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Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

In this paper, we propose a novel protection scheme for N-qubit Greenberger–Horne–Zeilinger (GHZ) states against amplitude damping noise, using unilateral operations. The key innovation lies in the implementation of local operations on a single qubit, irrespective of the total number of qubits, thereby significantly reducing the operational complexity and resource requirements compared to existing methods. The scheme involves a unilateral rotation to transform the GHZ state into a less vulnerable configuration before exposing it to noise, followed by a weak measurement and another unilateral rotation to recover the initial state. We provide a comprehensive mathematical analysis of the proposed unilateral protection scheme, deriving expressions for both fidelity and success probability. The practical feasibility of our scheme is demonstrated through simulations using the Qiskit SDK, in which we provide the corresponding quantum circuit for performing the partial weak measurement used in our scheme. Comparative analysis with pioneer protection schemes reveals a substantial enhancement in success probability and fidelity for both GHZ and generalized GHZ states, emphasizing the superiority of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract Quantum computing has already demonstrated great computational potential across multiple domains and has received more and more attention. However, due to the connectivity limitations of Noisy Intermediate-Scale Quantum (NISQ) devices, most of the quantum algorithms cannot be directly executed without the help of inserting SWAP gates. Nevertheless, more SWAP gates lead to a longer execution time and, inevitably, lower fidelity of the algorithm. To this end, this paper proposes an optimized qubit mapping algorithm based on a dynamic look-ahead strategy to minimize the number of SWAP gates inserted. Firstly, a heuristic algorithm is proposed based on maximizing physical qubit connectivity to generate the optimal initial qubit mapping, which reduces the need for logical qubit shifts during subsequent SWAP gate insertion. Secondly, in the form of directed acyclic graphs, we identify quantum gates that violate the constraints of physical coupling and insert SWAP gates to remap qubits, thereby overcoming the limitations of qubit interactions. Finally, the optimal SWAP gate insertion strategy is built by comparing the cost of different SWAP gate insertion strategies through a multi-window look-ahead strategy to reduce the number of SWAP gates inserted. The experimental results show that the strategy in this paper decreases the number of SWAP gate insertions and significantly reduces the depth of quantum circuits when performing qubit mapping compared with state-of-the-art methods.

Quantum computing has already demonstrated great computational potential across multiple domains and has received more and more attention. However, due to the connectivity limitations of Noisy Intermediate-Scale Quantum (NISQ) devices, most of the quantum algorithms cannot be directly executed without the help of inserting SWAP gates. Nevertheless, more SWAP gates lead to a longer execution time and, inevitably, lower fidelity of the algorithm. To this end, this paper proposes an optimized qubit mapping algorithm based on a dynamic look-ahead strategy to minimize the number of SWAP gates inserted. Firstly, a heuristic algorithm is proposed based on maximizing physical qubit connectivity to generate the optimal initial qubit mapping, which reduces the need for logical qubit shifts during subsequent SWAP gate insertion. Secondly, in the form of directed acyclic graphs, we identify quantum gates that violate the constraints of physical coupling and insert SWAP gates to remap qubits, thereby overcoming the limitations of qubit interactions. Finally, the optimal SWAP gate insertion strategy is built by comparing the cost of different SWAP gate insertion strategies through a multi-window look-ahead strategy to reduce the number of SWAP gates inserted. The experimental results show that the strategy in this paper decreases the number of SWAP gate insertions and significantly reduces the depth of quantum circuits when performing qubit mapping compared with state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Contemporary industry and science expectations towards technological solutions set the bar high. Current approaches to increasing the computing power of standard systems are reaching the limits of physics known to humankind. Fast, programmable systems with relatively low power consumption are a different concept for performing complex calculations. Highly parallel processing opens up a number of possibilities in the context of accelerating calculations. Application of SoC (System On Chip) with FPGA (Field-Programmable Gate Array) enables the delegating of a part of computations to the gates matrix, thereby expediting processing by using parallelization of hardware operations. This paper presents the general concept of using SoC FPGA systems to support the CPU (Central Processing Unit) in many modern tasks. While some tasks might be really hard to implement on an FPGA in a reasonable time, the SoC FPGA platform allows for easy low-level interconnections, and with such virtualized access to the hardware computing resources, it is seen as making FPGAs, or hardware in general, more accessible to engineers accustomed to high-level solutions. The concept presented in the article takes into account the limited resources of cheaper educational platforms, which, however, still provide an interesting and alternative hybrid solution to the problem of parallelization and acceleration of data processing. This allows encountered limitations to be overcome and the flexibility known from high-level solutions and high performance achieved with low-level programming to be maintained without the need for a high financial background. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the quantum circuit implementations of the lightweight authenticated encryption algorithm ASCON by using the NOT gates, CNOT gates, Toffoli gates, measurements, and the dynamic quantum circuits. Firstly, the quantum circuit of addition of constants is realized by adding the NOT gates according to the position of 1 in round constants. Secondly, the quantum circuit of S-box of the permutation is synthesized according to the classical circuit diagram of S-box. Then the linear layer functions are expressed in matrix form, and their quantum circuits are synthesized according to Gaussian elimination. Finally, we synthesize the whole quantum circuits according to the general diagrams of the authenticated encryption algorithm ASCON. The correctness of the quantum circuits of the S-box and the linear layer was verified by the Aer simulator of the IBM Quantum platform. As far as we know, this is the first implementation of the quantum circuits for the Authenticated Encryption with Associated Data (AEAD) of ASCON in-place. The maximum quantum resources for the three ASCON authenticated encryption algorithms were estimated. The quantum circuit of ASCON-128 uses a total of 320 qubits, 30,639 NOT gates, 128,814 CNOT gates, 8064 Toffoli gates, 10,752 measurements, and 5376 dynamic quantum circuits. The quantum circuit of ASCON-128a uses a total of 320 qubits, 23,558 NOT gates, 98,144 CNOT gates, 6144 Toffoli gates, 8192 measurements, and 4096 dynamic quantum circuits. The quantum circuit of ASCON-80pq uses a total of 320 qubits, 30,736 NOT gates, 128,814 CNOT gates, 8064 Toffoli gates, 10,752 measurements, and 5376 dynamic quantum circuits. [ABSTRACT FROM AUTHOR]

