaluating the Quantum Geometric Tensor (QGT) and the Energy Gradient.

However, the resource requirements for this method can quickly become overwhelming, especially when dealing with circuits that have a large number of parameters. For example, evaluating the QGT and gradient involves running many circuits, which is not practical for the current generation of quantum computers.

#Introducing Stochastic Variational Imaginary Time Evolution (SA-QITE)

To address the issues faced by VarQITE, we introduce a new approach called Stochastic Variational Imaginary Time Evolution (SA-QITE). This method uses a stochastic approximation technique that reduces the number of measurements required while still maintaining accuracy.

Instead of calculating the full QGT and gradient at each step, SA-QITE estimates these values using a smaller number of samples drawn from random perturbations. This sampling method drastically cuts down the number of circuits that need to be run, making the process more efficient.

By leveraging this stochastic method, we can achieve similar levels of accuracy with much fewer measurements than with the traditional VarQITE method.

#How SA-QITE Works

SA-QITE starts with an accurate estimate of the QGT and energy gradient. Instead of recalculating these values in full at each time step, the algorithm corrects its estimations using random samples generated through a technique called simultaneous perturbation stochastic approximation (SPSA).

This innovative approach ensures that the number of circuits needed remains constant, regardless of the number of parameters in the model. By focusing on sampling rather than full evaluation, we can achieve more efficient evolution of quantum states.

In practical terms, we perform numerical simulations to demonstrate how well SA-QITE works compared to VarQITE. We apply it to known models, such as the transverse field Ising model, and compare the resource requirements of both methods.

#Advantages of SA-QITE

The primary advantage of SA-QITE is its efficiency. In our experiments, we found that SA-QITE typically requires an order of magnitude fewer measurements than VarQITE to achieve the same level of accuracy. This reduction in resource needs is vital as quantum computers scale up and tackle more complex problems.

SA-QITE also effectively avoids many of the pitfalls faced by other variational methods that rely heavily on high accuracy in measurements. By focusing only on relative differences in state fidelity, SA-QITE can easily adapt to current devices with their inherent noise and limited connectivity.

#Simulation Results

In our studies, we investigated how well SA-QITE performs in two tasks: the evolution of the transverse field Ising model and approximating ground states of Hamiltonians derived from optimization problems, like the Max Cut problem.

For the transverse field Ising model, we first set up the model with specified interaction strengths and then analyzed the performance of both SA-QITE and VarQITE. With different numbers of qubits, we observed the number of measurements required to achieve a target accuracy.

Our results showed that SA-QITE consistently used significantly fewer measurements compared to VarQITE, proving its efficiency and practicality.

#Ground State Approximation with SA-QITE

When the exact dynamics of a system are not required, one can approximate ground states using SA-QITE. We tested this approach by minimizing the energy of a Hamiltonian related to the Max Cut problem.

In this case, the goal was to find the lowest energy configuration for a set of interconnected nodes. We compared SA-QITE with other gradient-based optimization methods and found that SA-QITE showed better performance in terms of the number of measurements needed to converge to the optimal state.

#Experimental Validation

To prove that SA-QITE works well on actual quantum computers, we ran experiments using a 27-qubit IBM Quantum processor. Our experiments involved scaling up the Ising Hamiltonian and executing the imaginary-time evolution directly on the hardware.

We used a simple ansatz that matched the topology of the quantum device, allowing for low-depth and efficient execution. After performing the calculations, we found that SA-QITE was capable of producing results that were close to the expected values, even with noise from the hardware.

Using additional techniques for error mitigation, we were able to enhance the accuracy of our results. The energy values calculated aligned closely with those obtained from ideal simulations, demonstrating that SA-QITE effectively navigated the challenges posed by noisy quantum environments.

#Conclusion

In summary, we have developed SA-QITE as a more efficient alternative to traditional methods like VarQITE for quantum imaginary time evolution. By leveraging a stochastic sampling approach, we can significantly reduce the resource requirements while maintaining accuracy.

This new method paves the way for more practical applications of quantum computing in various fields, from physics and chemistry to finance and beyond. Our results highlight the potential for SA-QITE to handle larger and more complex quantum systems, improving the prospects for solving real-world problems with quantum technology.

The future of quantum computing looks bright as we continue to refine algorithms and explore new ways to harness the capabilities of quantum systems. By focusing on scalability and practicality, we can bridge the gap between quantum theory and real-world applications, paving the way for innovative solutions to challenging problems.