Quantum encoder for fixed Hamming-weight subspaces

TL;DR

This paper introduces an efficient quantum circuit for encoding data vectors into fixed Hamming-weight subspaces, enabling polynomial space compression and practical implementation on quantum hardware for applications in quantum computing.

Contribution

It presents a novel, optimal quantum encoder for fixed Hamming-weight subspaces, with explicit compilation and extension to sparse vectors, demonstrated experimentally and applicable to various quantum algorithms.

Findings

Successfully uploaded a q-Gaussian distribution on a 6-qubit system.

Demonstrated noise mitigation techniques improving encoding fidelity.

Showed potential for enhancing variational quantum algorithms.

Abstract

We present an exact $n$-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of $d=\binom{n}{k}$ 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 $d-1$ (controlled) Reconfigurable Beam Splitter (RBS) gates and 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 CNOTs and single-qubit gates is presented, with the total CNOT-gate count of $\mathcal{O}(k\, d)$ 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 $\mathcal{O}(d\,\log(d))$ CNOT gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. Experimentally, 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 showcase 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.