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

Claudio Altafini|ArXiv.org|Mar 2, 2002

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a framework to directly map continuous-time control fields in the Schrödinger equation to discrete quantum gates via exponential coordinates on SU(N), using the Wei-Norman formula to relate Magnus expansion to product-of-exponentials parameterization. The key contribution is a differential formalism that enables analytical tracking of gate dynamics under control perturbations, while identifying singular loci where standard coordinates fail but dynamics remain governable via implicit equations.

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.

Motivation & Objective

• To reconcile continuous-time quantum dynamics governed by the Schrödinger equation with the discrete gate model of quantum computation.
• To express elementary quantum gates directly in terms of physical control inputs from external fields.
• To provide a differential framework for tracking unitary evolution under control perturbations using exponential coordinates on SU(N).
• To analyze the singularities of the Wei-Norman formula and show that dynamics remain controllable even when coordinate charts fail.

Proposed method

• Uses the Wei-Norman formula to relate the Magnus expansion (continuous-time evolution) to the product-of-exponentials decomposition (discrete gates).
• Expresses the time evolution of the system in terms of exponential coordinates on SU(N), parameterized by angles γ₁, γ₂, γ₃.
• Derives a system of nonlinear differential equations linking the control fields u₁, u₂, u₃ to the rates of change of the γ-parameters.
• Identifies the singular locus Σ where the Jacobian matrix Ξ becomes rank-deficient, rendering exponential coordinates undefined.
• Applies implicit dynamics (via equation 5) to continue evolution through singularities, ensuring control authority is preserved.
• Uses Lie algebra structure constants to systematically compute the Wei-Norman transformation for SU(2) and SU(3) systems.

Experimental results

Research questions

• RQ1 How can continuous-time control fields in the Schrödinger equation be systematically mapped to discrete quantum gates in the circuit model?
• RQ2 What is the role of the Wei-Norman formula in connecting the Magnus expansion to the product-of-exponentials representation of unitary evolution?
• RQ3 How do singularities in the exponential coordinate chart affect controllability and state evolution?
• RQ4 Can dynamics be analytically tracked through singular loci where standard parameterization fails?
• RQ5 To what extent do free Hamiltonians and control fields influence the relative phase and population of superposition states?

Key findings

• The Wei-Norman formula provides a differential link between continuous-time dynamics and discrete quantum gates, enabling direct mapping of control inputs to gate parameters.
• Singular loci Σ in the parameter space correspond to points where the Jacobian Ξ loses rank, making exponential coordinates undefined.
• Despite singularity, the dynamics remain governed by implicit equations (e.g., equation 5), allowing continued control via u₁ and u₂.
• In the singular case, the free Hamiltonian only induces a relative phase shift on superposition states, while control fields can steer the system out of unobservable subspaces.
• The system can be steered out of the γ-isotropy subgroup Hγ(|ψ₀⟩) for non-trivial superpositions, ensuring full controllability.
• The method enables analytical monitoring of gate variations due to dynamical perturbations or field uncertainties, even in non-smooth regions of parameter space.