To emulate a quantum computation on a classical computer i.e. the evolution of the unitary operations on the wave function of the particle in quantum mechanics, we have to perform unitary matrix and normalized vector multiplications in the high-level programming languages of Python, C++, Java, etc. Quantum Libraries already available perform the matrix-vector multiplication in the backend using the numpy libraries of Python like Qiskit or use a C++ wrapper to further optimize the runtime it as in Qiskit-Aer Simulators. Since a fully functioning fault-tolerant computer is decades away, it is in our best interest to design new quantum algorithms and develop accelerator test beds for Quantum Emulations. All the quantum computer operations can be emulated on a classical computer, with the only downside being that the matrix multiplications scale up as O (N3) in runtime. In contrast, the quantum computer scales it up as O (N2log2N), where N = 2n, where N is the matrix dimension, where n is the number of qubits, so the runtime for quantum emulations on the classical computer increases exponentially with increase in number of qubits and increases linearly with increase in number of depths, complexity wise. Though it is not possible to change the exponential index, it is possible to reduce the runtime for quantum emulations on classical computers by use of GPU and Alveo Accelerator Cards, and also code optimization on the software side like using a C++ wrapper. In this paper, we will benchmark the matrix-matrix multiplications on HPC Accelerator Cards varying the qubit size and the depth of the quantum circuit and provide a universal mathematical equation for the runtime on the GPU and Alveo Vector Cards for two variables of qubit size and quantum circuit depth. So a theoretical limit on qubit size and qubit depth exactly can be established for quantum emulations on present classical supercomputers.
To emulate a quantum computation on a classical computer i.e. the evolution of the unitary operations on the wave function of the particle in quantum mechanics, we have to perform unitary matrix and normalized vector multiplications in the high-level programming languages of Python, C++, Java, etc. Quantum Libraries already available perform the matrix-vector multiplication in the backend using the numpy libraries of Python like Qiskit or use a C++ wrapper to further optimize the runtime it as in Qiskit-Aer Simulators. Since a fully functioning fault-tolerant computer is decades away, it is in our best interest to design new quantum algorithms and develop accelerator test beds for Quantum Emulations. All the quantum computer operations can be emulated on a classical computer, with the only downside being that the matrix multiplications scale up as O (N3) in runtime. In contrast, the quantum computer scales it up as O (N2log2N), where N = 2n, where N is the matrix dimension, where n is the number of qubits, so the runtime for quantum emulations on the classical computer increases exponentially with increase in number of qubits and increases linearly with increase in number of depths, complexity wise. Though it is not possible to change the exponential index, it is possible to reduce the runtime for quantum emulations on classical computers by use of GPU and Alveo Accelerator Cards, and also code optimization on the software side like using a C++ wrapper. In this paper, we will benchmark the matrix-matrix multiplications on HPC Accelerator Cards varying the qubit size and the depth of the quantum circuit and provide a universal mathematical equation for the runtime on the GPU and Alveo Vector Cards for two variables of qubit size and quantum circuit depth. So a theoretical limit on qubit size and qubit depth exactly can be established for quantum emulations on present classical supercomputers.
As quantum computers of non-trivial size become available in the near future, it is imperative to develop tools to emulate small quantum computers. This allows for validation and debugging of algorithms as well as exploring hardware-software co-design to guide the development of quantum hardware and architectures. The simulation of quantum computers entails multiplications of sparse matrices with very large dense vectors of dimension 2n, where n denotes the number of qubits, making this a memory-bound and network bandwidth-limited application. We introduce the concept of a quantum computer emulator as a component of a software framework for quantum computing, enabling a significant performance advantage over simulators by emulating quantum algorithms at a high level rather than simulating individual gate operations. We describe various optimization approaches and present benchmarking results, establishing the superiority of quantum computer emulators in terms of performance.
Field Programmable Gate Arrays (FPGAs) have been used in most of the physics sub-fields for various unique purposes. This includes particle physics, quantum optics, and, more recently, quantum computing. FPGAs boast many benefits over previous experimental and computational setups. They are versatile, easy to program, and cost-effective, leading to an understandable desire to incorporate them into the new field of quantum computing. While FPGAs have been used to help control the readout and control of physical qubits, they can also be a good tool for improving algorithm simulations, which is the focus of this paper. Different algorithms have different computational limits because of the varying operations and circuit depth levels. Due to the ease of incorporating parallel processes and the ability to optimize FPGAs on a deeper level, we can speed up computationally expensive operations, increasing the size of the quantum circuits we can simulate. This paper aims to provide an educational walkthrough for quantum computing physicists interested in taking advantage of FPGAs in their research. Our findings indicate that an inexpensive FPGA with a clock rate of 100MHz can perform simple, unoptimized matrix operations on the order of 107 ns faster than a CPU using optimized Python libraries. This can be further improved upon using more expensive hardware and optimized designs. This will provide many researchers and students with the tools needed to study quantum algorithms without the need for costly hardware.
