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Comparative Study of Quantum State Preparation Methods in Sparse Isometry Decomposition

Ana Sofia Moreira de LIRA

Published Feb 6, 2025• 0 citations

Abstract

Initializing an isometry in a quantum circuit is a fundamental yet challenging task, espe- cially when aiming for efficient state preparation and resource optimization. In this work, we present a comparative analysis of the Householder Decomposition for isometry implementation, leveraging three distinct state preparation methods starting from the original Pivot method and extending the study to two additional strategies: Merge and Low Rank. The study investigates how each method impacts the performance of the isometry decomposition, focusing on key metrics such as CNOT gate count and circuit depth. To provide a comprehensive evaluation, we examine variations in matrix size and sparsity levels, capturing the effects of structural complexity on resource requirements. Our results reveal that the Merge state preparation method generally outperforms the other two approaches, particularly in terms of scalability and gate efficiency. Building upon these findings, we further compare the best-performing method, Merge, with the state-of- the-art isometry decomposition implementation available in Qiskit, a widely used quantum computing framework. The analysis demonstrates that, for isometries involving up to 6 qubits, Qiskit’s implementation exhibits superior performance. However, beyond this threshold, our proposed decomposition method proves more effective, especially for highly sparse isometries or those characterized by a smaller number of columns. This work highlights the potential for optimizing isometry decompositions in scenarios where sparsity and structural constraints are critical factors. These findings contribute to advancing state preparation techniques and offer insights into improving the efficiency of quantum circuits for applications in quantum information processing.

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Initializing an isometry in a quantum circuit is a fundamental yet challenging task, espe- cially when aiming for efficient state preparation and resource optimization. In this work, we present a comparative analysis of the Householder Decomposition for isometry implementation, leveraging three distinct state preparation methods starting from the original Pivot method and extending the study to two additional strategies: Merge and Low Rank. The study investigates how each method impacts the performance of the isometry decomposition, focusing on key metrics such as CNOT gate count and circuit depth. To provide a comprehensive evaluation, we examine variations in matrix size and sparsity levels, capturing the effects of structural complexity on resource requirements. Our results reveal that the Merge state preparation method generally outperforms the other two approaches, particularly in terms of scalability and gate efficiency. Building upon these findings, we further compare the best-performing method, Merge, with the state-of- the-art isometry decomposition implementation available in Qiskit, a widely used quantum computing framework. The analysis demonstrates that, for isometries involving up to 6 qubits, Qiskit’s implementation exhibits superior performance. However, beyond this threshold, our proposed decomposition method proves more effective, especially for highly sparse isometries or those characterized by a smaller number of columns. This work highlights the potential for optimizing isometry decompositions in scenarios where sparsity and structural constraints are critical factors. These findings contribute to advancing state preparation techniques and offer insights into improving the efficiency of quantum circuits for applications in quantum information processing.

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

The preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number

s

of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares

s

-sparse states on

n

qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a

\lceil{\log(s)}\rceil

-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially

2s

for sufficiently large values of

n

, yielding roughly a

\log(s)/2

improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to

s

. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.

We consider the task of breaking down a quantum computation given as an isometry into C-nots and single-qubit gates, while keeping the number of C-not gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.

The preparation of

n

-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when

m

ancillary qubits are available. The reduction in complexity is due to the use of a single operator

Λ

for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.

Efficient state preparation is a challenging and important problem in quantum computing. In this work, we present a recursive state preparation algorithm that combines logarithmic-depth Dicke state circuits with Hamming weight encoders for efficiently preparing ``leaf-separable" quantum states. The algorithm is built on binary partition trees, generalized weight distribution blocks (gWDBs), and leaf-level encoders. We evaluate the performance of the algorithm by numerically simulating it on randomly generated target states with between 4 and 15 qubits. Compared to general state preparation approaches which require

O(2^n)

CX gates, our algorithm achieves a circuit depth of

O(k\log\frac{n}{k} + 2^k)

and uses

O(n(k+2^k))

two-qubit gates, where

k < n

denotes the subtree size. We also compare implementations of the algorithm with and without the use of ancilla qubits, providing a detailed analysis of the trade-offs in circuit depth and two-qubit gate counts. These results contribute to scalable state preparation for quantum algorithms that require structured inputs such as Dicke or near-Dicke states.