Quantum Adiabatic Theorem Now Holds for Finite Temperatures

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Absolute zero is the best temperature for quantum computing because it allows the system to be controlled more easily, thanks to several fundamental properties in quantum mechanics. The quantum adiabatic theorem, for example, implies that quantum systems have simpler dynamics provided there is a gradual change in external parameters. Absolute zero is physically impossible to achieve. Hence, expanding the range of control strategies to include non-zero temperatures is a hot research topic.

Despite progress, an underlying problem remains: Optimum control of the underlying physical structures or components, whether atoms or individual photons, is required in all quantum systems. Controlling atoms and photons requires a system in the desired quantum state.

By establishing the adiabatic theorem at a non-zero temperature and specifying the quantum conditions for dynamics in the adiabatic regime, Russian scientists reported an advance that could prove useful in developing the next-generation of quantum devices. That requires fine-tuning features of quantum superpositions across thousands of components.

Adiabatic theorem in quantum mechanics
A useful framework in quantum mechanics is the adiabatic theorem, to wit: If the Hamiltonian H(t) changes slowly with time, then a system that is initially in the nth energy level of H(0) will be at time (t) in the nth energy level of H(t). H(t) is the mechanism associated with a system’s total energy in quantum mechanics. If the Hamiltonian and the wave function of the system are known, the total energy can be determined.

Adiabatic in this case has a different meaning than in thermodynamics. In thermodynamics, it means “without heat exchange,” which often means “very rapidly.” Here, it has the same meaning as “adiabatic invariant,” that is, slow with regard to the system’s typical time scale.

The presence of an energy gap is required for the theorem’s validity: The nth level must be energetically isolated from the adjacent levels. As a result, if the change is gradual (infinitesimal), the eigenfunction will unable a sufficient jump to cover the gap in a short time, and will always return to its energy level. However, it makes little difference whether the overall change in the Hamiltonian is modest or large; what matters is that it occurs slowly.

Adiabatic evolution
Quantum physics promises to aid in the development of ultra-fast computers, ultra-precise measuring devices and secure communications systems, all of which require unique conditions to perform as designed. The most suitable temperature for system components to be motionless and silent is –273.15 ˚C. The quantum superposition principle can exhibit its greatest efficiency in that state. Furthermore, absolute zero simplifies the theoretical description of quantum processes by providing physicists and engineers with the means for predicting the results of quantum experiments and the design of quantum devices. One reason is the removal of variations that muddle mathematical equations.

Applying the third rule of thermodynamics, temperatures are always non-zero and capable of destroying quantum superpositions. This necessitates the development of finite-temperature control techniques.

According to the Russian researchers, quantum systems may be characterized by the density operator, but the changing of electric and magnetic fields over time makes it extremely complicated. The operator changes with time, and the intricacy of this complication is the foundation of a quantum computer’s vast potential.

Quantum adiabatic evolution contributes to “taming” by varying external fields gradually, in which case the quantum state can be made more predictable. In the early days of quantum mechanics, Max Born and Vladimir Fock proved the adiabatic theorem. The theorem ensures that the evolving quantum state always remains close to the instantaneous eigenstate if the external parameters change slowly enough. Its main limitation was that it worked only for pure states, but not for all quantum states, meaning it could be applied only to systems at absolute zero but not to finite temperatures.

A quantum spin moves in a circle around a wire. Due to current fluctuations, electrons in the wire are magnetically related to the spin. The electron-spin system’s many-body adiabaticity at finite temperature is unaffected by expanding the wire’s length. At every finite driving rate, pure state adiabaticity breaks down. (Source: Adiabatic theorem for closed quantum systems initialized at finite temperature)

Thanks to rapid development of quantum technologies, large quantum systems may now be manipulated using a variety of experimental approaches. In practice, the states of these systems are rarely even close to pure. Rather, they are usually prepared at some small but non-zero temperature at the initialization stage, then are driven by external fields to perform metrological, computational or communications tasks.

Researchers from Skoltech, the Russian Academy of Sciences’ Steklov Mathematical Institute and Moscow Institute of Physics and Technology extended the adiabatic theorem to finite temperature systems, discovering quantum conditions that ensure adiabatic development with reasonable precision. As an example, the researchers tested these conditions on many simulated systems and discovered that adiabatic dynamics were even more stable in several cases at limited temperatures than at absolute zero.

The researchers said the new theorem will be useful for optimizing control of large quantum devices that benefit from adiabaticity but must be operated at finite temperatures. The prime example is quantum adiabatic annealer, a type of quantum simulating device that relies heavily on quantum adiabatic evolution. Quantum annealers are being developed by D-Wave Systems Inc. of Canada.


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