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Quantum State Certification via Effective Parent Hamiltonians from Local Measurement Data

Guy-Philippe Nadon, Guanyi Heng, Pacôme Gasnier, Antoine Lemelin, Camille Coti, Zeljko Zilic, Mikko Möttönen, Ville Kotovirta, Toni Annala, Ernesto Campos, Jacob Biamonte

Abstract

The preparation and certification of quantum states is a fundamental challenge across quantum information technology. We introduce a tomography-free state certification method that lower-bounds the fidelity by estimating expectation values of engineered parent-Hamiltonian terms from local measurement data. We apply this framework to construct a parent Hamiltonian that enables certification and variational optimization across the Dicke-state family, which includes the single-excitation $W_n$ state. We experimentally validate the framework on IBM quantum hardware, certifying genuine multipartite entanglement for $W_n$ states up to six qubits and establishing positive lower bounds on the state fidelity up to thirteen qubits. For Dicke states with two- and three-excitations, we certify genuine multipartite entanglement up to seven qubits. Within this stringent certification framework, these results constitute among the largest witness-certified demonstrations of such states on a programmable quantum processor.

Quantum State Certification via Effective Parent Hamiltonians from Local Measurement Data

Abstract

The preparation and certification of quantum states is a fundamental challenge across quantum information technology. We introduce a tomography-free state certification method that lower-bounds the fidelity by estimating expectation values of engineered parent-Hamiltonian terms from local measurement data. We apply this framework to construct a parent Hamiltonian that enables certification and variational optimization across the Dicke-state family, which includes the single-excitation state. We experimentally validate the framework on IBM quantum hardware, certifying genuine multipartite entanglement for states up to six qubits and establishing positive lower bounds on the state fidelity up to thirteen qubits. For Dicke states with two- and three-excitations, we certify genuine multipartite entanglement up to seven qubits. Within this stringent certification framework, these results constitute among the largest witness-certified demonstrations of such states on a programmable quantum processor.
Paper Structure (10 sections, 7 theorems, 44 equations, 4 figures)

This paper contains 10 sections, 7 theorems, 44 equations, 4 figures.

Table of Contents

  1. Stability Bound as a Tomography-Free Certification Tool
  2. Parent Hamiltonians to Certify Dicke States
  3. Dicke Family Entanglement Witnesses
  4. Experimental Results
  5. Discussion
  6. Acknowledgment
  7. Variational principle
  8. Entanglement witnesses
  9. Dicke states parent Hamiltonians
  10. Dicke states result

Key Result

Lemma 1

Let $H$ be a positive semidefinite Hamiltonian with ground-state projector $\ket{\psi}\!\bra{\psi}$ of eigenvalue $0$, and let the spectral gap above the ground state be $\Delta\ge 1$. Then for any density operator $\rho$ on the same Hilbert space,

Figures (4)

  • Figure 1: State-preparation circuits for $W_4$ states. The lower circuit shows the compilation of the textbook construction into the native gate set of the superconducting processor.
  • Figure 2: Expectation value versus number of qubits for prepared $W_n$ states. The blue dashed line indicates the spectral gap $\Delta=1$, and the purple curve shows the entanglement-witness threshold $\alpha=1-1/n$.
  • Figure 3: Expectation value versus the number of qubits for prepared Dicke states. Panel (a) corresponds to excitation number $k=2$, and panel (b) to $k=3$. The blue dashed curve indicates entanglement-witness threshold as introduced in Proposition \ref{['prop:Dicke_max_biseparable_fidelity']}.
  • Figure 4: The layout of ibm_quebec

Theorems & Definitions (13)

  • Lemma 1: Stability lower bound
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Definition 1: Hamming weight operator
  • ...and 3 more