Quantum Imaginary Time Evolution - RPIQuantumComputing/QuantumCircuits GitHub Wiki
In a universe devoid of the constraint of time, how would quantum systems perform and respond to this absence? Although the idea of a timeless reality seems almost impossible, the employment of mathematical formulations provides a lens through which we can conceptualize it. This mathematical framework not only facilitates visualization but also serves as a valuable tool, enhancing the efficiency of specific quantum simulations.
What is the Concept of Imaginary Time?
Imaginary time is a introduced in quantum physics to simplify certain calculations. Rather than serving as a conventional construct of time, it functions primarily as a mathematical concept. This involves substituting the time variable t with i, where i represents the imaginary unit i = $\sqrt{-1}$.
What is Ground State and Hamiltonian?
QITE (Quantum Imaginary Time Evolution) presents an alternative approach to obtaining the ground state of a Hamiltonian. This technique emulates the progression of imaginary time evolution through the application unitary operations on a quantum computer. The ground state, representing the system's lowest energy configuration, holds significance in diverse quantum applications, underscoring the need for a thorough understanding of its properties.
Algorithmic Approach
QITE involves the application of a unitary transformation to the quantum state, aiming to identify the closest unitary transformation mirroring the evolution suggested by imaginary time. This algorithmic strategy allows researchers to simulate the progression of quantum systems towards their ground state, eliminating the necessity for direct physical manipulation.
The Math Behind It
The mathematical formalism of QITE typically involves concepts from linear algebra and quantum mechanics. The algorithm often employs techniques such as the Trotter-Suzuki decomposition to approximate the unitary evolution operator. A pure quantum state is said to k-UGS if it's the unique ground state of a k-local Hamiltonian $\hat{H}=\sum_{j=1}^m h[j]$, where each local term $\hat{h}[j]$ acts on at most k neighboring qubits. The QITE algorithm is well suited for preparing k-UGS states with a relatively small k.
How Can We Visually See It?
The best way to visualize QITE is using graphical tools, used to represent the unnormalized transition that QITE would induce on the superposition corresponding to specific circuit steps. These visualizations aid in intuitively understanding effects of QITE on quantum states.