This paper introduces the Hamiltonian-Adaptive Ternary Tree (HATT) framework to compile optimized Fermion-to-qubit mapping for specific Fermionic Hamiltonians. In the simulation of Fermionic quantum systems, efficient Fermion-toqubit mapping plays a critical role in transforming the Fermionic system into a qubit system. HATT utilizes ternary tree mapping and a bottom-up construction procedure to generate Hamiltonian aware Fermion-to-qubit mapping to reduce the Pauli weight of the qubit Hamiltonian, resulting in lower quantum simulation circuit overhead. Additionally, our optimizations retain the important vacuum state preservation property in our Fermion-toqubit mapping and reduce the complexity of our algorithm from $O\left(N^{4}\right)$ to $O\left(N^{3}\right)$. Evaluations on various Fermionic systems demonstrate $5 \sim 25 \%$ reduction in Pauli weight, gate count, and circuit depth, alongside excellent scalability to larger systems. Experiments on the Ionq device also show the advantages of HATT in noise resistance in quantum simulations.

Proposes some simple algorithms for decoding BCH codes. The authors show that the pruned FFT is an effective method for evaluating syndromes and for finding the roots of error-locator polynomials. They show that a simple variation of the basic Gaussian elimination procedure can be adapted to compute the error-locator polynomial efficiently for codes with small designed distance. Finally, they give a procedure for computing the error values that has half the complexity of the Forney algorithm.>

We present a general framework for constructing transforms in the field of the input which have a convolution-like property. The construction is carried out over the reals, but is shown to be valid over more general fields. We show that these basefield transforms can be viewed as "projections" of the discrete Fourier transform (DFT). Furthermore, by imposing an additional condition on the projections, one may obtain self-inverse versions of the basefield transforms. Applying the theory to the real and complex fields, we show that the projection of the complex DFT results in the discrete combinational Fourier transform (DCFT) and that the imposition of the self-inverse condition on the DCFT yields the discrete Hartley transform (DHT). Additionally, we show that the method of projection may be used to derive efficient basefield transform algorithms by projecting standard FFT algorithms from the extension field to the basefield. Using such an approach, we show that many of the existing real Hartley algorithms are projections of well-known FFT algorithms.>

Over the field of complex numbers, it is well-known that if the input is real then it is possible to compute two real DFTs with one complex DFT. The authors extend the result to finite fields and show how to compute m DFTs over GF(q) with one DFT over GF(q/sup m/).>

A general framework is presented for constructing transforms in the field of the input which have a convolution-like property. The construction is carried out over finite fields, but is shown to be valid over the real and complex fields as well. It is shown that these basefield transforms can be viewed as "projections" of the discrete Fourier transform (DFT) and that they exist for all lengths N for which the DFT is defined. The convolution property of the basefield transforms is derived and a condition for such transforms to have the self-inverse property is given. Also, fast algorithms for these basefield transforms are developed, showing gains when compared to computations using the FFT. Application of the methodology to Hartley transforms over R leads to a simple derivation of fast algorithms for computing real Hartley transforms.>