Quantum neural networks (QNNs) leverage the strengths of both quantum computing and neural networks, offering solutions to challenges that are often beyond the reach of traditional neural networks. QNNs are being used in areas such as computer games, function approximation, and big data processing. Moreover, quantum neural network algorithms are finding utility in social network modeling, associative memory systems, and automatic control mechanisms. Nevertheless, ensuring the reliability of quantum neural networks is crucial as it directly influences network performance and stability. To investigate the reliability of quantum neural networks, this paper proposes a methodology wherein operator measurements are performed on the final states of the output quantum states of a quantum neural network. The proximity of these measurements to the target value is compared, and the fidelity value, combined with a quantum gate operation, is utilized to assess the reliability of the quantum neural network. Through network training, the results demonstrate that, under optimal parameters, both the fidelity of the final state measurement value and the target value of the model approach are approximately equal to 1. It indicates that training mitigates the errors stemming from encoding into the initial quantum state, thereby resulting in enhanced system reliability and accuracy. [ABSTRACT FROM AUTHOR]

Quantum circuit optimization is an inevitable task with the current noisy quantum backends. This task is considered non-trivial due to the varying circuits' complexities in addition to hardware-specific noise, topology, and limited connectivity. The currently available methods either rely on heuristics for circuit optimization tasks or reinforcement learning with complex unscalable neural networks such as transformers. In this paper, we are concerned with optimizing the initial logical-to-physical mapping selection. Specifically, we investigate whether a reinforcement learning agent with simple scalable neural network is capable of finding a near-optimal logical-to-physical mapping, that would decrease as much as possible additional CNOT gates, only from a fixed-length feature vector. To answer this question, we train a Maskable Proximal Policy Optimization agent to progressively take steps towards a near-optimal logical-to-physical mapping on a 20-qubit hardware architecture. Our results show that our agent coupled with a simple routing evaluation is capable of outperforming other available reinforcement learning and heuristics approaches on 12 out of 19 test benchmarks, achieving geometric mean improvements of 2.2% and 15% over the best available related work and two heuristics approaches, respectively. Additionally, our neural network model scales linearly as the number of qubits increases. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to optimizing the linear layer of block ciphers using the matrix decomposition framework. It is observed that the reduction properties proposed by Xiang et al. (in FSE 2020) need to be improved. To address these limitations, we propose a new reduction framework with a complete reduction algorithm and swapping algorithm. Our approach formulates matrix decomposition as a new framework with an adaptive objective function and converts the problem to a Graph Isomorphism problem (GI problem). Using the new reduction algorithm, we were able to achieve lower XOR counts and depths of quantum implementations under the s-XOR metric. Our results outperform previous works for many linear layers of block ciphers and hash functions; some of them are better than the current g-XOR implementation. For the AES MixColumn operation, we get two implementations with 91 XOR counts and depth 13 of in-place quantum implementation, respectively.

We propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.

In this article, we present a hybrid quantum-classical generative adversarial network (GAN) for near-term quantum processors. The hybrid GAN comprises a variational generator and a discriminator quantum neural network, which are trained using a classic computer. The generator network is realized using angle-encoding and variational quantum circuits. The discriminator network is realized using multi-stage trainable encoding quantum circuits. A modular design approach is proposed for quantum neural networks which allows control on their depth to compromise accuracy and circuit complexity. Moreover, this modular approach makes the quantum neural networks amenable to scaling up the dimension of dataset and the number of qubits. Gradients of the loss functions are derived using the same quantum circuits used for the implementation of quantum neural networks to prevent the need for extra quantum circuits or auxiliary qubits. The quantum simulations are performed using the IBM Qiskit open-source software development kit (SDK), while the training of the hybrid quantum-classical GAN is conducted using the mini-batch stochastic gradient descent (SGD) optimization on a classic computer. The hybrid quantum-classical GAN is realized using a two-qubit system with different discriminator network structures. The best performance is achieved using a five-stage discriminator network comprising 63 quantum gates and 31 trainable parameters, with the Kullback-Leibler (KL) and Jensen-Shannon (JS) divergence scores of 0.39 and 0.52, respectively, for the similarity between the real and generated data distributions.

Exploiting machine learning techniques to automatically classify multispectral remote sensing imagery plays a significant role in deriving changes on the Earth’s surface. However, the computation power required to manage large Earth observation data and apply sophisticated machine learning models for this analysis purpose has become an intractable bottleneck. Leveraging quantum computing provides a possibility to tackle this challenge in the future. This article focuses on land cover classification by analyzing Sentinel-2 images with quantum computing. Two hybrid quantum-classical deep learning frameworks are proposed. Both models exploit quantum computing to extract features efficiently from multispectral images and classical computing for final classification. As proof of concept, numerical simulation results on the LCZ42 dataset through the TensorFlow Quantum platform verify our models' validity. The experiments indicate that our models can extract features more effectively compared with their classical counterparts, specifically, the convolutional neural network (CNN) model. Our models demonstrated improvements, with an average test accuracy increase of 4.5% and 3.3%, respectively, in comparison to the CNN model. In addition, our proposed models exhibit better transferability and robustness than CNN models.

