Exact chiral symmetry with quantum signal processing

Technical Summary: Exact Chiral Symmetry with Quantum Signal Processing

Problem Statement
Quantum simulation of nonperturbative, nonequilibrium observables in lattice Quantum Chromodynamics (QCD) faces significant challenges regarding fermion doubling and the preservation of chiral symmetry in Hamiltonian formulations. While the Ginsparg-Wilson (GW) relation and the overlap operator provide elegant solutions in Euclidean spacetime, a canonical Hamiltonian analogue for overlap fermions has been lacking. Existing approaches, such as domain-wall fermions, map well to Hamiltonian settings but require an explicit extra dimension, increasing qubit costs. Conversely, overlap fermions offer a formulation in the physical spatial dimensions but involve highly nonlocal all-to-all interactions, specifically the sign function of the Wilson-Dirac Hamiltonian, $\varepsilon(h_W)$, which is difficult to implement efficiently on quantum hardware. The central problem addressed is how to construct efficient quantum algorithms for chiral-symmetric lattice fermions that balance memory (qubit) costs against gate complexity while maintaining exact (or controllably broken) chiral symmetry.

Methodology
The authors propose a Quantum Signal Processing (QSP) algorithm to simulate the overlap fermion Hamiltonian. The methodology proceeds through several key steps:

1. Hamiltonian Formulation: The work utilizes the overlap Hamiltonian $H_{ov} = \psi^\dagger (\Gamma_0 + \varepsilon(h_W)) \psi$, where $h_W$ is the Wilson-Dirac single-particle Hamiltonian. The sign function $\varepsilon(X) = X/\sqrt{X^\dagger X}$ is approximated by a polynomial $p_M(x)$ of degree $M$.
2. Block Encoding: The authors construct a block encoding of the single-particle Wilson Hamiltonian $\hat{h}_W$ (including gauge fields) using a "prepare" operator $P$ and a "select" operator $S$. This encodes the Hamiltonian into a unitary operator $U_{\hat{h}_W}$ acting on an extended Hilbert space with ancilla qubits.
3. Quantum Signal Processing (QSP): To implement the sign function approximation, the authors employ QSP (specifically Quantum Singular Value Transformation). This allows the application of a degree-$M$ polynomial $p_M(U_{\hat{h}_W})$ to the block-encoded operator. The polynomial is chosen to approximate the sign function within a spectral gap $[\kappa, 1]$ with an error $\epsilon_e$.
4. Time Evolution: Once the overlap Hamiltonian is block-encoded, the time-evolution operator $e^{-iHt}$ is approximated using QSP, requiring a number of gates scaling with the block-encoding cost and the evolution time.

Key Contributions and Results

• Algorithmic Construction: The paper provides a concrete QSP-based algorithm for the overlap Hamiltonian that preserves the GW relation up to a controllable error $\epsilon_e$. The modified chiral operator $\hat{\gamma}_5$ is shown to commute with the approximate Hamiltonian up to an error of order $\epsilon_e$.
• Complexity Analysis:
  • Gate Complexity: The cost to block-encode the overlap Hamiltonian scales as $O(Q \kappa^{-1} \log(1/\epsilon_e))$, where $Q$ is the number of lattice sites (times internal degrees of freedom) and $\kappa$ is related to the spectral gap. The total gate complexity for time evolution scales as $O(Q \kappa^{-1} \log(1/\epsilon_e) [Qt + \log(1/\epsilon_t)])$.
  • Qubit Cost: The algorithm requires $O(Q + \log Q)$ qubits. This is a significant reduction compared to domain-wall fermions, which require $O(Q L_5)$ qubits, where $L_5$ is the extent of the extra dimension.
  • Comparison: While domain-wall fermions benefit from geometric locality (allowing nearly linear cost $O(Q t \text{polylog})$ via Suzuki-Trotter or similar methods), the overlap formulation incurs a higher gate depth due to the nonlocal nature of the sign function approximation. However, the overlap approach offers superior asymptotic scaling in memory.
• Physical Interpretation of QSP: The authors demonstrate that the polynomial degree $M$ required to achieve error $\epsilon_e$ scales as $M = O(\kappa^{-1} \log(1/\epsilon_e))$. They identify this scaling with the size of the extra dimension $L_5$ in domain-wall fermions, where the error scales as $e^{-c L_5}$. Thus, the QSP implementation effectively "constructs" an extra dimension through the depth of the circuit, mirroring the physical correspondence between the overlap operator and the boundary theory of domain-wall fermions.

Significance and Claims
The paper claims that quantum simulations of Dirac fermions with exact chiral symmetry are "nearly free" in the sense that applying the overlap Hamiltonian costs only a logarithmic factor more (in terms of error tolerance) than the Wilson-Dirac Hamiltonian.

The central insight is that QSP provides a quantum-algorithmic realization of the known correspondence between the overlap operator and the extra dimension of domain-wall fermions. The trade-off is explicit:

• Domain-wall fermions: Higher qubit cost ($O(Q L_5)$) but lower gate depth due to locality.
• Overlap fermions (via QSP): Lower qubit cost ($O(Q)$) but higher gate depth ($O(Q^2)$ worst-case scaling for time evolution due to nonlocality).

The authors conclude that the choice between formulations depends on hardware constraints (qubit count vs. circuit depth). They note that while classical lattice QCD often uses rational approximations (e.g., Zolotarev) for the sign function which are more efficient, these do not have direct analogues in standard QSP, suggesting a direction for future work. The paper also clarifies that the scaling results are robust against the specific encoding of gauge fields, though constant factors may vary.