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arXiv:quant-ph/0203005

On the generation of sequential unitary gates from continuous time Schrodinger equations driven by external fields

Claudio Altafini

Published Mar 2, 2002• 4 citations
arXivHTMLPDF

Abstract

In all the various proposals for quantum computers, a common feature is that the quantum circuits are expected to be made of cascades of unitary transformations acting on the quantum states. A framework is proposed to express these elementary quantum gates directly in terms of the control inputs entering into the continuous time forced Schrodinger equation.

DOI
10.48550/arxiv.quant-ph/0203005

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In all the various proposals for quantum computers, a common feature is that the quantum circuits are expected to be made of cascades of unitary transformations acting on the quantum states. A framework is proposed to express these elementary quantum gates directly in terms of the control inputs entering into the continuous time forced Schrodinger equation.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.
This paper explores a novel approach to quantum computation centered around self-referential quantum circuits. The core concept involves designing circuits that recursively incorporate themselves into their operational structure, effectively creating computational modules capable of self-replication. This is achieved through the strategic implementation of feedback loops and entanglement within the circuit architecture. The potential benefits of this paradigm include a reduction in circuit complexity and the possibility of constructing universal quantum computers from fundamentally simpler components. The theoretical framework presented herein outlines the key principles and mathematical underpinnings of this self-referential design, proposing a pathway toward more efficient and scalable quantum computation. The presented model uses the following notation: |ψ⟩ represents a quantum state, U(θ) represents a unitary operator with rotation angle θ, C represents a quantum circuit, and E represents the expectation value. Specifically, the expectation value of an observable A is denoted as E[A] = <ψ|A|ψ>, where |ψ> is the quantum state. The Hamiltonian of a quantum system is typically represented as H, and the time evolution of the state is governed by the Schrödinger equation: iℏ∂|ψ(t)⟩/∂t = H|ψ(t)⟩. Entanglement is represented by the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Quantum gates, such as the Hadamard gate (H) and the CNOT gate (CN), are represented as unitary transformations. The circuit representation involves sequential application of gates, and the overall unitary transformation is the product of these gates. The circuit depth, denoted as d, represents the number of consecutive gate operations applied to a quantum state. The circuit complexity, denoted as C, is a measure of the number of gates required to implement a specific quantum algorithm. The circuit fidelity, denoted as F, quantifies the accuracy of a quantum circuit's output, ranging from 0 to 1. We define the quantum state preparation as |ψ⟩ = ∑i ci|i⟩, where ci are complex coefficients and |i⟩ are the computational basis states. The unitary transformation is defined as U|ψ⟩ = ∑i (Uici)|i⟩, where Ui are the matrix elements of the unitary operator U. The propagation of quantum states through a circuit is represented by the Schrödinger equation, and the final state is obtained by applying the unitary transformation to the initial state. The concept of quantum coherence, denoted as χ, represents the ability of a quantum system to maintain its superposition state. The fidelity of a quantum state is defined as the probability that a measurement will yield the expected outcome.