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arXiv:2606.31598

Memory-Scalable and Hardware-Adaptive Matrix-Free Quantum Simulation

Uriel Shafir, Ronnie Kosloff

quant-phSubmitted Jun 30, 2026• Announced Jul 1, 2026• 0 citations
arXivHTMLPDF

Abstract

The core step in quantum simulations is typically matrix vector multiplication $φ= \Hmat ψ$. Executing this step is limited by memory requirement to store the Hamiltonian. We present a memory-scalable, hardware-adaptive matrix-free framework for applying large operators on vectors without materializing the full matrix on a single accelerator. The operator is represented through a block-procedural interface: blocks may be generated, loaded, cached, distributed, or applied directly only when their action is needed. For quantum simulation, it provides the core kernel for quantum operations. An adaptive planner selects block size, cache strategy, GPU grouping, row distribution, and task parallelization from memory and workload estimates. We describe analytic, measured, and learned planning strategies that choose between procedural generation, partial caching, full caching, and row-distributed caching. The method removes the requirement that the full dense matrix fit in the accelerator memory. This shifts large simulations from a fixed memory barrier to a tunable balance between block generation, cache reuse, data movement, parallel scheduling, and numerical accuracy.

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The classical simulation of quantum algorithms is a crucial tool for circuit development, testing, and validation. Although acceleration using GPUs significantly reduces simulation time, most high-performance simulators rely on vendor-specific frameworks that target data-center hardware. To broaden access to quantum simulation, this work proposes a vendor-agnostic approach targeting the integrated GPUs commonly found in consumer-grade laptops. A primary challenge in state-vector simulation is its inherently poor spatial locality, which creates a memory bandwidth bottleneck. Consequently, baseline implementations experience a severe degradation in relative GPU speedup as the number of simulated qubits increases. To address this limitation, we introduce a state partitioning optimization that reorganizes the quantum state vector to maximize the last-level cache locality and minimize costly main memory fetches. We evaluate this strategy using a Quantum Phase Estimation algorithm across diverse architectures from Intel, AMD, and Apple. The experimental results demonstrate that the proposed optimization successfully mitigates performance degradation at larger qubit scales. In particular, for a 28-qubit simulation, the optimization reversed a performance deficit on an Intel Core i5, improving the GPU speedup over the CPU from 0.95x to 1.89x, and increased the Apple M1 Pro speedup from 3.71x to 5.88x. Overall, this approach yields consistent execution time improvements, demonstrating the viability of integrated GPUs for efficient quantum circuit simulation.