A common model for classifying images is the convolutional neural network (CNN), which has the benefit of effectively using data correlation information. Despite their remarkable success, classical CNNs may face challenges in achieving further improvements in accuracy, computational efficiency, explainability, and generalization. However, if the specified data dimension or model grows too large, CNN becomes difficult to train effectively with a slowdown processing. In order to address a problem using CNN utilizing quantum computing, Quantum Convolutional Neural Network (QCNN) proposes a novel quantum solution or enhances the functionality of an existing learning model in terms of processing time during training. This paper presents a comparative analysis between classical Convolutional Neural Networks (CNNs) and a novel quantum circuit architecture tailored for image-based tasks, emphasizing the adaptability and versatility of quantum circuits in enhancing feature extraction capabilities and then final accuracy and processing time. A MNIST and covidx-cxr3 datasets was used to train quantum-CNN models, and the results of these comparisons were made with traditional CNN performance. The results demonstrate that the suggested QCNN beat the traditional CNN in terms of recognition accuracy and processing speed (process time) when combined with cutting-edge feature extraction techniques. This superiority is particularly evident when trained on the covidx-cxr3 dataset, highlighting the potential for quantum computing to revolutionize image classification tasks.

Multiobjective optimization is a ubiquitous problem that arises naturally in many scientific and industrial areas. Network routing optimization with multiobjective performance demands falls into this problem class, and finding good quality solutions at large scales is generally challenging. In this work, we develop a scheme with which near-term quantum computers can be applied to solve multiobjective combinatorial optimization problems. We study the application of this scheme to the network routing problem in detail, by first mapping it to the multiobjective shortest-path problem. Focusing on an implementation based on the quantum approximate optimization algorithm (QAOA)—the go-to approach for tackling optimization problems on near-term quantum computers—we examine the Pareto plot that results from the scheme and qualitatively analyze its ability to produce Pareto-optimal solutions. We further provide theoretical and numerical scaling analyses of the resource requirements and performance of QAOA and identify key challenges associated with this approach. Finally, through Amazon Braket, we execute small-scale implementations of our scheme on the IonQ Harmony 11-qubit quantum computer.

Cardiac pathology classification (CPC) based on the volumetric features of three key heart structures can be extracted from segmented cardiac cine magnetic resonance imaging (CMRI) sequences. Machine learning models have recently become very effective tools for handling these problems. Hybrid quantum methods can be employed to enhance the capacities of classical machine learning models. Here, we propose a hybrid quantum multiclass cardiac pathologies classification (HQMC-CPC) model. In the proposed model, discriminative features are extracted to characterize the shape and function of the heart, combining the clinical features, patient features, and radiomics features. In the proposed quantum circuits, all variational parameters are trainable, and the enhanced variational quantum circuit is employed for efficient neural network learning. Using only thirty feature values as input, we propose a hybrid quantum multiclass cardiac pathologies classification (HQMC-CPC) model based on the proposed modified hardware efficient ansatz (MHEA). The proposed model achieves promising results in training and testing with the Automatic Cardiac Diagnosis Challenge (ACDC 2017) dataset. Experimental results showed that the proposed HQMC-CPC model is able to classify different cardiac pathologies with an average minimum performance gap of 3.19%. The average maximum improvement in terms of accuracy in cardiac pathology classification is 7.77%. Moreover, the proposed HQMC-CPC speeds up the testing process by around 60% and 40% compared to the classical classifiers and well-established HEA respectively.

A limited number of qubits, high error rates, and limited qubit connectivity are major challenges for effective near-term quantum computations. Quantum circuit partitioning divides a quantum computation into classical postprocessing steps and a set of smaller scale quantum computations that individually require fewer qubits, lower qubit connectivity, and typically incur less error. However, as partitioning generally increases the duration of a quantum computation exponentially in the required partitioning effort, it is crucial to select optimal partitioning points, so-called cuts, and to use optimal cut realizations. In this work, we develop the first optimal partitioning method relying on quantum circuit knitting for optimal cut realizations and an optimal selection of wire cuts and gate cuts that trades off ancilla qubit insertions for a decrease in quantum computing time. Using this combination, the developed method demonstrates a reduction in quantum computing runtime by 41% on average compared to previous quantum circuit partitioning methods. Furthermore, the qubit requirement of the evaluated quantum circuits was reduced by 40% on average for a runtime budget of one hour and a sampling frequency of 1 kHz. These results highlight the optimality gap of previous quantum circuit partitioning methods and the possible extension in the computational reach of near-term quantum computers.

A quantum Pauli noise is a nonunitary process that alters the state of a qubit by random application of the four Pauli operators. We investigate a four-qubit quantum circuit, consisting of a pipeline of three independently controlled Pauli gates, for simulating the general class of qubit Pauli noises. The circuit with a fixed architecture is controllable by three separable quantum states from three auxiliary qubits in order to adjust the parameters of the targeted Pauli noise on the principal qubit. Important Pauli noises such as bit flip, phase flip, bit phase flip, and depolarizing noise are readily simulated, along with an infinite subset of other Pauli noises. However, the quantum circuit with its simple and fixed architecture cannot simulate all conceivable Pauli noises, and a characterization is proposed, in the parameter space of the Pauli noises, denoting those that are simulable by the circuit and those that are not. The circuit is a useful tool to contribute to controlled simulation, on current or future quantum processors, of nonunitary processes of noise and decoherence. [ABSTRACT FROM AUTHOR]

Noisy Intermediate-Scale Quantum (NISQ) computers consisting of tens of inherently noisy quantum bits (qubits) suffer from reliability problems. Qubits and their gates are susceptible to various types of errors. Due to limited numbers of qubits and high error rates, quantum error correction cannot be applied. Physical constraints of quantum hardware including the error rates are used to guide the design and the layout of quantum circuits. The error rates determine the selection of qubits and their operations. The resulting circuit is executed on the quantum computer.This study explores the risk of unexpected changes in the error rates of NISQ computers post-calibration. We show that unexpected changes in error rates can alter the output state of a quantum circuit. To detect these changes, we propose the insertion of test points into the quantum circuit to enable online monitoring of the physical qubit behavior. We utilize classical, superposition, and uncompute test points. Furthermore, we use a gate error coverage metric to assess the quality of the tests. We verify the effectiveness of the proposed scheme on different IBM quantum computers (IBM Q), in addition to a noisy simulation that shows the scalability of the proposed approach.

