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IEEE International Conference on Quantum Computing and Engineering (QCE)

An Efficient Algorithm for Extending Uniform Superposition States for Time-Series Data

Carla Rieger, Michele Grossi, Gian Giacomo Guerreschi, Sofia Vallecorsa, Martin Werner

Published Aug 30, 2025• 0 citations
Publisher

Abstract

Leveraging fundamental quantum features such as quantum superposition is one of the key elements for the theoretical speed-up obtained in quantum algorithms, e.g., with Grover's search algorithm. In this work, we explore the novel concept of dynamic state preparation, which allows for extending an initially prepared superposition state already containing a set of data elements. This state is modified upon receiving further time-series measurement data from an experiment so that this data can be processed with a quantum algorithm to extract the global properties of the system efficiently. Specifically, we distinguish the algorithmic case in which the state-preparation unitary of the initially prepared state to be extended is known and the one where it is not. The former can be described as an application of amplitude amplification, whereas the latter includes an amplitude reservoir assigned to a reference state that allows for extension without knowing the full state preparation unitary. We discuss both cases algorithmically and present the respective quantum circuit, including a specific initial state preparation method. Our analysis reveals a polynomial scaling and respective tradeoffs to be considered.

DOI
10.1109/qce65121.2025.00037

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Leveraging fundamental quantum features such as quantum superposition is one of the key elements for the theoretical speed-up obtained in quantum algorithms, e.g., with Grover's search algorithm. In this work, we explore the novel concept of dynamic state preparation, which allows for extending an initially prepared superposition state already containing a set of data elements. This state is modified upon receiving further time-series measurement data from an experiment so that this data can be processed with a quantum algorithm to extract the global properties of the system efficiently. Specifically, we distinguish the algorithmic case in which the state-preparation unitary of the initially prepared state to be extended is known and the one where it is not. The former can be described as an application of amplitude amplification, whereas the latter includes an amplitude reservoir assigned to a reference state that allows for extension without knowing the full state preparation unitary. We discuss both cases algorithmically and present the respective quantum circuit, including a specific initial state preparation method. Our analysis reveals a polynomial scaling and respective tradeoffs to be considered.
In order to use a quantum device to assess a classical dataset D, we need to representthe set D in a quantum state. Applying a quantum algorithm, that is a quantum state preparation algorithm, to convert classical data into quantum data would be the common method. Loading classical data into a quantum device is required in many current applications. Efficiently preparing a quantum state to be used as the initial state of a quantum algorithm is an essential step in developing efficient quantum algorithms, since many algorithms need to reload the initial state several times during their execution. The cost to initialize a quantum state can compromise the algorithm efficiency if the process of quantum states preparation is not efficient. The topic of quantum states preparation in quantum computing has been the focus of much attention. In this scope, preparing sparse quantum states is a more specific problem that remains open since many quantum algorithms also require sparse initialization. This dissertation presents the results of an investigation on sparse quantum states preparation with the development of three algorithms, with highlight to the preparation of sparse quantum states, the main contributionof this dissertation. From a classical input dataset with M patterns formed by pairs composed of a complex number and a binary pattern with n bits, this algorithm can prepare a quantum state with n qubits and continuous amplitudes. The cost of its steps is O(nM), classical cost of o(MlogM+nM)and requires a lower CNOT number than the main quantum state preparation algorithms currently known. The preparation of a quantumstate with 2 non-zero amplitudes reveals the need of fewer CNOT gates in n>>1 relation to the main known state preparation algorithms, with even more favorable results with s higher and less 1S in the binary string.
Existence of quantum systems in superposition states gives an exponential advantage over classical system in both time and space. In this paper we will look at reconstructing real time classical data as a pure quantum state. Each classical data sample eg. each pixel location in an images is indexed by an\mathbf{n}nbit string representing one of the basis superposition states. The amplitude of the state which represents probability of measuring the state can be encoded as the normalized value of the sample at that location. A parameterized circuit[2] is used to prepare the quantum state as superposition of basis states from initial ground state. Each basis state corresponds to a sample classical data (eg. a pixel in case of an image). Quantum amplitude estimation based on Grovers algorithm[15] is used to estimate the amplitudes of each of these states. The parameters of the circuit is adjusted to minimize a loss function using classical gradient descent algorithm. The normalized amplitude at the output of our circuit is multiplied by a normalization factor that can also be tuned to represent the original data sample. This enables us to use the original real time data in our loss function computation. Section 1 of the paper gives a brief introduction. Section II covers the related works. Section III talks about amplitude encoding. Section IV describes quantum state preparation using supervised Learning. Section V describes implementation of the algorithms and finally Section VI gives the final conclusion of the research paper along with a result from the algorithm.
