Efficient circuits for leaf-separable state preparation

TL;DR

The paper addresses efficient deterministic preparation of leaf-separable states, enabling structured inputs such as Dicke states for quantum algorithms. It introduces a leaf-separable framework that combines logarithmic-depth Dicke-state circuits with generalized weight distribution blocks on binary partition trees, followed by leaf-level Hamming-weight encoders. The main contributions are the depth and gate-count characterizations, ancilla-optimized variants, and extensions to mixed Hamming weight states, supported by numerical simulations showing high fidelity for moderate system sizes. This approach offers scalable state preparation for quantum applications requiring symmetric or near-symmetric inputs, with clear trade-offs between ancilla usage and circuit depth.

Abstract

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.

Figures (6)

• Figure 1: Conceptual diagram of the gWDB tree for Hamming weight distribution. An initial Hamming weight $\ell$ on $n$ qubits is recursively distributed across sub-registers down to leaf nodes.
• Figure 2: Circuit diagram to create the target state $\ket{\psi_{\text{target}}}$ from Sec. \ref{['sec:example']} using Algorithm \ref{['alg:state-prep-algorithm']}.
• Figure 3: The 2-qubit gate count required to implement a HW-k encoder in the worst case when $k = \lceil \frac{n}{2} \rceil$ for $n$ between 4 and 15 qubits. The WDB with Ancilla line shows our algorithm performance with $2$ ancilla qubits.
• Figure 4: The total gate count required to implement a HW-k encoder in the worst case when $k = \lceil \frac{n}{2} \rceil$ for $n$ between 4 and 15 qubits.
• Figure 5: The average fidelity when using the leaf-separable state preparation algorithm to prepare 200 random states with real amplitudes on between 4 and 9 qubits.

Theorems & Definitions (10)

• Lemma 2.1
• Definition 2.2
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Definition 3.4
• Lemma 3.5
• Theorem 1
• Theorem 2
• proof