Fast-forwarding of Hamiltonians and exponentially precise measurements

Area of Science:

• Quantum mechanics
• Computational complexity theory
• Quantum information science

Background:

• The time-energy uncertainty relation (TEUR) is fundamental in quantum mechanics.
• TEUR typically holds when the Hamiltonian is unknown but can be violated under specific conditions.
• Existing studies lack a rigorous framework for quantifying TEUR violations.

Purpose of the Study:

• To rigorously investigate the conditions and extent of time-energy uncertainty relation (TEUR) violations.
• To propose and analyze a computational version of the TEUR (cTEUR).
• To explore the relationship between computational complexity and energy measurement accuracy.

Main Methods:

• Development of the computational time-energy uncertainty relation (cTEUR).
• Analysis of conditions for cTEUR violations using computational complexity.
• Identification of specific quantum systems exhibiting cTEUR violations, such as those with fast-forwardable Hamiltonians.

Main Results:

• cTEUR violations occur if and only if the Hamiltonian can be fast forwarded (FF).
• Demonstration of exponential cTEUR violations using Shor's algorithm, commuting local Hamiltonians, and quadratic fermion Hamiltonians.
• Established a rigorous framework linking computational complexity to energy measurement limitations.

Conclusions:

• The study initiates a rigorous theory of efficiency versus accuracy in quantum energy measurements.
• Connections between cTEUR violations, fast forwarding, and quantum sensing/gravity are discussed.
• Further research is needed for experimental demonstrations and discovering new FF examples.