Quantum circuit decomposition of the tangent-fermion Dirac operator

Abstract: The Dirac operator on a lattice cannot be both local and free of fermion doubling, at least not without breaking fundamental symmetries. Non-local, symmetry-preserving discretizations that avoid doubling have a quantum circuit representation as a linear-combination-of-unitaries (LCU) in which both the number of terms and their norm (the subnormalization factor) grow with the lattice size, compromising the efficiency of a quantum algorithm. We show that the tangent-fermion discretization escapes this obstruction when the Dirac equation is written as a generalized eigenvalue problem with a local operator pencil: Each member of the pencil has an exact LCU, with term count that is independent of lattice size and with subnormalization factor of order unity, on a par with elliptic operators. This provides an efficient block-encoding primitive for Dirac spectra and Green functions without fermion doubling.

• The paper introduces a novel tangent-fermion discretization reformulated as a local Hermitian pencil, achieving constant LCU scaling and avoiding fermion doubling.
• It employs an LCU decomposition with O(1) term count and unit subnormalization independent of lattice size, streamlining quantum simulation of Dirac operators.
• Results enable efficient spectral and Green function computations that facilitate practical quantum studies of topological and condensed matter systems.

Quantum Circuit Decomposition of the Tangent-Fermion Dirac Operator: Technical Analysis

Background and Motivation

The challenge of efficiently representing the Dirac operator on a quantum computer is deeply intertwined with the fermion doubling problem, which stems from discretizing first-order differentials on a lattice. Conventional local discretizations, such as the central difference, suffer from spurious zeros at the Brillouin zone boundary, generating unwanted low-energy excitations and breaking topological protection. Nonlocal schemes like SLAC and tangent fermion discretizations can circumvent fermion doubling without breaking chiral symmetry but typically at a prohibitive quantum circuit cost: their LCU decompositions scale with lattice size $N$, incurring large subnormalization factors, which compromise quantum algorithm efficiency.

This paper demonstrates that the tangent-fermion discretization, when reformulated as a generalized eigenvalue problem with a local Hermitian operator pencil, admits an LCU decomposition with term count and subnormalization independent of $N$. This aligns its quantum circuit complexity with that of elliptic operators, enabling efficient block encoding for spectrum and Green function calculations without fermion doubling (2606.19020).

Operator Discretization and LCU Representation

The Dirac operator in one dimension ($H_{\rm Dirac} = v p_x \sigma_x$) is discretized in three ways:

• Central Difference (Local): Couples only nearest neighbors; its Fourier symbol $\sin k$ vanishes at both $k=0$ and $k=\pi$, yielding a doubler.
• SLAC Fermion (Nonlocal): Implements a sawtooth (linear until boundary, then jumps); coupling decays as $1/$distance; avoids doubling but incurs $O(N)$ terms with $O(n)$ subnormalization.
• Tangent Fermion (Nonlocal): Coupling is all-to-all and does not decay; standard LCU decomposition yields $O(N)$ terms and exponential subnormalization ($N$0); unusable for large systems.

The innovation is to recast the tangent discretization as the generalized eigenproblem $N$1, where $N$2 and $N$3 are local, sparse Hermitian operators constructed from nearest-neighbor shift gates with antiperiodic boundary conditions. Both $N$4 and $N$5 admit exact LCU decompositions with unit subnormalization and $N$6 terms, regardless of lattice size.

This construction ensures invertibility (no zero modes at Brillouin boundaries) and avoids fermion doubling, matching the efficiency of elliptic operators and surpassing SLAC discretization.

Extension to Higher Dimensions and Inclusion of Fields

In $N$7 dimensions, the operator pencil generalizes to $N$8 with:

• $N$9: Sums over Clifford algebra generators $H_{\rm Dirac} = v p_x \sigma_x$0 and shift operators per dimension, composed with products of averaging operators in other dimensions.
• $H_{\rm Dirac} = v p_x \sigma_x$1: Product of averaging operators across all dimensions.

