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arXiv:2603.00845 · Physical Review A

Three-Qubit State Preparation: Classification and Explicit Circuits

Yonghae Lee, Taewan Kim

quant-phSubmitted Mar 1, 2026• Announced Mar 3, 2026• 1 citation
arXivHTMLPDF

Abstract

We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a $1|2$ bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.

Journal reference
Phys. Rev. A 113, 062444 (2026)
DOI
10.1103/5bw6-339b
Comments
19 pages, 5 figures

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We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1∣21|21∣2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
Quantum state preparation is the process of producing a target quantum state that will be used as the input to a quantum circuit. Many quantum algorithms require an input state where amplitudes, phases, or basis probabilities encode problem data, and the cost of preparing this state can be significant in gate count and circuit depth. This review summarizes common goals, assumptions, and methods for quantum state preparation, with emphasis on preparing states from classical vectors, probability distributions, and feature data used in quantum machine learning. We organize approaches by the information they load and by the resources they require, including gate count, circuit depth, qubit overhead, and classical preprocessing. It also compares exact and approximate preparation procedures, and discusses how precision targets affect cost. The review highlights links between families of methods, typical sources of resource estimates, and criteria that help match a preparation method to a task and hardware constraints.
For pure three-qubit states the classification of entanglement is both non-trivial and well understood. In this work, we study the quantum algorithmic complexity introduced in [1] of three-qubit pure states belonging to the most general class of entanglement. Contrary to expectations we find out that the degree of entanglement of states in this class quantified by the measure of 3-tangle, does not correlate with the quantum algorithmic complexity, defined as the length of the shortest circuit needed to prepare the state. For a given entangled state the evaluation of its quantum complexity is done via a pseudo random evolutionary algorithm. This algorithm allows us not only to determine the complexity of a quantum circuit in terms of the number of required quantum gates, but also to estimate another type of complexity related to the time required to obtain the correct answer.
We propose a theory of classifying quantum circuits with qubit functional configurations. Any quantum circuit can be decomposed into alternating sequences of 1-qubit unitary gates and CNOT gates. Each CNOT sequence prepares the current quantum state into a layer of qubit functional configuration to specify the rule for the next 1-qubit unitary sequence on how to collectively modify the state vector entries. The functional configuration layers on a quantum circuit define its circuit type. These functional configuration types characterize important properties of the corresponding quantum circuits such as the circuit complexity, providing systematic knowledge of large collections of quantum circuits. We demonstrate the theory by analyzing the quantum Fourier transform circuit with the functional configuration picture. For potential applications, the functional configuration picture may allow systematic understanding of quantum algorithms based on their quantum circuit types, which can benefit the development of new algorithms.