Quantum Simulation of Nuclear Inelastic Scattering

• tBFQ targets nuclear inelastic scattering problems in which time-dependent external interactions excite internal reaction-system degrees of freedom.Source: §1 · p. 1
• The approach is expected to achieve an exponential advantage over classical algorithms for this problem subset, potentially enabling more complicated simulations.Source: §1 · pp. 1–2
• The algorithm prepares and qubitizes a basis representation, then designs time-dependent quantum evolution for nuclear inelastic scattering.Source: §1 · p. 2
• Reference-Hamiltonian eigenbases are computed classically, importance-truncated, and used to construct the Hamiltonian and time-evolution operator before Trotterization.Source: §1 · p. 2
• IBM Qiskit simulations produce transition probabilities and observables for benchmarking the tBFQ algorithm on the model problem.Source: §1 · p. 2
• The demonstration treats Coulomb excitation of the deuteron in peripheral heavy-ion scattering, a nontrivial case involving strong time-dependent fields and higher-order transitions.Source: §1 · p. 2

2.1 Hamiltonian dynamics

• 2.1 Hamiltonian dynamics: The full Hamiltonian is divided into a time-independent reference term and a time-dependent external interaction that drives the reaction dynamics.Source: §2.1 · p. 3
• 2.1 Hamiltonian dynamics: The external interaction is modeled as a sum of terms, each factorized into a time-dependent external coefficient and a time-independent operator describing intrinsic properties.Source: §2.1 · pp. 3–4

2.2 Basis representation

• 2.2 Basis representation: The reference-Hamiltonian eigenbasis represents intrinsic reaction-system degrees of freedom and can incorporate both bound and scattering channels.Source: §2.2 · pp. 4–5
• 2.2 Basis representation: Importance truncation regulates the formally infinite basis while retaining predominant inelastic-scattering physics, with prior work verifying basis-parameter independence.Source: §2.2 · p. 4
• 2.2 Basis representation: Basis amplitudes define state probabilities, which are combined with diagonal observable matrix elements to evaluate expectation values.Source: §2.2 · p. 5
• 2.2 Basis representation: Asymptotic observables after the external field weakens are experimentally interrogable, whereas intermediate-time values expose virtual excitations not directly measurable.Source: §2.2 · p. 5

3 Algorithm

• The algorithm targets a subset of nuclear inelastic scattering problems using a time-dependent quantum simulation protocol.The discussion covers initial-state preparation, time-evolution approximation, basis decomposition, complete evolution, circuit construction, simulation and measurement, and complexity.Source: §3 · p. 6

3.1 Preparation of the initial state

• Basis states are compactly encoded as qubit configurations, requiring at least ⌈log2 Nd⌉ qubits for a basis space of dimension Nd.The initial superposition is prepared from a default register using elementary Pauli-X and/or Hadamard gates.Source: §3.1 · p. 6

3.2 Approximation of the time-evolution operator

• Time evolution is discretized into n equal partitions of duration ∆t, with first-order Trotterization accuracy O(∆t2) requiring ||H(ti)|| · ∆t ≪ 1 during scattering.Source: §3.2 · pp. 6–7

3.3 Decomposition of the time-evolution operator in the basis representation

• Each time-step unitary is decomposed into a time-independent diagonal free-evolution unitary and interaction unitaries whose time dependence enters through simple diagonal phases.The fixed transformation matrices connect basis and eigen-representations, while only the diagonal factors need updating with scattering time.Source: §3.3 · pp. 7–9

3.4 The construction of the complete time evolution

• The complete scattering evolution is assembled as a sequence of time-step modules, and the tBFQ algorithm uses only quantum computation without hybrid intermediate computation.The sequence approximates the full evolution with O(∆t2) error.Source: §3.4 · p. 9

3.5 Quantum circuit construction

• The quantum circuit represents Nd bases with ⌈log2 Nd⌉ qubits and applies a sequence of gates eU(tk) that vary between time steps but remain fixed within each short step.Each gate eU(tk) is constructed from the unitary decomposition of the corresponding time-step evolution operator.Source: §3.5 · p. 9
• Fixed-structure modules with scattering-time-dependent gate parameters let the quantum computer automatically construct the full circuit without intermediate classical-computer assistance.Time-independent unitaries use fixed gates, while diagonal time-dependent unitaries retain the same structure with parameters determined by scattering time.Source: §3.5 · pp. 10–11

3.6 Quantum simulation and measurement

• Repeated circuit modules evolve the initial state, and measurements yield basis-state probabilities from which final and intermediate-time observables can be computed.Intermediate-time probabilities and observables track the dynamics but are quantal effects, not experimentally measurable quantities.Source: §3.6 · p. 11

3.7 The complexity of the tBFQ algorithm

• Each time-evolution unitary decomposes into 3m + 1 unitaries, each simulable with circuit size poly(log2 N_d, T, D_max, 1/Δt, 1/ε).The construction relies on diagonal H_0 and interaction matrices with sparsity arising from transition-operator selection rules.Source: §3.7 · pp. 11–12
• The complete circuit size is bounded by poly(log2 Nd, T, Dmax, 1/∆t, 1/ϵ), whereas classical evolution is approximated by (Nd)^2 operations per step times T/∆t, suggesting an expected exponential speedup.Source: §3.7 · p. 12