quant-pharXiv:2605.15098•May 15, 2026•0 citations
The core step in quantum simulations is typically matrix vector multiplication φ=\Hmatψφ= \Hmat ψφ=\Hmatψ. Executing this step is limited by memory requirement to store the Hamiltonian. We present a memory-scalable, hardware-adaptive matrix-free framework for applying large operators on vectors without materializing the full matrix on a single accelerator.
The operator is represented through a block-procedural interface: blocks may be generated, loaded, cached, distributed, or applied directly only when their action is needed. For quantum simulation, it provides the core kernel for quantum operations. An adaptive planner selects block size, cache strategy, GPU grouping, row distribution, and task parallelization from memory and workload estimates. We describe analytic, measured, and learned planning strategies that choose between procedural generation, partial caching, full caching, and row-distributed caching. The method removes the requirement that the full dense matrix fit in the accelerator memory. This shifts large simulations from a fixed memory barrier to a tunable balance between block generation, cache reuse, data movement, parallel scheduling, and numerical accuracy.
Large-scale quantum circuit simulation on high-performance computing (HPC) systems is crucial for developing and verifying quantum algorithms to overcome the limitations of current noisy quantum computers. However, existing simulators face scalability bottlenecks due to memory limits and communication overhead. Many state-of-the-art approaches rely on static circuit partitioning, which makes it difficult to address the exponential growth in problem size as the number of qubits increases. To overcome these challenges, we present ScaleQsim, a highly scalable quantum circuit simulation framework for large-scale HPC systems. ScaleQsim focuses on providing a unified representation of the full qubit state, yet provides scalability through efficient synchronization and communication. ScaleQsim adopts a novel full state vector partitioning strategy that evenly distributes the full state vector across multiple nodes and GPUs. This distributed structure enables efficient parallel gate execution without costly synchronization, and ScaleQsim applies adaptive kernel configuration, which adjusts execution parameters based on GPU resources and task granularity to enhance simulation efficiency. Our evaluation on the Perlmutter supercomputer with up to 512 GPUs demonstrates that ScaleQsim simulates quantum circuits with up to 42 qubits and outperforms state-of-the-art simulators by up to 77.40× across various quantum circuits.
Classical simulation of quantum operations is essential for algorithm design, noise characterisation, and benchmarking of quantum hardware. The most general physically realisable operation can be described by a positive linear map acting on a hermitian operator, representing either a density matrix or an observable. Established simulators vectorise the density matrix on an nnn-qubit Hilbert space and reuse state-vector kernels, storing all 22n2^{2n}22n elements and forgoing the benefits of hermitian symmetry. In this work, I introduce \emph{Orkan}, a simulation library that uses a tiled memory layout storing only the lower triangle of the hermitian matrix at tile granularity, roughly halving both the memory footprint and the wall time to simulate the evolution of quantum states under generic quantum operations. The implementation treats any hermitian operator uniformly and is agnostic to whether the Schrödinger or Heisenberg picture is used. Dedicated kkk-local conjugation algorithms update all entries of the hermitian matrix in a single pass. Benchmarks against Qiskit Aer, QuEST, and Qulacs show consistent wall-clock speedups of 222-4×4{\times}4× partly attributable to the reduced memory footprint.
Classical computers require large memory resources and computational power to simulate quantum circuits with a large number of qubits. Even supercomputers that can store huge amounts of data face a scalability issue in regard to parallel quantum computing simulations because of the latency of data movements between distributed memory spaces. Here, we apply a cache blocking technique by inserting swap gates in quantum circuits to decrease data movements. We implemented this technique in the open source simulation framework Qiskit Aer. We evaluated our simulator on GPU clusters and observed good scalability.
Matrix multiplication (MatMul) is the computational backbone of modern machine learning, yet its classical complexity remains a bottleneck for large scale data processing. We propose a hybrid quantum classical algorithm for matrix multiplication based on an adaptive configuration of Hadamard tests. By introducing classical memoization that caches state preparation blocks outside the main compilation loop, we reduce the total classical pre processing overhead for all N^2 element circuits to O(N^2). This decouples the heavy gate synthesis overhead from the core quantum processing loop, enabling execution complexities that strictly match classical input/output boundaries. We introduce an "Adaptive Stacking" framework that allows the algorithm to dynamically reconfigure its execution pattern - from sequential horizontal stacking to massive vertical parallelism - based on available qubit resources. This flexibility enables a tunable time complexity range, theoretically reaching O(N^2) on fault tolerant systems while maintaining compatibility with near term hardware. Our core theoretical contribution is the formalization of the "Entropy Dividend": an information theoretic concentration bound proving that the effective measurement variance sigma^2_eff <= (1/S) (1 - (1/2N) e^[H_max - H(psi)]) approaches its maximum. This makes AQ-Stacker uniquely suited for stabilizing the stochastic weight distributions of deep neural networks. We validate the numerical stability of our approach through Quantum Machine Learning (QML) Statevector simulations, achieving 96% accuracy on the MNIST handwritten digit dataset. Our results suggest that entropic noise suppression and parallel Hadamard stacking provide a scalable path toward super-classical efficiency in next-generation quantum-enhanced AI.
In response to the challenges in large-scale quantum state simulation on classical computing platforms, including memory limits, frequent disk I/O, and high computational complexity, this study builds upon a previously proposed hierarchical storage-based quantum simulation system and introduces an optimization framework, the Quantum Vector Optimization Framework (QVecOpt). QVecOpt integrates four strategies: amplitude pairing, cache optimization, block storage optimization, and parallel optimization. These collectively enhance state vector storage and computational scheduling. The amplitude pairing mechanism locates relevant amplitude pairs via bitwise XOR, reducing traversal complexity of single-qubit gates from O(2n)O(2^n)O(2n) to O(1)O(1)O(1). Cache optimization pre-allocates buffers and loads only required data, cutting disk I/O. Block storage optimization partitions the state vector for on-demand loading and local updates, reducing redundant access. Parallel optimization distributes the state vector across nodes for collaborative computation, achieving near-linear speedup. Complexity analysis shows that, compared with hierarchical storage simulation, the method reduces state vector traversals for single-qubit gates from 2n2^n2n to 1, removing the main bottleneck. It also lowers computational and I/O complexity from O(2n)O(2^n)O(2n) to O(2n/C)O(2^n/C)O(2n/C) and O(2n/B)O(2^n/B)O(2n/B). In simulations of 16-29 qubits, efficiency improves nearly tenfold, breaking the memory bottleneck of existing tools and enabling high-bit quantum circuit simulations beyond traditional methods. This work provides an efficient, scalable solution for classical simulation of large-scale quantum computation with significant academic and practical value.
The classical simulation of quantum algorithms is a crucial tool for circuit development, testing, and validation. Although acceleration using GPUs significantly reduces simulation time, most high-performance simulators rely on vendor-specific frameworks that target data-center hardware. To broaden access to quantum simulation, this work proposes a vendor-agnostic approach targeting the integrated GPUs commonly found in consumer-grade laptops. A primary challenge in state-vector simulation is its inherently poor spatial locality, which creates a memory bandwidth bottleneck. Consequently, baseline implementations experience a severe degradation in relative GPU speedup as the number of simulated qubits increases. To address this limitation, we introduce a state partitioning optimization that reorganizes the quantum state vector to maximize the last-level cache locality and minimize costly main memory fetches. We evaluate this strategy using a Quantum Phase Estimation algorithm across diverse architectures from Intel, AMD, and Apple. The experimental results demonstrate that the proposed optimization successfully mitigates performance degradation at larger qubit scales. In particular, for a 28-qubit simulation, the optimization reversed a performance deficit on an Intel Core i5, improving the GPU speedup over the CPU from 0.95x to 1.89x, and increased the Apple M1 Pro speedup from 3.71x to 5.88x. Overall, this approach yields consistent execution time improvements, demonstrating the viability of integrated GPUs for efficient quantum circuit simulation.