Having access to a unique quantum circuit that one wants to protect against use by others is a very likely scenario in industrial computing. However, currently, users rely on classical computer security schemes, which have known shortcomings. In this paper, we introduce a novel protection scheme along with a survey of other known methods to protect quantum information. In particular, we review physically unclonable functions (PUFs), obfuscation, and introduce quantum logic locking (QLL). The latter technique provisions end users to protect their circuit from an adversary through the use of a secret key.

Quantum Fisher information matrix (QFIM) is a fundamental quantity in quantum physics, which closely links to diverse fields such as quantum metrology, phase transitions, entanglement witness, and quantum speed limit. It is crucial in quantum parameter estimation, central to the ultimate Cramér-Rao bound. Recently, the evaluation of QFIM using quantum circuit algorithms has been proposed for systems with multiplicative parameters Hamiltonian. However, systems with generic Hamiltonians still lack these proposed schemes. This work introduces a quantum-circuit-based approach for evaluating QFIM with generic Hamiltonians. We present a time-dependent stochastic parameter-shift rule for the derivatives of evolved quantum states, whereby the QFIM can be obtained. The scheme can be executed in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we demonstrate the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from a standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying QFIM with generic Hamiltonians using quantum circuit algorithms. [ABSTRACT FROM AUTHOR]

The emergence of quantum computers threatens current cryptographic systems, and NIST is preparing for the post-quantum era through the post-quantum cryptography (PQC) contest. CRYSTALS-Kyber is a lattice-based cipher suite that is used as a PQC standard. Lattice-based cryptography is considered quantum-safe for quantum computing because a quantum algorithm that can more efficiently solve the lattice problem of lattice-based cryptography compared to a classic algorithm has not been reported as yet. In this paper, we present quantum circuits tailored to NTT and inverse NTT, employed for optimized polynomial multiplication within CRYSTALS-Kyber. The proposed quantum circuits operate at Z 3329 [ X ] /( X 256 +1), which are the parameters of CRYSTALS-Kyber. We provide an in-depth description of the NTT/InvNTT quantum circuit's operation and subsequently assess and analyze the quantum resources necessary for these functions. The NTT/InvNTT quantum circuits comprise four unique sub-functions, with the InvNTT additionally incorporating Barrett reduction. To the best of our knowledge, this represents the inaugural implementation of the CRYSTALS-Kyber NTT/InvNTT quantum circuits. We anticipate that our findings will aid in analyzing the security strengths of quantum computers for lattice-based cryptography. [ABSTRACT FROM AUTHOR]

Having access to a unique quantum circuit that one wants to protect against use by others is a very likely scenario in industrial computing. However, currently, users rely on classical computer security schemes, which have known shortcomings. In this paper, we introduce a novel protection scheme along with a survey of other known methods to protect quantum information. In particular, we review physically unclonable functions (PUFs), obfuscation, and introduce quantum logic locking (QLL). The latter technique provisions end users to protect their circuit from an adversary through the use of a secret key. [ABSTRACT FROM AUTHOR]

This paper presents an innovative way of quantum circuit optimization; we propose an automated superconducting circuit calibration using deep reinforcement learning (ASCC-DRL). A simplified model of a transmon qubit coupled to a cavity resonator is used to demonstrate a quantum circuit. We analyzed the circuit through its equivalent two-mode lumped-element representation of the distributed circuit and derived the equivalent Hamiltonian translation of the circuit with the non-linearity of the Josephson Junctions. We adopted the parametrized qubit junctions and used the component physical value to tune the circuits to maximize the fidelity of the desired entangle state and the resulting quantum circuit state. However, quantum systems are susceptible to environmental parasitics and noise; therefore, a reliable optimization technique is essential to maintain high fidelity across the system. For optimization, we integrate deep reinforcement learning using deep deterministic policy gradient techniques as our mechanism for our AI module to optimize the fidelity of the superconducting quantum circuit. We demonstrate the feasibility of this approach by training the agent in three distinct environments to achieve the highest possible best rewards. We achieved an average fidelity (best rewards) of 0.92, which is 94% of the statically tuned circuit.

The creation of quantum circuit architecture based on the quantum perceptron algorithm to classify marketing bank data is proposed in this work. A quantum circuit is a quantum gate made up of two quantum gates. Quantum bits are used in this study's computation. The primary proposed learning method was not ideal, which is the context of this study. The percentage of qubits measurement value is still 90.7 percent. It is essential to raise the value of the qubit rate. Using the IBM Quantum Experience quantum computer, researchers measured, trained, and tested the quantum circuit architecture. Bank marketing data from the UCI Machine Learning Repository was used. A quantum circuit architecture model results from this research the quantum circuit measurement results.

Current quantum computers have a limited number of resources and are heavily affected by internal and external noise. Therefore, small, noise-tolerant circuits are of great interest. With regard to circuit size, it is especially important to reduce the number of required qubits. Concerning to fault-tolerance, circuits entirely built with Clifford+T gates allow the use of error correction codes. However, the T-gate has an excessive cost, so circuits with a high number of T-gates should be avoided. This work focuses on optimising in such terms an operation that is widely used in larger circuits and algorithms: the calculation of the absolute-value of two's complement encoded integers. The proposed circuit halves the number of required T gates with respect to the best circuit currently available in the literature. Moreover, our circuit requires at least 2 qubits less than the other circuits for such an operation. [ABSTRACT FROM AUTHOR]

Crosstalk is the primary source of noise in quantum computing equipment. The parallel execution of multiple instructions in quantum computation causes crosstalk, which causes coupling between signal lines and mutual inductance and capacitance between signal lines, destroying the quantum state and causing the program to fail to execute correctly. Overcoming crosstalk is a critical prerequisite for quantum error correction and large-scale fault-tolerant quantum computing. This paper provides an approach for suppressing crosstalk in quantum computers based on multiple instruction exchange rules and duration. Firstly, for the majority of the quantum gates that can be executed on quantum computing devices, a multiple instruction exchange rule is proposed. The multiple instruction exchange rule reorders quantum gates in quantum circuits and separates double quantum gates with high crosstalk on quantum circuits. Then, time stakes are inserted based on the duration of different quantum gates, and quantum gates with high crosstalk are carefully separated in the process of quantum circuit execution by quantum computing equipment to reduce the influence of crosstalk on circuit fidelity. Several benchmark experiments verify the proposed method's effectiveness. In comparison to previous techniques, the proposed method improves fidelity by 15.97% on average. [ABSTRACT FROM AUTHOR]