We introduce Unitaria, a Python library that brings the simplicity of classical linear algebra toolkits such as NumPy and SciPy to the implementation of quantum algorithms based on block encodings, a general-purpose abstraction in which a matrix is embedded as a sub-block of a larger unitary operator. Their implementation has so far required deep knowledge of low-level circuit construction, which Unitaria aims to eliminate. The library provides a composable, array-like interface through which users can define block encodings of matrices and vectors, combine them through standard operations such as addition, multiplication, tensor products, and the Quantum Singular Value Transformation, and extract the resulting quantum circuits automatically. A key feature is a matrix-arithmetic evaluation path in which every operation can be computed directly on encoded vectors and matrices without dependence on ancilla qubits or circuit simulation. This enables correctness verification and classical simulation that scale well beyond what state vector simulation permits and also allows resource estimation, including gate counts, qubit counts, and normalization constants, without executing any circuit. Together, these capabilities allow researchers to develop, verify, and analyze quantum linear algebra algorithms today, ahead of the availability of error-corrected hardware. Unitaria is open source and available at https://github.com/tequilahub/unitaria.
Evaluating quantum algorithms at utility-scale—involving more than 100 qubits—is a key step toward advancing real-world applications of quantum computing. In this study, we benchmark seven state-of-the-art quantum emulators employing techniques such as tensor networks, matrix product states (MPS), decision diagrams, and factorized ket based methods, running on CPU based hardware and focusing on effectively exact simulations. Performance is assessed on 13 benchmark circuits from the MQTBench library, spanning circuit sizes from 4 to 1,024 qubits. Our results reveal that MPS-based emulators outperform other approaches overall, successfully solving 8 benchmarks up to the maximum size of 1,024 qubits and 12 benchmarks up to at least 100 qubits in less than 5 minutes. We find evidence that all circuits except a random one can be simulated in polynomial time. This work demonstrates that quantum emulators can faithfully simulate a broad range of large and complex universal quantum circuits with high fidelity, far beyond the limits of statevector simulators and today’s quantum hardware.
The increasing complexity of quantum algorithms and the limitations of current Noisy Intermediate-Scale Quantum (NISQ) hardware underscore the importance of efficient classical simulators. To support informed decision-making by users of quantum circuit simulators, we benchmark seven statevector-based quantum circuit simulators (Qiskit, Qulacs, Qibo, Qsimov, Cirq, Pennylane and the Intel Quantum Simulator (IQS)) on a multicore node of the Lusitania high-performance computing (HPC) system. We evaluate their performance in terms of execution time, memory usage and core scalability using Grover’s algorithm, the quantum Fourier transform (QFT), and quantum volume (QV) circuits, across qubit counts ranging from 3 to 30. Our results reveal that Qulacs offers the best performance for circuits below 22 qubits, while Qiskit becomes the fastest for larger and more complex circuits. Qiskit and Qulacs achieve the most efficient parallel performance across multiple cores, while others display limited scaling benefits. IQS shows the lowest memory consumption in QFT and QV benchmarks for systems under 24 qubits; however, it suffers from higher execution times, particularly for Grover’s algorithm. The experimental implementation of Qsimov consistently underperforms in both runtime and scalability; for this reason, it is employed as a baseline in our measurements, serving to highlight the importance of performance optimizations in statevector-based quantum circuit simulators. Previous findings provide a comprehensive performance landscape to guide researchers in selecting appropriate simulators for both standard and large-scale quantum workloads on HPC infrastructures.
Matrix multiplication is a fundamental operation in compute-intensive tasks and a key component of modern quantum acceleration frameworks. Here we present a quantum matrix multiplication algorithm based on quantum kernels (QKMM), achieving an elementary gate complexity of O(N2log2N), with amplitude encoding overhead explicitly included and without assuming a QRAM oracle. This scaling is asymptotically lower than that of the best-known classical matrix multiplication algorithm O(N2.371339). Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V2V), Quantum Vector-Matrix Multiplication (V2M), QKMM (M2M), Quantum One-to-Many Matrix Multiplication(O2M) and Quantum Sequential Matrix Multiplication (SMM), providing a unified framework from vector operations to parallel and sequential matrix transformations. Through noiseless simulations, realistic noise modelling and experiments on a superconducting quantum processor, we systematically characterize the numerical accuracy, resource requirements and hardware execution limits of this operator framework. Furthermore, we integrate SMM into deep neural-network inference, enabling intermediate features to propagate coherently across layers without repeated measurement and re-encoding. These results establish a pathway from quantum circuit-level algorithm design to end-to-end coherent computation, providing a quantum computing framework for matrix-centric compute-intensive applications.