We (@ManjulaGandhi, @Harimakesh and @Barath-T) aim to implement a quantum algorithm for solving linear differential equations (LDEs) based on A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment (arXiv:1807.04553). Specifically, we will construct a quantum circuit that solves the harmonic oscillator equation using the approach outlined in this paper.

We focus on solving the differential equation of harmonic oscillator:

$$y'' + \omega^2 y = 0, \quad y(0) = 1, \quad y'(0) = 1$$

with $$\omega = 1$$.

Technical Approach

This equation can be rewritten as a first-order system:

$$ x(t) = \begin{bmatrix} y \\ y' \end{bmatrix}. $$

$$ \frac{dx}{dt} = \begin{bmatrix} 0 & 1 \\ -\omega^2 & 0 \end{bmatrix} x. $$

For LDE given by:

$$ \frac{dx(t)}{dt} = Mx(t) + b $$

the solution is given by:

$$ x(t) = e^{Mt} x(0) + (e^{Mt} - I) M^{-1} b $$

If the exponential evolution $e^{Mt}$ and the inverse operator $M^{-1}$ can be effectively realized, one can easily obtain the solution $x(t)$.
The $k$-th order approximate solution converts to:

$$ |x(t)⟩ ≈ \sum_{m=0}^{k} \frac{‖x(0)‖ (‖M‖ A t)^m}{m!} |x(0)⟩ + \sum_{n=1}^{k} \frac{‖b‖ (‖M‖ A)^{n-1} t^n}{n!} |b⟩ $$

up to normalization.

The algorithm

• The quantum system is initialized with work qubits (to store vectors) and two sets of ancilla qubits.
• The initial vectors |x(0)⟩ and |b⟩ are prepared based on the first ancilla qubit state.
• A sequence of controlled operations evolve the system while ensuring that necessary transformations on |x(0)⟩ and |b⟩ occur in parallel. This step prepares the system for solving the differential equation.
• The encoding operations are reversed to extract useful information.
• Specific unitary transformations ensure that the final system state contains the desired solution components.
• The final state is measured in a specific subspace where all ancilla qubits are |0⟩.
• The solution to the LDE, |x(t)⟩, is obtained up to a normalization factor N².

when A is non-unitary

• it can be expressed as a linear combination of unitary operators: $A = \sum \alpha_i A_i$
• This requires an additional ancilla register to label the different $A_i$​'s.