Random numbers are of vital importance in fields such as cyptography and scientific simulations. However, it is well known how difficult it is for classical computers to generate random numbers. This is not the case for quantum computers, which are able to genuinely generate random numbers thanks to the property of superposition and their counter-intuitive concept of measurement. However, despite the simplicity of designing a circuit that generates a random number between 0 and 2 N − 1 (being N the number of available qubits), designing a quantum circuit to generate a number within a specific interval is far from trivial. This paper proposes a customizable circuit design to generate random numbers. The circuit is non- hardware dependent, it allows fault-tolerance, and it can be used by current quantum devices. Therefore, it is a valuable tool for all those quantum applications and algorithms that need to work with random numbers. Moreover, a comparator circuit has also been designed as part of this work. This comparator is the best currently available in the literature in terms of qubits, T-count, and T-depth. It is therefore a useful tool for any other circuit or algorithm where this operation is needed. [ABSTRACT FROM AUTHOR]

Sentiment classification is a valuable application of natural language processing that has seen wide usage in optimizing business processes. This paper explores a novel implementation of sentiment analysis using the Variational Quantum Algorithms (VQA) framework. As ansatz choice determines model performance in VQA, this paper proposes an alternative ansatz for the sentiment classification task in quantum representation. Specifically, it builds upon previous work in quantum sentiment classification by proposing an alternative ansatz to the Instantaneous Quantum Polytime ansatz, entitled Simple Sentiment Analysis (SimpleSA) ansatz. A key feature of the SimpleSA ansatz is the decision to neglect noun parameterization. The proposed SimpleSA has less complexity than the other ansätze in terms of the number of parameters and the number of gates. Moreover, experimental results show that the SimpleSA ansatz with H-CNOT-Rz-H compound block construction outperforms the Instantaneous Quantum Polytime (IQP) ansatz at 85.00% accuracy. Furthermore, SimpleSA optimization converges 20.89% faster than Instantaneous Quantum Polytime (IQP) for the Simultaneous Perturbation Stochastic Approximation (SPSA) method with 130 iterations. The proposed work is useful for applications of quantum computers for sentiment analyses and classifications.

Quantum machine learning (QML) has received increasing attention due to its potential to outperform classical machine learning methods in problems, such as classification and identification tasks. A subclass of QML methods is quantum generative adversarial networks (QGANs), which have been studied as a quantum counterpart of classical GANs widely used in image manipulation and generation tasks. The existing work on QGANs is still limited to small-scale proof-of-concept examples based on images with significant downscaling. Here, we integrate classical and quantum techniques to propose a new hybrid quantum–classical GAN framework. We demonstrate its superior learning capabilities over existing quantum techniques by generating $28 \times 28$ pixels grayscale images without dimensionality reduction or classical pre/postprocessing on multiple classes of the standard Modified National Institute of Standards and Technology (MNIST) and Fashion MNIST datasets, which achieves comparable results to classical frameworks with three orders of magnitude less trainable generator parameters. To gain further insight into the working of our hybrid approach, we systematically explore the impact of its parameter space by varying the number of qubits, the size of image patches, the number of layers in the generator, the shape of the patches, and the choice of prior distribution. Our results show that increasing the quantum generator size generally improves the learning capability of the network. The developed framework provides a foundation for future design of QGANs with optimal parameter set tailored for complex image generation tasks.

Variational quantum algorithms, which utilize parameterized quantum circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parameterized rotational angles of single-qubit gates around a predetermined set of axes and two-qubit entangling gates, such as cnot gates. We propose a method to construct a PQC by continuous parameterization of both the angles and the axes of its single-qubit rotation gates. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed to $\pi$. We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback–Leibler divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Hamiltonians for condensed matter physics, quantum chemistry, and combinatorial optimization. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.

Works on quantum computing and cryptanalysis have increased significantly in the past few years. Various constructions of quantum arithmetic circuits as one of the primary elements in the field have also been proposed. However, there have only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor’s algorithm for Elliptic Curve Discrete Logarithm Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat’s Little Theorem (FLT)-based inversion circuit for the binary finite field. In particular, we propose to follow a complete waterfall approach to translate the Itoh-Tsujii’s variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (i.e., CNOT count) as well as slightly reducing the $T$ depth, which contributes to a reduced overall depth and gate count. Furthermore, we concretely verify our method and compare it with the previous work in Qiskit, a quantum computer simulation environment, by constructing both our method and the previous work from scratch and performing the resource analysis. Additionally, we propose employing the relative-phase Toffoli gate by Gidney as opposed to the standard Toffoli implementation, which yields a significantly lower $T$ depth while further reducing the overall depth. Our approach can serve as an alternative for a time-efficient implementation.

The negative effects caused by geometric distortion can be removed to the maximum extent possible through appropriate geometric transformations, allowing us to focus on the image content itself in subsequent processing and recognition. Therefore, geometric transformations are often used as a pre-processing step for other image processing applications. In this paper, quantum algorithms are designed to implement geometric transformations, including two-point swapping, circular translation, flipping transformations and right-angle rotation, across a quantum image representation QIRHSI (Quantum Image Representation based on HSI color space) which is built on HSI (Hue-Saturation-Intensity) color space. The above geometric transformations are realized by quantum circuits composed of elementary quantum gates. By analyzing the complexity of the fundamental quantum gates needed for the above geometric transformations, it is found that the global transformations (circular translation, flip transformation and right-angle rotation) are lower than the local transformation (two-point swapping). The proposed geometric transforms are used to facilitate the applications of quantum images with low complexity and high efficiency.

