Distributed Quantum Least Squares Protocol

• The paper presents a framework that enables collaborative least squares regression across partitioned quantum data blocks, achieving exponential and polynomial speedups.
• It leverages block-encoding unitaries and advanced inversion subroutines, such as gapped phase estimation, to efficiently solve both ordinary and regularized least squares problems.
• The protocol outputs classical model parameters with reduced communication complexity, ensuring scalable and practical deployment for large-scale quantum networks.

A distributed quantum least squares protocol is an algorithmic framework enabling collaborative least squares regression across a quantum network or multi-party quantum system, where data and computational resources are partitioned among multiple nodes. Such protocols leverage quantum computational advantages—most notably, exponential and polynomial speedups in data size and dimensionality—while addressing classical bottlenecks in scalability, communication, and aggregation. Modern variants output classical model parameters, efficiently estimate solution quality, accommodate regularization, and optimize communication complexity through advanced quantum signal processing and distributed quantum linear algebra.

1. Mathematical Formulation and Problem Setting

Distributed quantum least squares protocols address the minimization of quadratic objectives over partitioned data. The core task is to solve either the ordinary least squares (OLS) or regularized least squares problem across multiple parties:

Ordinary Least Squares:

Given a data matrix $A \in \mathbb{R}^{N \times d}$ and response vector $b \in \mathbb{R}^{N}$, distributed such that each node $P_i$ holds $A_i, b_i$, the objective is

$\min_{x \in \mathbb{R}^{d}} \|A x - b\|^2,$

with block-wise data

$$A = \begin{bmatrix} A_0 \ \vdots \ A_{r-1} \end{bmatrix}, \quad b = \begin{bmatrix} b_0 \ \vdots \ b_{r-1} \end{bmatrix}.$$

L₂-Regularized Least Squares (Tikhonov/Ridge):

For regularization parameter $\lambda > 0$ and full-rank $L \in \mathbb{R}^{d \times d}$,

$$\min_{x} \,\mathcal{L}_{l_2}(x) = \|A x - b\|^2 + \lambda \|L x\|^2,$$

which is equivalent to solving with the augmented matrix

$$A_L := \begin{bmatrix} A \ \sqrt{\lambda}L \end{bmatrix}$$

and extended response $(b, 0)^\top$.

In distributed quantum settings, these tasks are approached by constructing quantum representations (block-encodings) of $A$ or $A_L$, followed by quantum subroutines for inversion and solution extraction.

2. Quantum Algorithmic Framework and Protocol Architecture

Protocols operate under the quantum coordinator model: multiple parties exchange quantum information with a central referee or distributed controller. Key algorithmic components include:

• Block-Encoding Unitaries: Data blocks at each party are encoded into unitaries $U_{A_i}$ such that $$A_i \approx \alpha \langle 0^{a_i} | U_{A_i} |0^{a_i}\rangle$$ (see block-encoding constructions). Aggregation yields a global block-encoding for $A$ or regularized $A_L$ with norm $\sqrt{\alpha^2 + \lambda \|L\|^2}$.
• Matrix Inversion Subroutines: Quantum algorithms (e.g., Childs-Kothari-Somma, amplitude amplification, quantum signal processing) approximate $A^+$ by simulating Hamiltonian evolution $e^{-iAt}$, followed by quantum phase estimation and amplitude extraction. For regularized protocols, inversion is performed on $A_L$.
• Branch Marking and Gapped Phase Estimation (GPE): Recent protocols (Matsushita, 22 Aug 2025) integrate branch marking (mapping eigenphase sign onto ancilla qubits) and branch-marked GPE, which enables sharp spectral filtering and improved separation of low singular values, facilitating efficient inversion and reducing iterations needed for high-accuracy results.
• Oracle Communication and Aggregation: Each party locally implements oracles accessing $A_i$ and $b_i$. Aggregated quantum operations enable the coordinator to simulate global matrix actions when constructing block-encodings and executing phase estimations.

3. Output Characteristics and Classical Model Synthesis

Unlike earliest quantum regression methods, contemporary distributed quantum least squares protocols produce classical parameter outputs:

• After quantum simulation and amplitude estimation, each component of the regression vector $\beta$ (solution to $A^+ b$ or $(A_L)^+ (b,0)$) is extracted as a classical number, up to precision $\epsilon$.
• Classical outputs enable straightforward aggregation and deployment: model parameters $\beta$ can be transmitted, stored, and directly used for prediction and further data analysis with minimal postprocessing.
• Distributed nodes may independently estimate partial solution components and aggregate results with classical communication.

