Distributed VQLS · Chapter 03

What the numerical studies establish

Three studies separate representation capacity, the cost of finer partitions, and optimizer design. All are classical simulations of quantum algorithms, averaged over ten random initializations—not hardware experiments.

A structured 51-qubit instance

The large-instance experiment uses an Ising-inspired matrix with a compact Pauli representation. Four agents each represent 50 system qubits. Low-entanglement structure permits classical matrix-product-state simulation; overlaps are evaluated directly from that representation rather than by simulating every Hadamard-test circuit.

Model

The large-scale test system

$X_\ell$ and $Z_\ell$ are Pauli operators on qubit $\ell$ ; $H$ is the Hadamard gate. $\zeta$ is a positive normalization constant and $\lambda$ a spectral shift. The global matrix is $2^{51}\times2^{51}$ . The setup is structured and classically simulable, not an arbitrary dense matrix at this scale.

\begin{aligned} A&=\frac1\zeta\left(\sum_{\ell=1}^{n}X_\ell +\kappa\sum_{\ell=1}^{n-1}Z_\ell Z_{\ell+1}+\lambda I\right),\\ b&=H^{\otimes n}|0\rangle^{\otimes n},\\ n&=51,\qquad \kappa=0.1,\qquad \operatorname{cond}_2(A)=50. \end{aligned}

Manuscript · Large-Scale Demonstration

Two plots from the large-scale distributed VQLS study: global residual norm and parameter consensus error against 0 to 100 iterations, both with logarithmic vertical axes. — Original manuscript figure, Large-Scale Demonstration: global residual norm (top) and parameter consensus error (bottom), on logarithmic vertical axes. Results are averaged over ten random initializations for the 51-qubit system with four 50-qubit local representations. This is a classical matrix-product-state simulation, not a hardware run. Parameter consensus is not solution error; no interpretation of the shaded bands is inferred here.

Large-scale demonstration: setup and reported observations
Quantity	Reported value or observation
Block partition / agents	2 × 2 / 4
Global / local system qubits	51 / 50
Base stepsize / initializations	0.02 / 10
Global residual norm	Nearly two orders of magnitude lower over 100 iterations
Parameter consensus error	Decreases to approximately 10⁻⁴

Source: manuscript, Large-Scale Demonstration. Consensus here measures parameter disagreement, not state-vector error. The reported reductions are approximate descriptions from the manuscript; they are not reconstructed curve samples.

Finer partitions reduce local size and slow optimization

A separate 13-qubit perturbed cluster-state system has condition number 20 and six Pauli terms. This smaller experiment explicitly simulates Hadamard-test circuits. Both communication graphs are paths, and the base stepsize is 0.01.

Definition

Residual and state-vector agreement are different metrics

$x_{ij}$ is the scaled local state, so $u_i$ stacks all estimates in block row $i$ . $R$ measures equation error; $E$ measures agreement between those row estimates. In this study agreement is reached sooner than a small residual. Unlike the 51-qubit study, $E$ is defined in vector space, not parameter space.

\begin{aligned} u_i(t)&=(x_{i1}(t)^\top,\ldots,x_{im}(t)^\top)^\top,\\ \bar x(t)&=\frac1m\sum_{i=1}^{m}u_i(t),\\ R(t)&=\|A\bar x(t)-b\|_2,\\ E(t)&=\left(\frac1m\sum_{i=1}^{m}\|u_i(t)-\bar x(t)\|_2^2\right)^{1/2}. \end{aligned}

Manuscript · Effect of Block Partitioning

Same 13-qubit problem, four representation sizes
Partition	Agents	Local system qubits	Reported behavior
1 × 1	1	13	Fastest residual reduction; centralized version of this objective
2 × 2	4	12	Slower reduction than the centralized case
4 × 4	16	11	Residual eventually fluctuates around 10⁻²
8 × 8	64	10	Slowest case; residual eventually around 10⁻²

Source: manuscript, Effect of Block Partitioning; ten initializations per case. Qubit counts exclude measurement ancillas. The objective differs from the original centralized VQLS objective, so the 1 × 1 case is not an independent implementation of the original VQLS baseline.

The classical optimizer materially changes the outcome

The ablation uses a 7-qubit Ising-inspired system, a 4 × 4 partition (16 agents, five local system qubits), condition number 200, and base stepsize 0.01. It changes one component at a time while keeping the test problem fixed.

Optimizer ablation: reported residual-norm behavior
Variant	Change from the full method	Reported observation
Track + AdamX + AdamZ	None	More than 100× reduction by approximately 10,000 iterations
Track + AdamZ	Gradient descent replaces Adam for x parameters	Residual remains above 1 at 20,000 iterations
Track + AdamX	Gradient descent replaces Adam for z parameters	Residual remains above 1 at 20,000 iterations
Consensus + AdamX + AdamZ	Remove gradient tracking	Residual remains above 1 at 20,000 iterations

Source: manuscript, Failure of Distributed Gradient Descent; ten random initializations and simulated Hadamard tests. This supports the full method in the tested setting. It is neither a general convergence theorem nor a proof that other optimizers cannot succeed.

Which costs determine practical usefulness?

The reformulation is exact, but the experiments do not establish a quantum advantage or a general convergence guarantee. The next stage should measure the resources and reliability of the complete hybrid algorithm.

Conditioning and connectivity: vary the matrix condition number, agent count, and row/column graphs independently.
Communication: reduce the present two transmission rounds; account for latency, delays, and asynchronous updates.
Hardware: measure circuit, sampling, and readout costs under realistic noise before claiming improved resilience or speed.

Last updated September 26, 2026