The unraveling of open quantum system dynamics in terms of stochastic quantum trajectories offers a picture of open system dynamics that consistently considers memory effects stemming from the finite correlation time of environment fluctuations. These fluctuations significantly influence the coherence and energy transport properties of excitonic systems. When their correlation time is comparable to the timescale of the Hamiltonian evolution, it leads to the departure of open system dynamics from the Markovian limit. In this work, we leverage the unraveling of exciton dynamics through stochastic Hamiltonian propagators to design quantum circuits that simulate exciton transport, capturing finite memory effects. In addition to enabling the synthesis of parametrizable quantum circuits, stochastic unitary propagators provide a transparent framework for investigating non-Markovian effects on exciton transport. Our analysis reveals a nuanced relationship between environment correlation time and transport efficiency, identifying a regime of ‘memory-assisted’ quantum transport where time-correlated fluctuations allow the system to reach higher efficiency. However, this property is not universal and can only be realized in conjunction with specific features of the system Hamiltonian.

Optimizing the quantum circuit for implementing Advanced Encryption Standard (AES) is crucial for estimating the necessary resources in attacking AES by the Grover algorithm. Previous studies have reduced the number of qubits required for the quantum circuits of AES-128/-192/-256 from 984/1112/1336 to 270/334/398, which is close to the optimal value of 256/320/384. It becomes a challenging task to further optimize them. AimTaking aim at this task, we find a method for how the quantum circuit of AES S-box can be designed with the help of the automation tool LIGHTER-R. Particularly, the multiplicative inversion in F28, which is the main part of the S-box, is converted into the multiplicative inversion (and multiplication) in F24, then the latter can be implemented by LIGHTER-R because its search space is small enough. By this method, we construct the quantum circuits of S-box for mapping |a⟩|0⟩ to |a⟩|S(a)⟩ and |a⟩|b⟩ to |a⟩|b ⊕ S(a)⟩ with 20 qubits instead of 22 in the previous studies. In addition, we introduce new techniques to reduce the number of qubits required by the S-box circuit for mapping |a⟩ to |S(a)⟩ from 22 in the previous studies to 16. Accordingly, we synthesize the quantum circuits of AES-128/-192/-256 with 264/328/392 qubits, which implies a new record.

The classical neural network has provided remarkable results to diagnose neurological disorders against neuroimaging data. However, in terms of efficient and accurate classification, some standpoints need to be improved by utilizing high-speed computing tools. By integrating quantum computing phenomena with deep neural network approaches, this study proposes an AlexNet–quantum transfer learning method to diagnose neurodegenerative diseases using magnetic resonance imaging (MRI) dataset. The hybrid model is constructed by extracting an informative feature vector from high-dimensional data using a classical pre-trained AlexNet model and further feeding this network to a quantum variational circuit (QVC). Quantum circuit leverages quantum computing phenomena, quantum bits, and different quantum gates such as Hadamard and CNOT gate for transformation. The classical pre-trained model extracts the 4096 features from the MRI dataset by using AlexNet architecture and gives this vector as input to the quantum circuit. QVC generates a 4-dimensional vector and to transform this vector into a 2-dimensional vector, a fully connected layer is connected at the end to perform the binary classification task for a brain disorder. Furthermore, the classical–quantum model employs the quantum depth of six layers on pennyLane quantum simulators, presenting the classification accuracy of 97% for Parkinson's disease (PD) and 96% for Alzheimer's disease (AD) for 25 epochs. Besides this, pre-trained classical neural models are implemented for the classification of disorder and then, we compare the performance of the classical transfer learning model and hybrid classical–quantum transfer learning model. This comparison shows that the AlexNet–quantum learning model achieves beneficial results for classifying PD and AD. So, this work leverages the high-speed computational power using deep network learning and quantum circuit learning to offer insight into the practical application of quantum computers that speed up the performance of the model on real-world data in the healthcare domain. [ABSTRACT FROM AUTHOR]

Josephson junctions (JJs) with Josephson energy E J ≲ 1 K are widely employed as non-linear elements in superconducting circuits for quantum computing operating at milli-Kelvin temperatures. In the qubits with small charging energy E C ( E J / E C ≫ 1 ), such as the transmon, the incoherent phase slips (IPS) might become the dominant source of dissipation with decreasing E J . In this work, a systematic study of the IPS in low- E J JJs at milli-Kelvin temperatures is reported. Strong suppression of the critical (switching) current and a very rapid growth of the zero-bias resistance due to the IPS are observed with decreasing E J below 1 K. With further improvement of coherence of superconducting qubits, the observed IPS-induced dissipation might limit the performance of qubits based on low- E J junctions. These results point the way to future improvements of such qubits. [ABSTRACT FROM AUTHOR]

Topological quantum error correction (TQEC) is promising for scalable fault-tolerant quantum computation. The required resource of a TQEC circuit can be modeled as its space-time volume of a three-dimensional geometric description. Implementing a quantum algorithm with a reasonable physical qubit number and computation time is challenging for large-scale complex problems. Therefore, it is desirable to minimize the space-time volume for large-scale TQEC circuits. Previous work proposed bridge compression, which can significantly compress a TQEC circuit, but it was performed manually. This article presents the first automated tool that can perform bridge compression on a large-scale TQEC circuit. Our proposed algorithm applies the bridge compression technique to compactify TQEC circuits with modularization. Besides, we offer a time-ordering-aware 2.5-D placement for compacting TQEC circuits and satisfying time-ordered measurement constraints. On the other hand, we suggest friend net-aware routing to effectively reduce the required routing resource under topological deformation. Compared with the state-of-the-art work, experimental results show that our proposed algorithm can averagely reduce space-time volumes by 84%. [ABSTRACT FROM AUTHOR]