The LCU decompositions scale polynomially in $H_{\rm Dirac} = v p_x \sigma_x$2 ($H_{\rm Dirac} = v p_x \sigma_x$3 and $H_{\rm Dirac} = v p_x \sigma_x$4 terms for $H_{\rm Dirac} = v p_x \sigma_x$5 and $H_{\rm Dirac} = v p_x \sigma_x$6 respectively), with subnormalization factors linear in $H_{\rm Dirac} = v p_x \sigma_x$7. Mass terms ($H_{\rm Dirac} = v p_x \sigma_x$8) and scalar potentials ($H_{\rm Dirac} = v p_x \sigma_x$9) can be efficiently incorporated as LCU sums if spatially localized or structured, maintaining locality and Hermiticity.

Importantly, translational invariance is not required for the efficient pencil decomposition, crucial for simulating materials with spatial inhomogeneity.

Applications: Spectral and Green Function Computation

Spectrum Calculation

The efficient LCU and block encoding of $\sin k$0 enable quantum algorithms for the generalized eigenvalue problem, focusing on energies near the Dirac point ($\sin k$1). Variational Rayleigh quotient minimization is ineffective for interior spectrum points, while landscape scanning [Rajchel-Mieldzioć et al., (Rajchel-Mieldzioć et al., 16 Jun 2025)] involves phase estimation to locate eigenvalues via minima in the singular value landscape of $\sin k$2.

This quantum approach offers quadratic speedup versus classical scanning, scaling as $\sin k$3 in system size and energy grid, at the price of polynomial dependence on precision.

Green Function Evaluation

The Green function is computed as $\sin k$4, which translates into an inhomogeneous equation in the pencil representation. Efficient quantum linear system solvers exploit the block encoding to compute selected matrix elements, with scaling polylogarithmic in $\sin k$5. A key requirement is adequate preconditioning to mitigate large condition numbers when $\sin k$6 is small.

This enables efficient computation of local density of states and scattering amplitudes, fundamental for simulating transport in topological materials.

Numerical Results and Claims

The paper establishes that the tangent-fermion discretization, when block-encoded via a generalized eigenvalue pencil, matches the circuit complexity of elliptic operators—$\sin k$7 term count and subnormalization—independent of lattice size. This sharply contradicts prior understandings for the tangent scheme, where exponential scaling renders the approach intractable. The claim is supported by rigorous circuit analysis and explicit LCU decompositions.

Practical and Theoretical Implications

The formalism enables quantum simulation of lattice Dirac operators retaining chiral symmetry and avoiding fermion doubling, essential for modeling topologically protected phases with unpaired Dirac cones. Efficient block encoding supports practical quantum algorithms for spectral and Green function analysis, relevant to condensed matter and field theory applications.

The theoretical advance demonstrates that nonlocal discretizations, previously deemed quantum-infeasible due to their LCU profile, can be rendered efficient by appropriate operator re-factorization. Further, the ability to handle spatial inhomogeneity implies applicability to materials with disorder and interfaces.

Future Directions

Key open problems include comprehensive resource estimates for quantum simulation using tangent fermions, particularly in presence of realistic preconditioners and multi-particle extensions to second quantization. Potential developments involve direct simulation of Luttinger liquids in quantum spin Hall systems, where the tangent discretization maintains massless spectrum in interacting regimes, in contrast to SLAC schemes [Zakharov et al., 2024; Wang et al., 2023]. Leveraging this approach may lead to efficient quantum algorithms for strongly correlated systems and topological matter.

Conclusion

The paper rigorously shows that the tangent-fermion Dirac operator, reformulated as a local Hermitian pencil, enables quantum circuit decomposition with optimal LCU properties, overcoming previously intractable scaling and preserving fundamental symmetries. This result facilitates efficient quantum simulation of fermionic systems free of doublers, with broad implications for quantum algorithms in lattice field theory and condensed matter physics (2606.19020).