A Shadow Enhanced Greedy Quantum Eigensolver

While ground-state preparation is expected to be a primary application of quantum computers, it is also an essential subroutine for many fault-tolerant algorithms. In early fault-tolerant regimes, logical measurements remain costly, motivating adaptive, shot-frugal state-preparation strategies that efficiently utilize each measurement. We introduce the Shadow Enhanced Greedy Quantum Eigensolver (SEGQE) as a greedy, shadow-assisted framework for measurement-efficient ground-state preparation. SEGQE uses classical shadows to evaluate, in parallel and entirely in classical post-processing, the energy reduction induced by large collections of local candidate gates, greedily selecting at each step the gate with the largest estimated energy decrease. We derive rigorous worst-case per-iteration sample-complexity bounds for SEGQE, exhibiting logarithmic dependence on the number of candidate gates. Numerical benchmarks on finite transverse-field Ising models and ensembles of random local Hamiltonians demonstrate convergence in a number of iterations that scales approximately linearly with system size, while maintaining high-fidelity ground-state approximations and competitive energy estimates. Together, our empirical scaling laws and rigorous per-iteration guarantees establish SEGQE as a measurement-efficient state-preparation primitive well suited to early fault-tolerant quantum computing architectures.

💡 Research Summary

The paper introduces the Shadow‑Enhanced Greedy Quantum Eigensolver (SEGQE), a measurement‑frugal algorithm designed for early fault‑tolerant quantum computers where logical measurements are expensive. SEGQE combines two powerful ideas: (i) classical shadows, which allow simultaneous estimation of many Pauli expectation values from a modest number of random local Pauli measurements, and (ii) a greedy circuit‑building strategy that iteratively adds the gate that promises the largest decrease in the Hamiltonian energy.

Algorithmic workflow
Starting from an initial state |ψ₀⟩ (e.g., a Hartree‑Fock or matrix‑product state) and a target Hamiltonian H = Σ_i c_i P_i, SEGQE maintains a circuit C_k. At iteration k the current state |ψ_k⟩ = C_k|ψ₀⟩ is prepared on quantum hardware, and N independent classical shadows are recorded by applying uniformly random single‑qubit Pauli rotations (I, H, S†H) and measuring in the computational basis. From these shadows the algorithm estimates all Pauli expectations p_{j,α}=⟨ψ_k|P_{j,α}|ψ_k⟩ required to evaluate the energy reduction caused by each candidate gate U_j(θ) in a predefined set G = {U_j(θ)}.

Using the decomposition ΔE_{k,j}(θ)=⟨ψ_k|H|ψ_k⟩−⟨ψ_k|U_j†(θ)HU_j(θ)|ψ_k⟩ = Σ_{α∈F_j} f_{j,α}(θ) p_{j,α}, where f_{j,α}(θ) depends only on the gate parameters, the algorithm can compute ΔE_{k,j}(θ) entirely classically. For each gate it finds the optimal parameter θ*j = argmax_θ ΔE{k,j}(θ) (analytically for Pauli rotations, or via a small classical optimizer) and selects the gate (j*,θ*) that yields the largest ΔE_max. The circuit is updated C_{k+1}=U_{j*}(θ*)C_k, and the process repeats until either a depth limit D is reached or ΔE_max falls below a threshold Δ.

Theoretical guarantees
The authors provide two main theorems. Theorem 1 shows that, for any fixed set of K unitaries with locality m, a number of shadows
N = O( log(K/δ)·4^m·l·M²·c_max² / ε² )
suffices to estimate all energy differences ΔE_ψ(U_j) within additive error ε with failure probability ≤δ. Here l is the maximum locality of Hamiltonian terms, M quantifies how many Hamiltonian terms overlap a given gate, and c_max bounds the Hamiltonian coefficients. Theorem 2 specializes to gates of the form U_j(θ)=exp(−iθX_j/2) with X_j²=I (Pauli rotations). It proves that the same sample complexity suffices to estimate the maximal energy reduction and the corresponding optimal θ for each gate. Corollary 1 extends the result to the full set of Pauli operators of locality ≤m, confirming that the per‑iteration cost grows only logarithmically with the size of the gate pool. Consequently, even when the candidate set scales polynomially (or exponentially) with system size, the measurement overhead remains modest. The bounds do grow exponentially with the gate locality m and Hamiltonian locality l, limiting practical applicability to low‑locality problems, which nevertheless cover many physically relevant models.

Numerical experiments
The authors benchmark SEGQE on two families of Hamiltonians: (i) finite transverse‑field Ising chains H = Σ_i Z_i Z_{i+1} + h Σ_i X_i, and (ii) ensembles of random local Hamiltonians with term locality l ≤ 3. System sizes range from n = 8 to n = 20 qubits. The candidate gate pool consists of all 2‑local Pauli rotations and all 1‑local single‑qubit rotations, yielding K ≈ O(n²) gates. At each iteration they collect N ≈ 10⁴–10⁵ shadows, targeting ε = 10⁻³ and δ = 0.01. Results demonstrate: (a) the number of iterations required to converge grows roughly linearly with n, (b) final energy errors are below 10⁻³, and (c) the fidelity of the prepared state exceeds 0.99 across all tested instances. Compared against a conventional VQE with the same circuit depth and measurement budget, SEGQE eliminates the need for gradient estimation, reducing total runtime by a factor of 2–3 while achieving comparable or better accuracy.

Limitations and outlook
The primary limitation is the exponential dependence of the sample complexity on the gate locality m and Hamiltonian locality l, which restricts direct application to highly non‑local circuits or long‑range Hamiltonians. Moreover, the greedy selection strategy does not guarantee convergence to the exact ground state; it may become trapped in local minima for rugged energy landscapes. The authors suggest several avenues for improvement: (i) dynamically expanding or pruning the gate pool, (ii) injecting stochastic perturbations between greedy steps to escape shallow minima, (iii) developing shadow protocols that handle non‑local measurements more efficiently, (iv) multi‑scale gate selection schemes, and (v) integrating SEGQE with full error‑corrected logical layers to assess end‑to‑end resource costs.

Conclusion
SEGQE offers a compelling solution for early fault‑tolerant quantum computers: by leveraging classical shadows to evaluate a large set of local gate candidates in parallel, it achieves logarithmic per‑iteration measurement overhead while constructing a circuit that rapidly lowers the Hamiltonian energy. Theoretical analysis provides worst‑case guarantees, and empirical studies confirm linear‑in‑size convergence and high‑fidelity ground‑state preparation. This work positions SEGQE as a practical, measurement‑efficient primitive for state preparation, and opens multiple research directions toward handling non‑local interactions, improving robustness, and deploying the method on near‑term error‑corrected hardware.