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arXiv:2405.20408 · Physical Review Applied

Quantum encoder for fixed-Hamming-weight subspaces

Renato M. S. Farias, Thiago O. Maciel, Giancarlo Camilo, Ruge Lin, Sergi Ramos-Calderer, Leandro Aolita

Published Apr 4, 2025• 7 citations
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Abstract

We present an exact $n$-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of $d=(\genfrac{}{}{0}{}{n}{k})$ components into a subspace of fixed Hamming weight $k$. This represents a polynomial space compression of degree $k$. The circuit is optimal in that it expresses an arbitrary data vector using only $d\ensuremath{-}1$ (controlled) reconfigurable beam splitter (RBS) gates. It is constructed by an efficient classical algorithm that sequentially generates all bitstrings of weight $k$ and identifies the gates that superpose the corresponding states with the correct amplitudes. An explicit compilation into $\mathrm{CNOT}\mathrm{s}$ and single-qubit gates is presented, with the total $\mathrm{CNOT}$-gate count of $\mathcal{O}(kd)$ provided in analytical form. In addition, we show how to load data in the binary basis by sequentially stacking encoders of different Hamming weights using $\mathcal{O}(d\mathrm{log}(d))\phantom{\rule{0.2em}{0ex}}\mathrm{CNOT}$ gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. We perform a proof-of-principle demonstration of our scheme on a commercial trapped-ion quantum computer. We successfully upload a $q$-Gaussian probability distribution in the non-log-concave regime with $n=6$ and $k=2$. We also illustrate how the effect of hardware noise can be alleviated by quantum error mitigation. Numerically, we show how our encoder can improve the performance of variational quantum algorithms for problems that include particle-preserving symmetries. Our results constitute a versatile framework for quantum data compression with various potential applications in fields such as quantum chemistry, quantum machine learning, and constrained combinatorial optimizations.

DOI
10.1103/physrevapplied.23.044014

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Compared to general quantum states, the sparse states arise more frequently in the field of quantum computation. In this work we consider the preparation for nnn-qubit sparse quantum states with sss nonzero amplitudes and propose two algorithms. The first algorithm uses O(ns/log2n+n)O(ns/{log}_{2}n+n)O(ns/log2​n+n) gates, improving upon previous methods by O(log2n)O({log}_{2}n)O(log2​n). Moreover, the classical runtime of this algorithm is optimal. We further establish a matching lower bound for any algorithm that is not amplitude aware and employs at most poly(n)poly(n)poly(n) ancillary qubits. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path. An application is encoding the input data into a state with specified Hamming weight kkk in quantum machine learning, for which our algorithm constructs a circuit of size O((nk)log2n)O(\left(\genfrac{}{}{0pt}{}{n}{k}\right){log}_{2}n)O((kn​)log2​n). This surpasses previous results by O(k/log2n)O(k/{log}_{2}n)O(k/log2​n) and is close to the lower bound O((nk))O(\left(\genfrac{}{}{0pt}{}{n}{k}\right))O((kn​)). The classical runtime is also nearly optimal. Both algorithms shrink the existing gap theoretically and provide increasing advantages numerically.
Quantum state preparation (QSP) for a general nnn-qubit state requires O(2n)O(2^n)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))O(poly(n))O(poly(n)) for a broad class of compressible data. The method first applies a classical compression step to obtain a ddd-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.
A pure state of fixed Hamming weight is a superposition of computational basis states such that each bitstring in the superposition has the same number of ones. Given a Hilbert space of the form $\mathcal{H}={({\mathbb{C}}_{2})}^{\ensuremath{\bigotimes}n}$, or an nnn-qubit system, the identity operator can be decomposed as a sum of projectors onto subspaces of fixed Hamming weight. In this work, we propose several quantum algorithms that realize a coherent Hamming weight projective measurement on an input pure state, meaning that the post-measurement state of the algorithm is the projection of the input state onto the corresponding subspace of fixed Hamming weight. We analyze a depth-width tradeoff for the corresponding quantum circuits, allowing for a depth reduction of the circuits at the cost of more control qubits. For an nnn-qubit input, the depth-optimal algorithm uses O(n)\mathcal{O}(n)O(n) control qubits and the corresponding circuit has depth O(log(n))\mathcal{O}(log(n))O(log(n)), assuming that we have the ability to perform qubit resets. Furthermore, the proposed algorithm construction uses only one- and two-qubit gates.
Hybrid variational quantum algorithms (VQAs) are promising for solving practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, with typical random ansatz or quantum alternating operator ansatz, derived variational quantum algorithms become a black box that cannot be trusted for model interpretation, not to mention deploying as applications in informing critical decisions: the results of these variational parameters are just rotational angles for the quantum gates and have nothing to do with interpretable values that a model can provide directly. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also take advantage of the encoded data structure to reduce the time complexity of computing the regression map. To shorten the circuit depth for nonlinear regression, our algorithm can be extended by building nonlinear features by classical preprocessing as the independent encoded column vectors. Even though the realization of compressed encoding in superconducting qubits has been achieved by the less noisy compressed encoding recently by the authors, we envision potential quantum utilities with multi-qubit gates implemented in neutral cold atoms and ions.
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