4. Communication Complexity and Efficiency Enhancements

Quantum protocols achieve marked improvements over classical distributed algorithms in both scaling and communication:

• Logarithmic Data Size Dependence: For estimated solution precision $\epsilon$, communication cost is $O(\log_2(N)/\epsilon)$, compared to classical $O(1/\epsilon^2)$ or $O(N \log(1/\epsilon))$ (Tang et al., 2022).
• Quadratic Improvement in Precision Scaling: Protocols utilizing branch marking and branch-marked GPE reduce the number of digits of precision required for quantum state generation by a quadratic factor, minimizing quantum communication overhead as $\log (1/\epsilon)$ instead of $\log^2 (1/\epsilon)$ (Matsushita, 22 Aug 2025).
• Variable-Time Amplitude Amplification (VTAA): Amplitude amplification steps adapt to eigenvalue regimes, balancing quantum resource usage.

Table: Communication Complexity Comparison

Protocol Variant	Precision Scaling	Data Size Scaling
Classical sampling	$O(1/\epsilon^2)$	$O(N)$
Quantum (standard phase/inner product)	$O(1/\epsilon)$	$O(\log_2(N))$
Quantum (branch-marked GPE)	$O(1/\epsilon)$	$O(\log_2(N))$

5. Robustness to Regularization, Nonsparsity, and Quality Estimation

Distributed quantum least squares protocols accommodate multiple data regimes and model requirements:

• L₂-Regularization: By augmenting $A$ with $\sqrt{\lambda} L$, protocols address Tikhonov-regularization and ridge regression, ensuring well-posedness even for ill-conditioned problems (Matsushita, 22 Aug 2025).
• Nonsparse Matrices: Advanced Hamiltonian simulation and signal processing techniques enable efficient handling of dense, nonsparse data blocks. No sparsity assumption is needed (Wang, 2014).
• Model Quality Estimation: Fast quantum subroutines estimate the fraction of variance explained by the model ($T = \|\text{II}(X)y\|^2/\|y\|^2$), providing a rapid pre-check of data suitability for regression before full solution computation (Wang, 2014).

6. Distribution, Aggregation, and Scalability on Quantum Networks

Distributed protocols are naturally suited to quantum network architectures:

• Local Oracle Execution: Each node executes its own PX and PY oracles, allowing for modular, parallel quantum processing.
• Model Parameter Aggregation: Classical outputs are securely and efficiently aggregated, reducing the requirement for quantum state transmission or state manipulation across nodes.
• Fault Tolerance: Since only classical parameters are collected for deployment, quantum coherence need not be maintained beyond the computation, increasing robustness and lowering operational overhead.
• Network Topology Impact: The protocol can flexibly handle arbitrary partitioning of data blocks, accommodating network heterogeneity.

7. Practical Significance and Application Domains

Distributed quantum least squares protocols have implications across statistical analysis, machine learning, and distributed signal processing:

• Large-Scale Regression Tasks: Protocols are practical for extremely large $N$ and moderate $d$, where classical algorithms are infeasible due to communication or computational cost.
• Secure Multi-Party Learning: Quantum secure aggregation (e.g., via GHZ states and Chinese remainder theorem) enables privacy-preserving federated regression in adversarial settings (Yu et al., 2022).
• High-Dimensional Data Analysis: Exponential speedup with respect to $N$ and quadratic speedup in $d$ allows handling of complex, distributed data without centralization.
• Quality Assessment: Model suitability can be evaluated quantumly prior to deployment, optimizing workflow efficiency.

Summary

Distributed quantum least squares protocols provide a resource-efficient and scalable framework for solving regression problems across quantum networks. By producing classical model outputs, accommodating regularization and nonsparsity, employing advanced quantum signal processing, and minimizing quantum communication complexity, such protocols are positioned for practical implementation in distributed machine learning, large-scale data analytics, and privacy-sensitive collaborative environments. Recent technical advancements ensure high accuracy and efficiency—crucial criteria for next-generation quantum computing applications.