• The demonstration models deuteron Coulomb excitation during peripheral scattering from a heavy ion, neglecting nuclear interactions and focusing on the external electromagnetic field.Source: §4 · pp. 12–13
• The target Hamiltonian separates intrinsic deuteron motion from a time-dependent interaction induced by the projectile’s Coulomb field.Source: §4 · p. 13
• The interaction is approximated by the electric-dipole E1 component at low incident speed and factorized into time-dependent and intrinsic matrix terms.Source: §4 · pp. 13–14
• The basis-representation time-step unitary is decomposed into fixed basis-dependent unitaries and scattering-time-dependent diagonal unitaries.Source: §4 · p. 14
• Each time-evolution step becomes a fixed-structure circuit module whose gate parameters are explicit functions of scattering time, and these modules compose the complete or partial scattering circuit.Source: §4 · pp. 14–15

• The model uses a bare uranium projectile, incident speed 0.1, impact parameter 5 fm, and an exposure interval from -3 to 3 MeV^-1.Source: §5 · p. 15
• The evolution begins and ends when the uranium is approximately 60 fm from closest approach, where induced intrinsic excitations become negligible outside the interval.Source: §5 · p. 15
• The simulation retains the lowest seven states in three interaction channels and directly maps the resulting basis representation to a qubit register.Source: §5 · p. 15
• The circuit is assembled by decomposing fixed and time-dependent gates, then the model is simulated on an ideal IBM Qiskit simulator without gate noise or decoherence.Source: §5 · pp. 15–16
• The initial deuteron state is (3S1,3 D1), M = −1, encoded as |000⟩, with evolution step ∆t = 0.01 MeV^-1.Source: §5 · p. 16

Transition probabilities

• The tBFQ results reproduce all retained-channel transition probabilities, with statistical errors below 1% for most states and below 4% for 3P0, M = 0 and 3P1, M = 0.Error bars are smaller than the plotted symbols and decrease with more simulations.Source: §6 · p. 16

Results

• Good agreement is obtained between tBFQ and classical tBF results for transition probabilities and selected deuteron observables.The comparison covers retained-channel populations, intrinsic energy excitation, and state-average r.m.s. point-charge radius.Source: §6 · p. 16, Figure 4 · p. 17

Numerical accuracy

• The calculation uses an ideal quantum simulator, with first-order Trotterization error of order (∆t)2 and statistical measurement variance.The reported study uses 10 simulations with 107 measurements per evolution time.Source: §6 · p. 16

Scattering observables

• The quantum and classical calculations also agree for intrinsic-energy excitation and the deuteron’s state-average r.m.s. point-charge-radius expansion during Coulomb excitation.These observables are computed according to Eq. (11).Source: §6 · pp. 16–17, Figure 4 · p. 17

Quantum evolution

• The tBFQ evolution is fully coherent until measurement, preserving the wave function’s coherent dynamics and entanglement during simulation.This coherence is presented as an appealing feature of simulating one quantum system with another.Source: §6 · p. 17

Non-perturbative effects

• Higher-order effects, including sequential E1 excitations, create a transition network that populates states forbidden by direct E1 selection from the initial state.The network can feed states such as (3S1,3 D1), M = +1.Source: §6 · pp. 17–18

7 Conclusion and Outlook

• The algorithm presents a time-dependent quantum treatment of nuclear inelastic scattering using basis functions encoded on qubits.It uses outputs from classical nuclear-structure calculations and targets reactions whose internal degrees of freedom are excited by external interactions.Source: §7 · p. 18
• The full Hamiltonian is split into a reference part governing available excitations and a time-dependent external interaction driving dynamical excitation.The interaction is represented through dominant terms coupling time-dependent external actions to time-independent intrinsic dynamics.Source: §7 · p. 18
• Importance truncation reduces the eigenbasis, after which the full Hamiltonian and time-evolution operator are solved in the trimmed basis and compactly qubitized.The circuit is parameterized according to scattering time and the resulting measurements provide transition probabilities and other observables.Source: §7 · p. 18
• The authors expect exponential speedup for the targeted subset, potentially enabling complicated scattering simulations that classical algorithms would find intractable.The algorithm is presented as a route toward studying many-body Hamiltonian dynamics in atomic and subatomic physics.Source: §7 · pp. 18–19
• IBM Qiskit simulations of peripheral deuteron Coulomb excitation agree well with corresponding classical transition probabilities and selected observables.The transition probabilities determine the inelastic scattering cross section.Source: §7 · p. 18
• Efficient quantum-circuit construction, including optimization of circuit depth and CNOT counts, remains necessary for general reaction simulations on real quantum computers.The present work focuses on illustrating the methodology of simulating nuclear inelastic scattering on quantum computers.Source: §7 · pp. 18–19

A Decomposition of diagonal unitary

• The decomposition targets a time-varying 2^3 × 2^3 unitary whose entries are parameterized by real functions of time.Rows and columns are labeled by binary configurations from 000 through 111.Source: A Decomposition of diagonal unitary · p. 20
• The leftmost, middle, and rightmost qubits are treated as the most, second, and least significant qubits, with the first two controlling the least significant qubit.This convention determines how the matrix rows and columns map onto the circuit controls.Source: A Decomposition of diagonal unitary · p. 20
• A circuit identity decomposes the diagonal unitary into general unitary, special unitary, and phase-gate components.The remaining matrix parts are decomposed using controlled-gate transformations, including conversion between controls on |1⟩ and |0⟩.Source: A Decomposition of diagonal unitary · p. 21
• For the considered special case, the time-dependent functions are real and the resulting construction uses identity and Rz rotational gates with time-dependent angles.The gate decomposition is applied to the matrix P(t).Source: A Decomposition of diagonal unitary · pp. 20–21