Variational-Adiabatic Quantum Solver for Systems of Linear Equations with Warm Starts

TL;DR

The paper tackles solving linear systems on NISQ devices by merging adiabatic parametrization with warm-started variational optimization. It introduces an interpolating family $A(s)$ and a ground-state cost $C_s(\theta)$, plus a Hessian-guided, adaptive step schedule $s(v)$ to follow the solution from $s=0$ to $s=1$ while avoiding barren plateaus. A Householder transformation is used to simplify state preparation, and the method is validated on discretized 1D heat-flow problems across several conductivity and source configurations. Results show improved convergence toward near-global minima with shallow circuits compared to direct VQLS optimizations, highlighting practical potential for quantum linear solvers on noisy hardware, albeit with open questions about scalability and hardware noise effects.

Abstract

We propose a revisited variational quantum solver for linear systems, designed to circumvent the barren plateau phenomenon by combining two key techniques: adiabatic evolution and warm starts. To this end, we define an initial Hamiltonian with a known ground state which is easily implemented on the quantum circuit, and then "adiabatically" evolve the Hamiltonian by tuning a control variable in such a way that the final ground state matches the solution to the given linear system. This evolution is carried out in incremental steps, and the ground state at each step is found by minimizing the energy using the parameter values corresponding to the previous minimum as a warm start to guide the search. As a first test case, the method is applied to several linear systems obtained by discretizing a one-dimensional heat flow equation with different physical assumptions and grid choices. Our method successfully and reliably improves upon the solution to the same problem as obtained by a conventional quantum solver, reaching very close to the global minimum also in the case of very shallow circuit implementations.

Figures (11)

• Figure 1: Increase of $s$ as a function of $T=100$ values of $v$ for condition numbers $\kappa = 10$ (solid blue line), $\kappa=10^2$ (dashed red line) and $\kappa = 10^3$ (dot-dashed green line). Note the concentration of $s(v)$ at values closer to $1$ for higher $\kappa$.
• Figure 2: Structure of the parametric quantum circuit $U(\theta)$ for the case $n=3$. The circuit starts with a $R_y$ rotation for each qubit, followed by $d=2$ repetitions of the layer detailed in the first box, which combines more single-qubit rotations and a set of two-qubit $\text{CNOT}$ gates arranged in a ring configuration.
• Figure 3: A slice of the energy landscape along a preferred direction $\Theta$: at a given $s$, the blue cross locates the minimum of the cost function (solid blue line). If we evolve the system by $\delta s_1$, the new global minimum (green cross) is still in the same valley as the previous one (dashed green line), but if we evolve it by $\delta s_2$ we overshoot the convexity region and the new global minimum (red cross) appears in another valley (dot-dashed red line), with no descending path joining it to the previous minimum (red dot).
• Figure 4: (a) Representation of a block of some heterogeneous material with conductivity increasing along the $z$ coordinate up to some noise $\zeta$: $\lambda(z) = c z+\zeta$. A heat source located at the bottom of the bulk, where $\lambda(z)=\lambda_\mathrm{min}$, is shown and can be represented in Eq. \ref{['eq:stationary']} by a pointlike source term $Q_r(z)\propto\delta(z)$. Our goal is to find the stationary state of the heat flow equation for this problem on a finite lattice. (b) Condition number $\kappa$ of the matrix $A$ for the heat flow equation in the discretized form given in Eq. \ref{['eq:heat_matrix']}, for the cases $\lambda(z)=1$ (blue line), $\lambda(z)=1+\zeta(z)$ with $\zeta$ normally distributed with $\sigma = 0.2$ (red line) and $\lambda(z)=2z$ (green line). $\kappa$ increases exponentially with the number of qubits for all the analyzed cases.
• Figure 5: A comparison of the different variational strategies considered in this paper, applied to the heat equation Eq. \ref{['eq:stationary']} with $\lambda=1$, $\mathbf{b}=\mathbf{e}_1$ and an ansatz of depth $d=1$. (a) Minimum value of the cost function for all qubit numbers $n$ considered, obtained using the adiabatic approach $s(v)$ with 10 timesteps (solid blue line), fixed $\delta s$ equal to 0.1 (dashed red line) and 0.01 (dot-dashed green line) and the best direct optimization at $s=1$ obtained from 10 random initialization points, averaged over 100 trials (dotted purple line). (b) Number of adiabatic steps needed to reach $s=1$ using the convexity analysis (blue stars) or $s(v)$ with $T=100$ (red crosses). The resulting minimum is the same. (c) Components $x_i$ of exact and variational solutions using constant $\delta s=0.01$ (dashed black line), $s=s(v)$ with $T=10$ (solid red line), and exact solution (dotted green line).