Probabilistic imaginary-time evolution via compiled diagonal representations

Abstract

We introduce a probabilistic imaginary-time evolution scheme based on a compiled approximate diagonal representation of the Hamiltonian. A basis transformation is optimized offline. For a fixed compiled representation, the online circuit structure and gate count per attempt are independent of the imaginary time, and the same representation can be reused across different input states and evolution times. We derive a condition for enhanced success probability. Numerical tests show reduced online cost relative to termwise Pauli-based methods at a common accuracy threshold. We also examine how approximation errors and offline preparation costs affect the overall benefit.

Probabilistic imaginary-time evolution (PITE) incurs product-formula costs and postselection losses when implemented termwise. This paper asks whether an offline-compiled approximate diagonal representation can reduce online costs. It fits a short-time real-time propagator as $WDW^\dagger$, retains the basis circuit $W$, and recovers diagonal Hamiltonian coefficients by trace projection from the physical Hamiltonian, avoiding phase-branch ambiguities. For $\widetilde H=W(\sum_k\xi_k Z_k)W^\dagger$, commuting ancilla-assisted filters implement $e^{-\beta\Lambda}e^{-\beta\widetilde H}$, where $\Lambda=\sum_k|\xi_k|$. At fixed representation, changing $\beta$ changes only rotation angles, not circuit structure or gate count per attempt. With exact representation and vanishing baseline Trotter error, the success-probability ratio against termwise PITE is $\exp[2\beta(\lambda-\Lambda)]$, with $\lambda$ the original Pauli coefficient one-norm. Noiseless benchmarks use a two-qubit Hubbard model, five-qubit XY chain and three-qubit XYZ chain; most results use phase-derived coefficients, with projection checked separately. At $\beta=1.1$, mean Haar-input infidelities are approximately $0.00245$, $0.00158$ and $0.00408$, with success gains of $21$, $263$ and $123$ over first-order termwise PITE. Matched-accuracy accounting includes early abort and retries. Grouped two-site filtering, however, costs about $3.3\times10^3$ CX per accepted XYZ output versus $2.36\times10^5$ for the compiled method. The stated limitations include unquantified total offline cost, potentially tiny absolute success probabilities, input-dependent symmetry leakage, and failed 1% accuracy targets for tested four- and five-qubit XYZ fits. Efficient scaling, hardware performance and overall amortized advantage remain unestablished.