A Shadow Enhanced Greedy Quantum Eigensolver

TL;DR

The Shadow Enhanced Greedy Quantum Eigensolver (SEGQE) is introduced as a greedy, shadow-assisted framework for measurement-efficient ground-state preparation and is established as a measurement-efficient state-preparation primitive well suited to early fault-tolerant quantum computing architectures.

Abstract

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.

Figures (7)

• Figure 1: Schematic overview of the Shadow-Enhanced Greedy Quantum Eigensolver (SEGQE). At iteration $k$, a quantum computer prepares the state $|\psi_k\rangle\xspace=C_k|\psi_0\rangle\xspace$ and $N$ independent classical shadows are collected by applying random unitaries $V\in \mathcal{U}$ and measuring in the computational basis. In this work, $\mathcal{U}=\{I, H, S^\dagger H\}^{\otimes n}$, corresponding to uniformly random single-qubit Pauli measurements. The shadows are then used to estimate all Pauli expectation values needed to evaluate, for each candidate gate $U_j(\theta)\in\mathcal{G}$, the energy decrease $\Delta E_{k,j}(\theta)$. A classical optimizer finds $(j^*, \theta^*_{j^*})$ maximizing the expected decrease and appends $U_{j^*}(\theta^*_{j^*})$ to obtain $C_{k+1}$. The procedure repeats until convergence or until a predefined circuit depth is reached.
• Figure 2: Convergence properties of SEGQE applied to the open-boundary transverse-field Ising model at criticality ($J=w=1$, blue circles) and a gapped, field-dominated regime ($J=w/2$, orange circles). (Top) Relative energy error $\delta E=(E_\mathrm{exact}{-}E_\mathrm{SEGQE})/E_\mathrm{exact}$ at convergence as a function of system size $n$. (Middle) Infidelity with the exact ground state $1-\mathcal{F}$. (Bottom) Number of appended two-qubit Pauli rotations required for convergence $N_\mathrm{Gates}$. In all cases, SEGQE terminates once no candidate gate yields an energy reduction exceeding $\Delta=10^{-3}$.
• Figure 3: Qualitative comparison between SEGQE and a variational quantum eigensolver (VQE) based on a variational Hamiltonian ansatz (VHA) applied to the open-boundary transverse-field model at the critical point $J=w=1$. (Top) Relative energy error $\delta E = (E_\mathrm{exact}{-}E)/E_\mathrm{exact}$ at convergence as a function of system size $n$. (Bottom) Number of iterations $N_\mathrm{iter}$ required for convergence. Results are shown for SEGQE (blue circles) and for VHA-VQE with one ($L=1$, orange squares) and two ($L=2$, green squares) ansatz layers. For VHA-VQE, circuit parameters are initialized randomly and results are averaged over 300 independent initializations. Error bars indicate the standard error of the mean but are smaller than the marker size. Both algorithms terminate once the energy difference between two successive iterations falls below $\Delta=10^{-3}$.
• Figure 4: Convergence properties of SEGQE applied to 500 instances of the random local Hamiltonians defined in \ref{['eq:RLH']}. (Top) Relative energy error $\delta E=(E_\mathrm{exact}{-}E_\mathrm{SEGQE})/E_\mathrm{exact}$ at convergence as a function of system size $n$. (Middle) Ground-state fidelity $\mathcal{F}$. (Bottom) Number of appended two-qubit gates required for convergence $N_\mathrm{Gates}$. Results are shown for SEGQE using all two-qubit Pauli rotations ($\mathcal{P}^2$, orange) and for the restriction to nearest-neighbor two-qubit Pauli rotations (NN-$\mathcal{P}^2$, blue). Solid lines indicate the mean over all considered instances, dashed lines the median. Shaded regions indicate the two-sigma confidence interval of the mean. In all cases, SEGQE terminates once no candidate gate yields a maximal energy reduction exceeding $\Delta=10^{-3}$.
• Figure 5: Dependence of SEGQE convergence on gate-set for random local Hamiltonians at fixed system size $n=5$. Shown is the relative energy error $\delta E=(E_\mathrm{exact}{-}E_\mathrm{SEGQE})/E_\mathrm{exact}$ as a function of the number of appended gates $N_\mathrm{Gates}$ for four gate sets: nearest-neighbor two-qubit rotations (NN-$\mathcal{P}^2$, blue), arbitrary-distance two-qubit Pauli rotations ($\mathcal{P}^2$, orange), nearest-neighbor arbitrary two-qubit unitaries (NN-$\mathrm{SU}(4)$, green), and arbitrary-distance two-qubit unitaries ($\mathrm{SU}(4)$, red). Solid lines indicate the mean over 100 independent random Hamiltonian instances, and shaded regions indicate the one-sigma confidence interval of the mean. Horizontal lines show the average final energy error at convergence for each gate set, while vertical dashed lines indicate the corresponding average number of appended gates at convergence. SEGQE terminates once no candidate gate yields a maximal energy reduction exceeding $\Delta=10^{-3}$.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 1
• Lemma 1: Second Moments of Pauli Shadows
• proof
• Corollary 2: Covariances of Pauli Shadows
• Lemma 2: Variance of Energy-Difference Estimators
• proof
• Lemma 3: Single-Shot Bound for Energy-Difference Estimators
• proof