Weakly Relativistic Lattice Hamiltonians

• The paper demonstrates that truncating the fourth-order expansion of the positive-energy kinetic operator yields lattice Hamiltonians with weakly relativistic corrections.
• It shows that using boundary-consistent discretization under periodic and Dirichlet conditions results in distinct momentum operators and explicit boundary correction terms.
• The measurement protocol combines cyclic translation moments with boundary overlap measurements to accurately reconstruct kinetic energies and validate convergence to continuum models.

Weakly relativistic lattice Hamiltonians are first-quantized finite-grid Hamiltonians obtained by starting from the positive-energy relativistic kinetic operator and truncating its low-momentum expansion at order $\hat p^4$. On a one-dimensional finite domain, this construction requires boundary-consistent lattice realizations of both the second and fourth momentum moments, so the Hamiltonian depends explicitly on the boundary condition. In the formulation presented in "First-Quantized Relativistic Quantum Simulation with Periodic and Dirichlet Boundary Conditions" (Bang et al., 19 Jun 2026), the relevant cases are periodic boundary conditions (PBC) and Dirichlet boundary conditions (DBC), with the corresponding weakly relativistic kinetic operator written as

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$

The central structural point is that weakly relativistic simulation on a lattice requires reconstruction of both $\langle \hat P_\tau^2\rangle$ and $\langle \hat P_\tau^4\rangle$, and that these operators are boundary-dependent rather than universal.

1. Relativistic origin and lattice definition

The continuum starting point is the positive-energy single-particle kinetic operator

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$

In the weakly relativistic regime, the expansion is truncated at fourth order,

$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$

so the lattice problem is reduced to constructing boundary-consistent versions of $\hat p^2$ and $\hat p^4$ on a finite grid (Bang et al., 19 Jun 2026).

On the lattice, the total Hamiltonian is

$\hat H_\tau^{\rm tot}=\hat T_\tau+\hat V_\tau,$

with diagonal potential operator

$$\hat V_\tau=\sum_{j=0}^{N-1}V\!\left(x_j^{(\tau)}\right)|j\rangle\langle j|.$$

The paper emphasizes that the boundary condition is part of the Hamiltonian definition, not an afterthought: PBC and DBC correspond to different lattice momentum operators. This is decisive for the relativistic correction, because the leading correction depends on $$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$0 in addition to $$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$1.

2. Finite-domain discretization and first-quantized encoding

The wavefunction is encoded in a first-quantized $$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$2-qubit register,

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$3

The finite domain is discretized differently for the two boundary conditions (Bang et al., 19 Jun 2026).

For PBC, the grid is

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$4

so the lattice is cyclic and the last point is connected back to the first.

For DBC, the grid is

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$5

with virtual endpoints

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$6

In this case the computational basis stores only the interior points of an open chain; the physical boundary points are not represented as basis states.

This difference in grid definition is not merely notational. Under PBC the natural discrete kinetic structure is generated by cyclic translation. Under DBC the desired operator is the open-chain finite-difference Laplacian, and the finite register must therefore exclude the unphysical wrap-around link that would otherwise couple the two ends of the chain.

3. Periodic boundary conditions and cyclic-translation reconstruction

For PBC, the basic operator is the unitary cyclic translation

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$7

with

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$8

The standard second-order finite-difference momentum-squared operator is

$$\hat T_\tau=\frac{\hat P_\tau^2}{2m}-\frac{\hat P_\tau^4}{8m^3c^2},\qquad \hat P_\tau^4\equiv (\hat P_\tau^2)^2,\qquad \tau\in\{\mathrm P,\mathrm D\}.$$9

and its expectation value reduces to a single real translation moment,

$\langle \hat P_\tau^2\rangle$0

The fourth moment is obtained by squaring the same operator,

$\langle \hat P_\tau^2\rangle$1

which implies

$\langle \hat P_\tau^2\rangle$2

Defining translation moments

$\langle \hat P_\tau^2\rangle$3

the PBC kinetic sector is completely determined by $\langle \hat P_\tau^2\rangle$4 and $\langle \hat P_\tau^2\rangle$5. The corresponding weakly relativistic kinetic expectation value is

$\langle \hat P_\tau^2\rangle$6

Operationally, this means that under PBC the relativistic correction can be reconstructed entirely from cyclic-translation measurements. That feature gives the PBC case a particularly compact estimator structure: no boundary-specific observables are required.

4. Dirichlet boundary conditions and boundary-local correction terms

For DBC, the desired kinetic operator is the open-chain finite-difference Laplacian rather than the cyclic one. The construction in (Bang et al., 19 Jun 2026) expresses the open-chain operator using the same cyclic translation $\langle \hat P_\tau^2\rangle$7, supplemented by boundary-local terms that remove the unphysical wrap-around link.

