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.
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(2n) CX gates, our algorithm achieves a circuit depth of O(klogkn+2k) and uses O(n(k+2k)) 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.
Hamming-weight computation and symmetric Boolean functions are basic primitives for quantum arithmetic and state preparation, where circuit depth and qubit overhead matter. We construct an exact circuit computing the Hamming weight of an n-qubit register with depth O(logn) and size O(nlogn), using only the n input qubits and m=⌈log2(n+1)⌉ binary output qubits. The circuit uses arbitrary one-qubit gates and CNOTs with all-to-all connectivity. Both depth and total width are optimal. Our constructions combine tree structures and pipelining, temporarily using input qubits as workspace. We also give exact circuits for every symmetric Boolean function with the same depth and size bounds, using only m ancillary qubits returned to zero. Finally, we apply the counter to exact preparation of balanced Dicke states for even n. With only O(logn) ancillary qubits, we obtain expected depth O(log2n) using measurements and classical feedforward, and depth O(n1/4logn) using unitary gates alone.
Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree−d polynomial, using circuits with logarithmic depth in the number n of qubits and only O(n) ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only n terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state ∣1⟩. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using 14 qubits and more than 500 primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.
We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of O(klogkn) for all-to-all connectivity (such as on current ion trap devices); (ii) depth of O(k√kn)=O(nk) for Grid connectivity on grids of size Ω(√sn)×O(ns) with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of O(kn), need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth O(n) on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth O(logn) on all-to-all connectivity for k = 1 (AQT 2019 [9]).
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.
Dicke states serve as a critical resource in quantum metrology, communication, and computation. However, unitary preparation of Dicke states is limited to logarithmic depth in standard circuit models and existing constant-depth protocols require measurement and feed-forward. In this work, we present the first unitary, constant-depth protocols for exact Dicke state preparation. We overcome the logarithmic-depth barrier by moving beyond the standard circuit model and leveraging global interactions (native to architectures such as neutral atoms and trapped ions). Specifically, utilizing unbounded CZ gates (i.e. within the QAC0 circuit class), we offer circuits for exact computation of constant-weight Dicke states, using polynomial ancillae, and approximation of weight-1 Dicke states (i.e. W states), using only constant ancillae. Granted additional access to the quantum FAN-OUT operation (i.e. upgrading to the QACf0 circuit class), we also achieve exact and clean preparation of arbitrary-weight Dicke states, with polynomial ancillae. These protocols distinguish the constant-depth capabilities of quantum architectures based on connectivity and offer a novel path toward resolving a long-standing quantum complexity conjecture.
Hamming-Weight-Preserving (HWP) states, ∣ψH⟩=∑HW(x)=kαx∣x⟩, represent a fundamental class of quantum states on n qubits with fixed excitation number k, playing a pivotal role in quantum simulation, machine learning, and optimization, with prominent examples including the Bethe eigenstates of integrable models and Configuration Interaction states. However, existing preparation methodologies typically rely on sequential architectures whose depth scales linearly with the subspace dimension, whereas practical deployment on realistic quantum hardware demands shallow circuits to mitigate decoherence. In this work, we present quantum circuits that achieve the information theoretic limit of depth logarithmic in the number of populated computational basis states while maintaining asymptotically optimal size. Specifically, for general HWP states, we provide a construction that matches the lower bounds of Θ(log(kn)) depth and Θ((kn)) size. Furthermore, we propose a resource efficient scheme utilizing only O(logk) ancillary qubits. For the k=2 case, where HWP states naturally admit a graph representation, we show that states whose support graphs are trees or grids admit ancilla free preparation with Θ(logn) depth and Θ(m) size. Overall, our work provides logarithmic depth state preparation algorithms across various regimes while establishing a connection between quantum state synthesis and Krawtchouk polynomials.