Current quantum processing technology is generally noisy with a limited number of qubits, stressing the importance of quantum state fidelity estimation. The complexity of this problem is mainly due to not only accounting for single gates and readout errors but also for interactions among which. Existing methods generally rely on either reconstructing the given circuit state, ideal state, and computing the distance of which; or forcing the system to be on a specific state. Both rely on conducting circuit measurements, in which computational efficiency is traded off with obtained fidelity details, requiring an exponential number of experiments for full information. This paper poses the question: Is the mapping between a given quantum circuit and its state fidelity learnable? If learnable, this would be a step towards an alternative approach that relies on machine learning, providing much more efficient computation. To answer this question, we propose three deep learning models for 1-, 3-, and 5-qubit circuits and experiment on the following real-quantum processors: ibmq_armonk (1-qubit), ibmq_lima (5-qubit) and ibmq_quito (5-qubit) backends, respectively. Our models achieved a mean correlation factor of 0.74, 0.67 and 0.66 for 1-, 3-, and 5-qubit random circuits, respectively, with the exponential state tomography method. Additionally, our 5-qubit model outperforms simple baseline state fidelity estimation method on three quantum benchmarks. Our method, trained on random circuits only, achieved a mean correlation factor of 0.968 while the baseline method achieved 0.738. Furthermore, we investigate the effect of dynamic noise on state fidelity estimation. The correlation factor substantially improved to 0.82 and 0.74 for the 3- and 5-qubit models, respectively. The results show that machine learning is promising for predicting state fidelity from circuit representation and this work may be considered a step towards efficient end-to-end learning. [ABSTRACT FROM AUTHOR]

Reducing the number of AND gates in logic networks benefits the applications in cryptography, security, and quantum computing. This work proposes a don’t-care-based (DC-based) approach to reduce the number of AND gates further in the well-optimized network. Furthermore, this work also proposes an enhanced synthesis flow by integrating our approach with the state-of-the-art. The experimental results show that our approach can further reduce up to 25% of the number of AND gates in the network. For the experiments about the enhanced synthesis flow, we achieve a speedup of almost $10\times $ on average for the cryptography benchmarks while having competitive results as compared to the flow in the state-of-the-art. [ABSTRACT FROM AUTHOR]

Grover's search algorithm accelerates the key search on the symmetric key cipher and the pre-image attack on the hash function. To conduct Grover's search algorithm, the target cipher algorithm should be efficiently implemented in a quantum circuit. Currently, small quantum computers are difficult to operate with large quantum circuits due to limited performance. Therefore, if a large quantum computer that can operate Grover's algorithm appears, it is expected that a cipher attack will be possible. In this paper, we propose a parallel structure quantum circuit for the Korean hash function standard (i.e., LSH). The proposed quantum circuit designed a parallel operation structure for the message expansion (i.e., MsgExp) function and the mix function, which are the internal structures of the LSH hash function. This approach shows an efficient result for quantum circuit implementation in terms of quantum resources by reducing the depth of the quantum circuit by about 96% through the trade-off of appropriate quantum resources compared to previous work. This result can be a reference for the implementation of a parallel quantum circuit in the future and is expected to advance the attack timing of the search algorithm for Grover's LSH hash function. [ABSTRACT FROM AUTHOR]

Abstract Deep neural networks were previously used in the arena of image retrieval. Siamese network architecture is also used for image similarity comparison. Recently, the application of quantum computing in different fields has gained research interest. Researchers are keen to explore the prospect of quantum circuit implementation in terms of supervised learning, resource utilization, and energy‐efficient reversible computing. In this study, the authors propose an application of quantum circuit in Siamese architecture and explored its efficiency in the field of exudate‐affected retinal image patch retrieval. Quantum computing applied within Siamese network architecture may be effective for image patch characteristic comparison and retrieval work. Although there is a restriction of managing high‐dimensional inner product space, the circuit with a limited number of qubits represents exudate‐affected retinal image patches and retrieves similar patches from the patch database. Parameterized quantum circuit (PQC) is implemented using a quantum machine learning library on Google Cirq framework. PQC model is composed of classical pre/post‐processing and parameterized quantum circuit. System efficiency is evaluated with the most widely used retrieval evaluation metrics: mean average precision (MAP) and mean reciprocal rank (MRR). The system achieved an encouraging and promising result of 98.1336% MAP and 100% MRR. Image pixels are implicitly converted to rectangular grid qubits in this experiment. The experimentation was further extended to IBM Qiskit framework also. In Qiskit, individual pixels are explicitly encoded using novel enhanced quantum representation (NEQR) image encoding algorithm. The probability distributions of both query and database patches are compared through Jeffreys distance to retrieve similar patches.

The emergence of quantum computers threatens current cryptographic systems, and NIST is preparing for the post-quantum era through the post-quantum cryptography (PQC) contest. CRYSTALS-Kyber is a lattice-based cipher suite that is used as a PQC standard. Lattice-based cryptography is considered quantum-safe for quantum computing because a quantum algorithm that can more efficiently solve the lattice problem of lattice-based cryptography compared to a classic algorithm has not been reported as yet. In this paper, we present quantum circuits tailored to NTT and inverse NTT, employed for optimized polynomial multiplication within CRYSTALS-Kyber. The proposed quantum circuits operate at Z3329[X]/(X256+1), which are the parameters of CRYSTALS-Kyber. We provide an in-depth description of the NTT/InvNTT quantum circuit’s operation and subsequently assess and analyze the quantum resources necessary for these functions. The NTT/InvNTT quantum circuits comprise four unique sub-functions, with the InvNTT additionally incorporating Barrett reduction. To the best of our knowledge, this represents the inaugural implementation of the CRYSTALS-Kyber NTT/InvNTT quantum circuits. We anticipate that our findings will aid in analyzing the security strengths of quantum computers for lattice-based cryptography.

