We propose a hybrid quantum state preparation method based on the Walsh--Hadamard transform within the dense-to-sparse quantum state preparation framework. The method approximately prepares structured classical data in a quantum circuit through an indirect approach. Instead of directly preparing the dense amplitude-encoded state, the method first transforms the classical data into the Walsh domain and prepares only the largest-magnitude coefficients using a sparse state-preparation algorithm. The original state can then be approximately recovered through a parallel layer of Hadamard gates. Since this reconstruction requires no CNOT gates and has circuit depth 1, the proposed method introduces no additional CNOT or circuit-depth overhead, and its quantum preparation cost is determined by the underlying sparse state-preparation algorithm. Numerical results on representative benchmark signals demonstrate that the proposed method enables accurate state preparation for signals admitting sparse or approximately sparse representations in the Walsh domain.
We propose a hybrid quantum state preparation method based on the Walsh--Hadamard transform within the dense-to-sparse quantum state preparation framework. The method approximately prepares structured classical data in a quantum circuit through an indirect approach. Instead of directly preparing the dense amplitude-encoded state, the method first transforms the classical data into the Walsh domain and prepares only the largest-magnitude coefficients using a sparse state-preparation algorithm. The original state can then be approximately recovered through a parallel layer of Hadamard gates. Since this reconstruction requires no CNOT gates and has circuit depth 1, the proposed method introduces no additional CNOT or circuit-depth overhead, and its quantum preparation cost is determined by the underlying sparse state-preparation algorithm. Numerical results on representative benchmark signals demonstrate that the proposed method enables accurate state preparation for signals admitting sparse or approximately sparse representations in the Walsh domain.
Quantum state preparation (QSP) for a general n-qubit state requires O(2n) CNOT gates and circuit depth, making exact amplitude encoding (EAE) impractical for near-term quantum hardware. We introduce an ancilla-free hybrid classical-quantum strategy that reduces this cost to O(poly(n)) for a broad class of compressible data. The method first applies a classical compression step to obtain a d-sparse representation of the input, loads this sparse vector using a sparse-state preparation routine, and then reconstructs the target state through a polynomial-depth quantum inverse transform. We evaluate the framework on synthetic benchmark signals and real biomedical time series using Fourier and Haar transforms, demonstrating substantial reductions in CNOT counts and circuit depth compared to EAE, together with competitive performance relative to the Fourier Series Loader (FSL). The quantum simulation results show that combining classical data compression with quantum decompression provides a scalable framework for efficient QSP, reducing quantum overhead without requiring variational training or ancillary registers.
Quantum processing of classical data fundamentally relies on efficiently mapping classical datasets onto the amplitudes of quantum states. Preparing large amplitude-encoded states, however, remains a major bottleneck in quantum computing. In this paper, we introduce a novel state-preparation framework we call Q-MERGE, that addresses this bottleneck by partitioning a large target state into Mn-qubit segments, preparing these segments independently and in parallel, and then coherently combining them into a single amplitude-encoded state using SELECT-SWAP operations and measurement. Q-MERGE is agnostic to the segment-level preparation method, allowing existing techniques to be applied to smaller subproblems while providing a tunable trade-off between circuit depth and qubit count. With mid-circuit measurement and preparation-register reuse, the required ancilla qubits can be reduced from O(Mn) to O(n). Applied to a real-world 32x1024 ultrasound dataset, Q-MERGE achieves an infidelity of 3.5x10−9, compared with 6.2x10−3 for direct preparation using the same underlying method, a more than six orders of magnitude improvement at equal CNOT circuit depth. We demonstrate its feasibility experimentally with a 53 qubit circuit comprised of 954 1-qubit gates and 478 2-qubit gates, on the Quantinuum System Model H2 trapped-ion quantum computer. The prepared state is validated using shadow-overlap tomography. Numerical analysis of Haar-random states indicates consistent success probability for up to M=107 segments, supporting the scalability of Q-MERGE for encoding massive classical datasets.
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.
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 ⌈log(s)⌉-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.
An approximate quantum state preparation method is introduced, called the Walsh series loader (WSL). The WSL approximates quantum states defined by real-value functions of single real variables with a depth independent of the number n of qubits. Two approaches are presented. The first one approximates the target quantum state by a Walsh series truncated at $O(1/\sqrt{\ensuremath{\epsilon}})$, where $\ensuremath{\epsilon}$ is the precision of the approximation in terms of infidelity. The circuit depth is also $O(1/\sqrt{\ensuremath{\epsilon}})$, the size is $O(n+1/\sqrt{\ensuremath{\epsilon}})$, and only one ancilla qubit is needed. The second method accurately represents quantum states with sparse Walsh series. The WSL loads s-sparse Walsh series into n qubits with a depth doubly sparse in s and k, the maximum number of bits with value 1 in the binary decomposition of the Walsh function indices. The associated quantum circuit approximates the sparse Walsh series up to an error $\ensuremath{\epsilon}$ with a depth O(sk), a size O(n+sk), and one ancilla qubit. In both cases, the protocol is a repeat-until-success procedure with a probability of success $P=\mathrm{\ensuremath{\Theta}}(\ensuremath{\epsilon})$, giving an averaged total time of $O(1/{\ensuremath{\epsilon}}^{3/2})$ for the WSL and $O(sk/\ensuremath{\epsilon})$ for the sparse WSL. Amplitude amplification can be used to reach a probability of success $P=\mathrm{\ensuremath{\Theta}}(1)$, modifying the quantum circuit size to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+1/\sqrt{\ensuremath{\epsilon}})/\sqrt{\ensuremath{\epsilon}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+sk)/\sqrt{\ensuremath{\epsilon}})$ and the depth to $O([log{(n)}^{3}+1/\sqrt{\ensuremath{\epsilon}}]/\sqrt{\ensuremath{\epsilon}})$ and $O([log{(n)}^{3}+sk]/\sqrt{\ensuremath{\epsilon}})$, respectively. Amplitude amplification reduces by a factor $O(1/\sqrt{\ensuremath{\epsilon}})$ the total time dependence on $\ensuremath{\epsilon}$ but increases the size and depth of the associated quantum circuits, making them linearly dependent on n. These protocols give overall efficient algorithms with no exponential scaling in any parameter. They can be generalized to any complex-value, multivariate, almost-everywhere-differentiable function. The repeat-until-success Walsh series loader is so far the only method that prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits.
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state ∣ψ⟩, a Hermitian matrix H, a polynomial f, and a set of observables {O1,…,Om}, the goal is to estimate ⟨ψ∣f(H)†Ojf(H)∣ψ⟩ for all j∈{1,…,m}. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of f(H). It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary.
We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with H and ∣ψ⟩ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from H, f, and ∣ψ⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)∣ψ⟩. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
Emad Rezaei Fard Boosari