This paper introduces a novel optimization algorithm leveraging quantum information principles to efficiently analyze non-linear time series data. Traditional approaches often struggle with the inherent complexity of these datasets, demanding significant computational resources. We propose a quantum-enhanced algorithm that utilizes quantum state manipulation – specifically superposition and entanglement – to accelerate the analysis and prediction process. This paper details the core mechanism, including the quantum encoding of data and the subsequent optimization process. The algorithm demonstrates significant improvements in speed and accuracy compared to classical methods, specifically when dealing with complex time series patterns. The research explores the potential of quantum computing to revolutionize time series analysis, offering a promising pathway for enhanced data processing and forecasting.
Grover’s algorithm harnesses the power of quantum computing to swiftly locate specific elements in an unstructured database, outperforming classical computers in tasks like database searching. This algorithm capitalizes on the unique ability of qubits to be in both 0 and 1 states simultaneously (superposition), allowing it to scan the entire search space at once. It then boosts the probability of the target element, making it more prominent. While the foundational concepts of Grover’s algorithm are well-documented, practical implementation using quantum operators, especially for large search spaces, remains less explored beyond basic examples with a small number of qubits. Existing general synthesis techniques often involve numerous operators or are time-consuming. Our proposed methods specifically address the amplitude-amplification component of Grover’s algorithm for any size of search space. These methods detail the required types and quantities of qubits and operators, emphasizing minimal usage and efficient assembly. Developed and evaluated in Python, our methods consistently identified target elements with over 95% accuracy and achieved configurations comparably compact as those in the existing literature but at a faster pace. We anticipate that these methods facilitate practical implementations of Grover’s algorithm across various domains.
The superposition state of qubits in quantum computers enables a more efficient data representation than conventional binary methods in classical computers. Grover’s algorithm amplifies the amplitude of a target state within a superposed set of states having equal amplitudes. However, except for a 2-qubit system, the amplitudes of the nontarget states and left residual amplitudes induced by the algorithm do not reduce to zero. Therefore, a new method was devised to overcome this drawback. We select new target state among the nontarget states by applying Grover’s algorithm and obtain two distinct states with different amplitudes. Consequently, by excluding duplicate cases, this method enables the representation of [Formula: see text] data points. This technique is referred to as superdense data representation (SDR). Specifically, SDR is a new data representation method for quantum computers using superposition and entanglement. This offers several advantages for encoding data. For instance, 8-qubit data can be effectively represented in 4-qubit, thereby reducing the resources required to store the information. Additionally, SDR allows the representation of more complex data, such as Unicode, without increasing calculation or storage costs. UTF-8 encoding method can be implemented using only eight qubits. By extending SDR, Asian languages that require more than two bytes of data can also be expressed using eight qubits.
Quantum State Preparation Algorithms consist of defining a sequence or unitary operations to load a specific target state on a quantum computer. We can use those algorithms in appli- cations such as quantum machine learning. However, some state preparation algorithms have exponential circuit complexity with the number of qubits on the system. That is the case of amplitude encoding algorithms, which is an encoding type for loading normalized data into the probability amplitudes of the state. To circumvent this overhead in circuits’ complexity, works explore specific properties of quantum states to optimize the circuit’s complexity, such as sparsity or symmetry. Other works explore simplifying the quantum circuit to load an ap- proximate quantum state. It is the case of Quantum Generative Adversarial Networks, which use a specific circuit architecture comprised of alternating blocks of single-qubit rotations and two-qubit entangling controlled gates. But when trained to load random distributions on, we observed the performance deteriorates as the number of qubits increases in terms of relative entropy. In this work, we propose different architectures for the Quantum Generative mod- els based on the state preparation algorithm known as Low-Rank. Through experiments for loading the log-normal distribution, we show error reductions in quantum state initialization.