Beyond asymptotic reasoning: the practicalities of a quantum ground state projector based on the wall-Chebyshev expansion

TL;DR

The paper tackles ground-state preparation by introducing a wall-Chebyshev projector, a Chebyshev expansion of the wall function that targets the ground state while potentially reducing circuit depth. It develops both product-LCU and generalized quantum signal processing (GQSVT) encodings of the projector, analyzes their success probabilities and scalability, andBenchmark comparisons against leading approaches such as VITE, QITE, PITE, and eigenstate filtering. A key finding is a polynomial-depth speedup in the number of Hamiltonian applications needed to reach a given fidelity, balanced by potentially exponential decay in success probability for some encodings; GQSVT-based implementations can retain nontrivial success probabilities in practical regimes. The work suggests the wall-Chebyshev approach is promising for early fault-tolerant quantum computing and strongly correlated systems where accurate $E_0$ estimates are unavailable, offering a distinct trade-off between depth and repetitions.

Abstract

We consider a quantum algorithm for ground-state preparation based on a Chebyshev series approximation to the wall function. In a classical setting, this approach is appealing as it guarantees rapid convergence. We analyze the asymptotic scaling and success probabilities of different quantum implementations and provide numerical benchmarks, comparing the performance of the wall-Chebyshev projectors with current state-of-the-art approaches. We find that this approach requires fewer serial applications of the Hamiltonian oracle to achieve a given ground state fidelity, but is severely limited by exponentially decaying success probability. However, we find that some implementations maintain non-trivial success probability in regimes where wall-Chebyshev projection leads to a fidelity improvement over other approaches. As the wall-Chebyshev projector is highly robust to loose known upper bounds on the true ground state energy, it offers a potential resource trade-off, particulary in the early fault-tolerant regime of quantum computation.

Figures (11)

• Figure 1: Different order Chebyshev polynomial expansions of the wall function.
• Figure 2: Encoding circuits for wall-Chebyshev projector of order $m$. Top: Product of LCUs corresponding to each term in Eq. 34. Bottom: GQSVT implementation following Ref. Sunderhauf2023. $P_x$ can be either $P_1$ or $P_2$ as defined in Sec. IV, depending on the phases acting on the first ancilla qubit.
• Figure 3: Theoretical success probability of different implementations of the wall-Chebyshev projector. The product form assumes the upper bound on $\alpha_v = \alpha + |a_v|$.
• Figure 4: Projector convergence onto the ground state of the two-site Hubbard model with $t = 1$ and different values of $U$. The left panels show energy convergence, while the right show fidelity with the ground state wavefunction, as a function of polynomial order. In all cases, the ground state energy is known a priori and given as a parameter to the projector.
• Figure 5: Projector convergence onto the ground state of the two-site Hubbard model with $t = 1$ and different values of $U$. The left panels show energy convergence, while the right show fidelity with the ground state wavefunction, as a function of polynomial order. In all cases, the ground state energy is estimated as the energy of the Hartree--Fock state.