Neutron-nucleus dynamics simulations for quantum computers

TL;DR

This study provides first solutions of the neutron–alpha dynamics from quantum simulations suitable for noisy intermediate-scale quantum processors, using an optical potential rooted in first principles, as well as a study of the bound-state physics in neutron–Carbon systems, along with a comparison of the efficacy of the one-hot and Gray encodings.

Abstract

With a view toward addressing the explosive growth in the computational demands of nuclear structure and reactions modeling, we develop a novel quantum algorithm for neutron-nucleus simulations with general potentials, which provides acceptable bound-state energies even in the presence of noise, through the noise-resilient training method. In particular, the algorithm can now solve for any band-diagonal to full Hamiltonian matrices, as needed to accommodate a general central potential. While we illustrate the approach for exponential Gaussian-like potentials and ab initio inter-cluster potentials (optical potentials), it can also accommodate the complete form of the chiral effective-field-theory nucleon-nucleon potentials used in ab initio nuclear calculations. In this study, we provide a comprehensive analysis for the efficacy of this approach for three different qubit encodings, including the one-hot, binary, and Gray encodings, in terms of the number of Pauli strings and commuting sets involved. We also discuss the advantages of the algorithm for Hamiltonians of various band-diagonal widths, especially critical for potentials of perturbative nature, leading to a drastically reduced runtime of quantum simulations. We prove that the Gray encoding allows for an efficient scaling of the model-space size $N$ and is more resource efficient for band-diagonal Hamiltonians having bandwidth up to $N$. We introduce a new commutativity scheme called distance-grouped commutativity (DGC) and compare its performance with the well-known qubit-commutativity (QC) scheme. We lay out the explicit grouping of Pauli strings and the diagonalizing unitary under the DGC scheme, and we prove that it outperforms the QC scheme, at the cost of a more complex diagonalizing unitary. Lastly, we provide first solutions of the neutron-alpha dynamics from quantum simulations suitable for current quantum processors.

Figures (20)

• Figure 1: The kinetic and potential energy matrices: (a) for the contact potential used in Refs. PhysRevLett.120.210501PhysRevA.103.042405, (b) for the complete potential $V(r)$, and (c) for the truncated potential $V_K(r)=\sum_{k=0}^K v_k r^{2k}$ used in this work.
• Figure 2: Recursive circuit ansatz to generate the superposition of the one-hot basis states. The input to the circuit is $|0\rangle^{\otimes N}$, and $\theta_i$ denotes the $R_y(2\theta_i)=\exp{(-\mathrm{i} \theta_i Y_i)}$ rotation gate.
• Figure 3: Circuit ansatz to generate a parameterized real superposition of all basis states. The input to the circuit is $|0\rangle^{\otimes n}$, where $n = \operatorname{log}_2(N)$. Each layer (marked with dotted lines) is repeated $L$ times. Thus, the total number of parameters is $nL$.
• Figure 4: Number of Pauli terms for one-hot, binary, and Gray encodings for a general potential of the form $V_K(r) = \sum_{k=0}^K v_k r^{2k}$ and a general (tridiagonal to full) Hamiltonian matrix. Note that in the one-hot encoding, each term acts on $N$ qubits, while in the binary and Gray encodings, each term acts on $n= \operatorname{log}_2(N)$ qubits.
• Figure 5: Number of qubit-wise commuting Pauli sets terms for one-hot, binary, and Gray encodings for a general potential of the form $V_K(r) = \sum_{k=0}^K v_k r^{2k}$ and a general (tridiagonal to full) Hamiltonian matrix.

Theorems & Definitions (29)

• Remark 1
• Remark 2
• Lemma A.1
• Definition A.1: Projection Operators
• Definition A.2: Distance-$k$ operators
• Remark 3
• Definition A.3
• Lemma A.2
• Corollary A.3
• Lemma A.4