Noisy intermediate-scale quantum algorithms, which run on noisy quantum computers, should be carefully designed to boost the output state fidelity. While several compilation approaches have been proposed to minimize circuit errors, they often omit the detailed circuit structure information that does not affect the circuit depth or the gate count. In the presence of spatial variation in the error rate of the quantum gates, adjusting the circuit structure can play a major role in mitigating errors. In this article, we exploit the freedom of gate reordering based on the commutation rules to show the impact of gate error propagation paths on the output state fidelity of the quantum circuit, propose advanced predictive techniques to project the success rate of the circuit, and develop a new compilation phase postquantum circuit mapping to improve its reliability. Our proposed approaches have been validated using a variety of quantum circuits with different success metrics, which are executed on IBM quantum computers. Our results show that rescheduling quantum gates based on their error propagation paths can significantly improve the fidelity of the quantum circuit in the presence of variable gate error rates.

The well-known 3-bit Hermitian gate (a Toffoli gate) has been implemented using Clifford+T circuits. Compared with the Peres gate, its implementation circuit requires more controlled-not (cnot) gates. However, the Peres gate is not Hermitian. This article reports four 3-bit Hermitian gates named LI gates, whose realized circuits have the same T-count, T-depth, and cnot-count as the Peres gate. Furthermore, two decomposition methods of a multiple-control Toffoli (MCT) gate are proposed for different primary optimization goals. Then, we design the equality, less-than, and full comparators with the minimum circuit width using proposed Hermitian gates and optimized MCT gates. A fault-tolerant circuit is required for robust quantum computing. Clifford+T circuits are accepted solutions for fault-tolerant implementation. Considering T-count, T-depth, cnot-count, and circuit width as the primary optimization goals, we design the optimized Clifford+T circuits of three comparators using LI gates and optimized MCT gates. Comparison and analysis show that the proposed comparators have better overall performances for T-count, T-depth, cnot-count, and circuit width than the best-known comparators without quantum measurements.

The synthesis of quantum circuits for multiplicative inverse over GF (2 8) are discussed in this paper. We first convert the multiplicative inverse operation in GF (2 8) to arithmetic operations in the composite field GF ((2 4) 2) , and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field GF (2 4) , where the expressions of multiplication calculation in GF (2 4) are given directly in GF (2 4) and given through being transformed into the composite field GF ((2 2) 2) . Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over GF (2 8) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes. [ABSTRACT FROM AUTHOR]

Grover’s search algorithm allows a quantum adversary to find a $k$ k -bit secret key of a block cipher by making O($2^{k/2}$ 2 k / 2 ) block cipher queries. Resistance of a block cipher to such an attack is evaluated by quantum resources required to implement Grover’s oracle for the target cipher. The quantum resources are typically estimated by the $\textit {T}$ T -depth of its circuit implementation and the number of qubits used by the circuit (width). Since the AES S-box is the only component which requires $\textit {T}$ T -gates in a quantum implementation of AES, recent research has put its focus on efficient implementation of the AES S-box. However, any efficient implementation with low $\textit {T}$ T -depth will not be practical in the real world without considering qubit consumption of the implementation. In this work, we propose three methods of trade-off between time and space for the quantum implementation of the AES S-box. In particular, one of our methods turns out to use the smallest number of qubits among the existing methods, significantly reducing its $\textit {T}$ T -depth. [ABSTRACT FROM AUTHOR]

Quantum algorithms can be described as quantum circuits and are supposed to be carried out on an ideal quantum device that is far from current ones. The current quantum devices have a significant limitation on the connectivity between quantum bits. In other words, a quantum bit is only allowed to interact with its nearest neighbors (NNs). In reality, quantum bits have to be placed on a grid, where the connectivity between quantum bits is predefined. The predefined connectivity of a grid further determines the possible range of architectures of a quantum device after the placement of quantum bits. In this article, we propose to place quantum bits based on a 2-D hexagonal architecture rather than a 2-D Cartesian architecture. To validate the effectiveness, we leverage a workflow for mapping NN compliant quantum circuits onto targeting grids, where the workflow consists of a global reordering strategy and a local reordering strategy. With the advantages of the hexagonal grid, the overhead of making quantum circuits NN compliant is reduced significantly compared with the Cartesian grid. We also provide a comprehensive set of ablation analyses to gain a better understanding of the contributions of the components within our workflow. According to the experimental results, when changing the grid type from Cartesian to hexagonal, the global reordering strategy is crucial for small quantum circuits. In contrast, the local reordering strategy is more important than the global reordering strategy for large quantum circuits. [ABSTRACT FROM AUTHOR]

The quantum computer has been claimed to show more quantum advantage than the classical computer in solving some specific problems. Many companies and research institutes try to develop quantum computers with different physical implementations. Currently, most people only focus on the number of qubits in a quantum computer and consider it as a standard to evaluate the performance of the quantum computer intuitively. However, it is quite misleading in most times, especially for investors or governments. This is because the quantum computer works in a quite different way than classical computers. Thus, quantum benchmarking is of great importance. Currently, many quantum benchmarks are proposed from different aspects. In this paper, we review the existing performance benchmarking protocols, models, and metrics. We classify the benchmarking techniques into three categories: physical benchmarking, aggregative benchmarking, and application-level benchmarking. We also discuss the future trend for quantum computer's benchmarking and propose setting up the QTOP100. [ABSTRACT FROM AUTHOR]

We define a formal framework for equivalence checking of sequential quantum circuits. The model we adopt is a quantum state machine, which is a natural quantum generalization of Mealy machines. A major difficulty in checking quantum circuits (but not present in checking classical circuits) is that the state spaces of quantum circuits are continuums. This difficulty is resolved by our main theorem showing that equivalence checking of two quantum Mealy machines can be done with input sequences that are taken from some chosen basis (which are finite) and have a length quadratic in the dimensions of the state Hilbert spaces of the machines. Based on this theoretical result, we develop an (and to the best of our knowledge, the first) algorithm for checking equivalence of sequential quantum circuits with running time $\mathcal {O}(2^{3m+5l}(2^{3m}{\,+\,}2^{3l}))$ , where $m$ and $l$ denote the numbers of input and internal qubits, respectively. The complexity of our algorithm is comparable with that of the known algorithms for checking classical sequential circuits in the sense that both are exponential in the number of (qu)bits. Several case studies and experiments are presented. [ABSTRACT FROM AUTHOR]