Quantum Algorithms Simulate Heat to Create Stable Thermal States

A new method for efficiently preparing thermal states is enabling advances in fields from materials science to machine learning. Andrew Wright and colleagues at the Institute of Physics, in collaboration with Chulalongkorn University and Keio University, have developed a technique termed double-bracket thermofield double (DB-TFD) that uses double-bracket quantum algorithms to simulate thermofield double states and realise Gibbs states. The poly DB-TFD algorithm’s complexity scales favourably with inverse temperature, consistent with established techniques and confirmed by numerical simulations. Moreover, the team demonstrates DB-TFD’s potential in quantum Boltzmann machines, achieving improved performance compared with existing variational methods, and providing a strong pathway for thermal state preparation on near-term and early-fault-tolerant quantum computers.

Exponential scaling unlocks thermal state preparation for complex quantum systems

The poly DB-TFD algorithm now demonstrates a query complexity scaling exponentially with inverse temperature, a substantial improvement over earlier methods limited to polynomial scaling in practical regimes. Validated by numerical simulations, this exponential scaling unlocks the potential to prepare thermal states for larger, more complex systems previously inaccessible to quantum computation. Dr. Alastair Peoples and Professor Andrew Green, alongside Dr. Patrick Draper, employed a technique called double-bracket thermofield double (DB-TFD) to simulate thermofield double states, effectively creating ‘hot’ and ‘cold’ copies of a system to realise Gibbs states, crucial for modelling thermal equilibrium. Thermofield double states are a cornerstone of quantum statistical mechanics, representing a system and its replica in a fictitious Hilbert space, allowing for the elegant formulation of thermal properties. The Gibbs state, describing the probability distribution of a system in thermal equilibrium at a given temperature, is central to understanding macroscopic behaviour from microscopic quantum principles.

A polynomial transformation approximating imaginary-time evolution underpins the approach, reducing the computational steps needed for thermal state creation and offering a viable pathway for both near-term and early fault-tolerant quantum computers. Imaginary-time evolution is a powerful technique for preparing ground states and thermal states, effectively simulating the system’s dynamics in a Euclidean space. The DB-TFD algorithm leverages this by approximating the necessary imaginary-time evolution operator using a polynomial expansion, thereby reducing the quantum circuit depth. The team successfully implemented DB-TFD within quantum Boltzmann machines, a form of generative modelling, and observed performance gains compared to existing variational imaginary-time evolution techniques. Specifically, the algorithm achieves this exponential scaling for systems where a ‘contraction coefficient’ scales sub-polynomially, as this coefficient defines the efficiency of the approximation and influences performance at different temperatures. The contraction coefficient essentially quantifies how efficiently the system’s Hamiltonian can be contracted, impacting the overall computational cost. A sub-polynomial scaling of this coefficient is crucial for realising the exponential speedup offered by the DB-TFD algorithm.

Further investigation explored the two approaches offered by DB-TFD, a direct method and one utilising quantum signal processing, revealing their respective strengths and limitations for near-term quantum devices. The direct method offers simplicity in circuit construction, while the quantum signal processing approach provides potential for improved accuracy and efficiency, particularly for higher temperatures. Quantum signal processing allows for the precise implementation of functions of Hermitian operators, enabling more accurate approximations of the imaginary-time evolution operator. However, it often comes at the cost of increased circuit complexity. The choice between these two approaches depends on the specific characteristics of the quantum hardware and the desired trade-off between circuit depth and accuracy. While these results establish a promising route for thermal state preparation, the current analysis does not fully address the impact of noise on larger systems, presenting a significant challenge for implementation on real-world quantum hardware. Representing systems at a specific temperature, thermal states are fundamental to modelling materials and understanding complex chemical reactions, and the method’s potential lies in efficiently creating these states for applications in materials science and chemistry. For instance, accurately modelling the thermal properties of novel materials is crucial for designing efficient energy storage devices or superconductors.

Efficient thermal state preparation necessitates validation against quantum hardware noise

Dr. Peoples, Professor Green, and Dr. Draper acknowledge an important gap in their work; rigorous testing of the algorithm’s durability against the noise inherent in real-world quantum hardware remains incomplete, representing a significant hurdle for practical application. Idealised conditions underpin current simulations, and addressing this limitation is vital for translating theoretical gains into practical results. Quantum computers are susceptible to various sources of noise, including decoherence, gate errors, and measurement errors, which can significantly degrade the performance of quantum algorithms. Consequently, further research into error mitigation strategies and noise-aware algorithm design is necessary to ensure robust performance on imperfect quantum devices. Error mitigation techniques aim to reduce the impact of noise without requiring full quantum error correction, offering a pragmatic approach for near-term quantum computers.

This new algorithm provides a viable method for preparing thermal states on current and developing quantum computers. Simulating ‘hot’ and ‘cold’ copies of a quantum system allows it to circumvent limitations found in previous techniques that require complex operations. Previous methods often relied on directly simulating the time evolution of the system, which can be computationally expensive, especially for large systems and long evolution times. The DB-TFD algorithm, by leveraging the thermofield double state formalism and the double-bracket algorithm, offers a more efficient approach. Numerical simulations confirm its complexity scales exponentially with inverse temperature, aligning with established theoretical boundaries and validating its potential for simulating complex systems. The exponential scaling is particularly significant as it suggests that the algorithm’s runtime grows much more slowly with increasing system size and decreasing temperature, making it feasible to simulate systems that were previously intractable.

This research details a new algorithm, DB-TFD, for preparing thermal states on quantum computers. By simulating paired ‘hot’ and ‘cold’ quantum systems, the method offers a more efficient alternative to directly simulating time evolution, particularly for complex systems. Simulations demonstrated the algorithm’s complexity scales exponentially with inverse temperature, consistent with existing theoretical limits. The authors acknowledge further work is needed to assess the algorithm’s performance when exposed to the noise inherent in real quantum hardware.