The boundary coupling operator is defined as

$\langle \hat P_\tau^2\rangle$8

This isolates the coupling between the first and last interior points. The DBC momentum-squared operator is then

$\langle \hat P_\tau^2\rangle$9

If one introduces the cyclic finite-difference operator on the DBC grid spacing,

$\langle \hat P_\tau^4\rangle$0

then

$\langle \hat P_\tau^4\rangle$1

Its expectation value is therefore

$\langle \hat P_\tau^4\rangle$2

with

$\langle \hat P_\tau^4\rangle$3

The correction is explicitly a boundary coherence.

The fourth moment is more intricate because $\langle \hat P_\tau^4\rangle$4 generates additional boundary terms. The derived expression is

$\langle \hat P_\tau^4\rangle$5

where

$\langle \hat P_\tau^4\rangle$6

The explicit endpoint forms are

$\langle \hat P_\tau^4\rangle$7

$\langle \hat P_\tau^4\rangle$8

$\langle \hat P_\tau^4\rangle$9

Hence

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$0

In measurable form, the paper writes

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$1

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$2

with

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$3

The DBC kinetic energy is therefore

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$4

The physical interpretation of the correction terms is localized: $$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$5 is endpoint-to-endpoint coherence, $$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$6 gives endpoint probabilities, and $$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$7 involve endpoints and near-endpoints. Their role is to remove the unphysical periodic wrap-around link and restore the open-chain DBC Laplacian exactly.

5. Measurement primitives and energy-estimation workflow

The energy-estimation protocol separates kinetic and potential contributions (Bang et al., 19 Jun 2026). For PBC, kinetic estimation requires only the translation moments

$$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$8

These are measured using a Hadamard-test-style controlled-unitary circuit: ancilla in $$\hat T_{\rm rel}=mc^2\left(\sqrt{1+\frac{\hat p^2}{m^2c^2}}-1\right).$$9, apply Hadamard, apply controlled-$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$0, apply Hadamard, and measure ancilla in $$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$1. The circuit yields

$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$2

and for $$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$3 gives $$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$4.

For DBC, the same $$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$5 are needed, together with the boundary terms $$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$6. The boundary coherences are obtained from overlap probabilities via

$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$7

where

$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$8

and

$$\hat T_{\rm rel}=\frac{\hat p^2}{2m}-\frac{\hat p^4}{8m^3c^2}+O\!\left(\frac{\hat p^6}{m^5c^4}\right),$$9

Consequently, DBC requires only a small number of endpoint and near-endpoint overlap measurements in addition to the translation measurements.

For the potential term, since $\hat p^2$0 is diagonal in the computational basis,

$\hat p^2$1

it is estimated by sampling the position register in the computational basis and classically post-processing the sampled positions. The total-energy estimate is

$\hat p^2$2

This workflow makes the estimator structure boundary-aware. Under PBC the kinetic sector is reconstructed from translation measurements alone, whereas under DBC the same cyclic estimator is supplemented by boundary-local terms.

6. Benchmarks, validation, and error structure

The framework is validated in four benchmark classes (Bang et al., 19 Jun 2026). For the free particle with PBC, the analysis uses Fourier modes $\hat p^2$3 and the exact lattice dispersion

$\hat p^2$4

The benchmark demonstrates convergence of the lattice relativistic energy to the continuum result as the grid is refined and separates discretization error, defined as the difference between lattice and continuum, from weak-relativistic truncation error, defined as the difference between the exact square-root relativistic energy and the $\hat p^2$5 expansion.

For the infinite square well with DBC, the states are sine modes

$\hat p^2$6

The open-chain spectrum is confirmed as

$\hat p^2$7

Most importantly, the boundary-corrected cyclic estimator exactly reproduces the open-chain matrix moments $\hat p^2$8 and $\hat p^2$9. This is the direct validation of the boundary-local correction scheme.

The smooth-potential benchmarks use the periodic cosine potential

$\hat p^4$0

for PBC and the harmonic confining potential

$\hat p^4$1

for DBC. The procedure diagonalizes the nonrelativistic Hamiltonian and then evaluates the relativistic correction on the ground state. The comparison between direct matrix evaluation and estimator reconstruction shows near-perfect agreement, validating the full workflow, including kinetic terms and diagonal potential sampling.

A finite-shot measurement test simulates the statistical behavior of the measurement primitives. The kinetic-energy estimator RMSE scales as

$\hat p^4$2

as expected for sampling error. PBC uses only translation moments, while DBC has a slightly larger constant prefactor because it includes the extra boundary overlap measurements. The reported agreement between estimator reconstruction and direct matrix evaluation, together with the explicit separation of finite-grid discretization, weak-relativistic truncation, and measurement errors, delineates the scope of the method: it is a boundary-consistent first-quantized framework for weakly relativistic quantum simulation on a one-dimensional finite grid.