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Active Sampling Sample-based Quantum Diagonalization from Finite-Shot Measurements

Rinka Miura

Abstract

Near-term quantum devices provide only finite-shot measurements and prepare imperfect, contaminated states. This motivates algorithms that convert samples into reliable low-energy estimates without full tomography or exhaustive measurements. We propose Active Sampling Sample-based Quantum Diagonalization (AS-SQD), framing SQD as an active learning problem: given measured bitstrings, which additional basis states should be included to efficiently recover the ground-state energy? SQD restricts the Hamiltonian to a selected set of basis states and classically diagonalizes the restricted matrix. However, naive SQD using only sampled states suffers from bias under finite-shot sampling and excited-state contamination, while blind random expansion is inefficient as system size grows. We introduce a perturbation-theoretic acquisition function based on Epstein--Nesbet second-order energy corrections to rank candidate basis states connected to the current subspace. At each iteration, AS-SQD diagonalizes the restricted Hamiltonian, generates connected candidates, and adds the most valuable ones according to this score. We evaluate AS-SQD on disordered Heisenberg and Transverse-Field Ising (TFIM) spin chains up to 16 qubits under a preparation model mixing 80\% ground state and 20\% first excited state. Furthermore, we validate its robustness against real-world state preparation and measurement (SPAM) errors using physical samples from an IBM Quantum processor. Across simulated and hardware evaluations, AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion. Detailed ablation studies demonstrate that physics-guided basis acquisition effectively concentrates computation on energetically relevant directions, bypassing exponential combinatorial bottlenecks.

Active Sampling Sample-based Quantum Diagonalization from Finite-Shot Measurements

Abstract

Near-term quantum devices provide only finite-shot measurements and prepare imperfect, contaminated states. This motivates algorithms that convert samples into reliable low-energy estimates without full tomography or exhaustive measurements. We propose Active Sampling Sample-based Quantum Diagonalization (AS-SQD), framing SQD as an active learning problem: given measured bitstrings, which additional basis states should be included to efficiently recover the ground-state energy? SQD restricts the Hamiltonian to a selected set of basis states and classically diagonalizes the restricted matrix. However, naive SQD using only sampled states suffers from bias under finite-shot sampling and excited-state contamination, while blind random expansion is inefficient as system size grows. We introduce a perturbation-theoretic acquisition function based on Epstein--Nesbet second-order energy corrections to rank candidate basis states connected to the current subspace. At each iteration, AS-SQD diagonalizes the restricted Hamiltonian, generates connected candidates, and adds the most valuable ones according to this score. We evaluate AS-SQD on disordered Heisenberg and Transverse-Field Ising (TFIM) spin chains up to 16 qubits under a preparation model mixing 80\% ground state and 20\% first excited state. Furthermore, we validate its robustness against real-world state preparation and measurement (SPAM) errors using physical samples from an IBM Quantum processor. Across simulated and hardware evaluations, AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion. Detailed ablation studies demonstrate that physics-guided basis acquisition effectively concentrates computation on energetically relevant directions, bypassing exponential combinatorial bottlenecks.
Paper Structure (19 sections, 13 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 13 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Statement
  4. Active SQD via Epstein--Nesbet Perturbation Theory
  5. Algorithm
  6. Overview
  7. Pseudocode
  8. Baselines
  9. Experimental Setup
  10. Model: Disordered Heisenberg Chain
  11. Imperfect Preparation and Finite Shots
  12. Protocol and Metrics
  13. Results
  14. Performance and Scaling under Contaminated Sampling
  15. Real Hardware Validation on IBM Quantum
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Energy error vs. system size for the Heisenberg model under exact contaminated sampling (median over 5 disorder instances).
  • Figure 2: Energy error vs. system size for the TFIM model under exact contaminated sampling (median over 5 disorder instances).
  • Figure 3: Energy error vs. system size using physical samples from IBM Quantum (ibmq_pittsburgh). Note the remarkable recovery at $n=8$ where AS-SQD successfully filters hardware noise to identify the exact ground state subspace.
  • Figure 4: Ablation study of acquisition functions at $n=16$ (Heisenberg model under exact contaminated sampling). The full perturbation score (en) and coupling-only significantly outperform energy-based heuristics (denom, diag) and standard baselines.
  • Figure 5: Representative error trace ($n=12$ Heisenberg model) demonstrating that extending candidate proposals to 2-hops slows down convergence compared to the standard 1-hop approach. The 1-hop connectivity effectively acts as a strong inductive bias under a fixed addition budget.