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 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 spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special ``staggered’’ grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form \bm{\mathcal H}\bm{\psi}=\bm{E}\bm{\mathcal P}\bm{\psi} ℋ 𝛙 = 𝐄 𝒫 𝛙 , with Hermitian tight-binding operators \bm{\mathcal H} ℋ , \bm{\mathcal P} 𝒫 , a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.
In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries (LCU)-based block-encodings for the first-order derivative and Laplacian operators in the non-local N=2n-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense LCU amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm (QLSA). For a d-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity O(dn3α(k)log(1/ε)), where α(k) is the subnormalisation factor of the order-k SLAC block-encoding and ε denotes the algorithmic approximation error.
Abstract Linear combination of unitaries (LCU) decompositions have become a key tool for encoding operators on quantum computers, enabling efficient implementations of arbitrary operators. In particular, LCU methods provide a way to encode the electronic structure Hamiltonian into quantum circuits. Over the years, various decomposition techniques have been developed for this purpose. Here, we introduce the Majorana tensor decomposition, a framework that unifies existing LCU approaches and introduces novel decompositions using low-rank tensor factorizations. We benchmark a range of decomposition techniques on small molecular systems and hydrogen chains of increasing sizes, evaluating their performance across different LCU methods.
The Dirac quantum walk (a 1+1 dimensional space-time discretization of the Dirac equation) has a 2μ mass gap both at the center and at the corner of the quasi-energy-momentum Brillouin zone. Gupta and Short recently noted [Quantum 9, 1845 (2025)] that the Dirac vacuum can create a particle-hole pair at the zone corner with the release of an energy 2μ to the environment. They removed this vacuum instability at the expense of fermion doubling, the appearance of a second low-energy Dirac cone. Here we show that an alternative discretization scheme, with a tangent rather than a sine dispersion relation, offers stability while retaining a single Dirac cone. The key step is the Cayley transformation from the unit circle of Floquet eigenvalues e−iε to the real line of unbounded energies E=2tan(ε/2). We compute the Schwinger effect (particle-hole pair creation in a uniform electric field) for tangent fermions and show that the pair-production rate agrees with the continuum result for a single Dirac cone. The zone corner is exactly decoupled if the scalar potential is coupled through the Hermitian generator of the quantum map. If it is coupled as a split operator, in order to preserve exact gauge invariance on the lattice, the corner does contribute - with a weight that vanishes quadratically with the lattice constants, in contrast to the Dirac quantum walk where the zone-corner instability survives the continuum limit.
We give the definitive proof that the Dirac Quantum Cellular Automaton (QCA) used for both quantum simulation and algorithmic foundations of Quantum Field Theory (QFT), and especially of Quantum Electrodynamics (QED), as put forward in References https://doi.org/10.1007/s11128-019-2555-4 and https://doi.org/10.22331/q-2023-11-08-1179, does exhibit Fermion Doubling (FD), albeit thrice as less severe as discrete-time standard Lattice Gauge Theories (LGTs) [arXiv:2505.0790], which are naive regarding the spacetime discretization of differential operators acting on fermionic fields. The proof is done for the (1 + 1)D Dirac-QCA model. We show that the (one-time-step) two-point correlation function, also called Green's function (GF), of the Dirac QCA, is of astonishing simplicity, which is in contrast with the GF of the Dirac equation. We also compare, both qualitatively and quantitatively, this Dirac QCA to the continuous-time-LGT spatial discretization of Dirac fermions regarding how well these two lattice models approximate their naive continuum limit$\unicode{x2014}$which is nothing but the Dirac equation$\unicode{x2014}$even when far away from that limit, a situation which must be considered because of experimental limitations in quantum simulation: the Dirac QCA is better for ultrarelativistic regimes, whereas continuous-time LGT is better for non-relativistic regimes. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)$\unicode{x2014}$which staggers an extra, artificial flavor only, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA two-point correlation function is of extreme simplicity, and can be expressed in a very simple manner in terms of the four chiral components of the FD-suffering, original-model GF.
Quantum algorithms based on Quantum Signal Processing (QSP) offer the potential for speedups across a broad range of applications, with block encodings serving as the central input model. In this framework, non-unitary matrices are embedded into larger unitary operators, and the cost of constructing these encodings often dominates the overall gate complexity. In this work, we introduce Tag-and-Restore Encoding (TARE), a block-encoding method for linear combinations of Pauli strings. In this method coefficient magnitudes are absorbed into a unitary built from a set of mutually anti-commuting Pauli strings acting on the system register. These Pauli strings are then mapped to the target Pauli strings through appropriate transformations, yielding a block encoding of the target operator. The ancilla register size scales logarithmically with the number of Pauli strings and can be extended to larger registers providing a width/depth tradeoff. We evaluate TARE through numerical simulations of the transverse-field Ising model, the Jordan-Wigner image of a fermionic star Hamiltonian, and random Pauli-string operators. Compared with standard Linear Combination of Unitaries (LCU), TARE substantially reduces the T-gate count while improving circuit depth in several cases. These results suggest that TARE can provide resource-efficient block encodings for a wide range of relevant systems.