Lecture Notes: Certifying Almost All Quantum States with Few Single Qubit Measurements

Introduction

• Host: Robert Wang
• Background:
  • Senior Research Scientist at Google Quantum AI
  • Visiting Scientist at MIT
  • Will join Caltech as Assistant Professor (2025)
  • PhD under John Preskill and Tomo Vidic
  • Research Areas: Quantum Information Theory, Learning Theory
  • Awards include: Milton and Francis Clauser Doctoral Prize, Google PhD Fellowship, Boeing Quantum Creator Prize, Ben PC True Doctoral Prize

Overview of Quantum State Certification

• Importance of creating quantum many-body systems with intricate entanglement for quantum computation.
• Challenges:
  • Experimental setups are subject to errors and noise.
  • Need to determine if the generated quantum state (ρ) is close to the target state (ψ).
• Definition of Certification:
  • Testing if ρ is close to ψ using measurement data.
  • Closeness measured using fidelity (F) or trace distance.

Challenges in Existing Certification Techniques

• Approach Zero:

  • Direct measurement using inverse operations.
  • Issues: Requires perfect implementation of inverse circuit, impractical.

• Approach One: Randomized Clifford measurements.

  • Efficiently predicts fidelity but requires linear depth circuits, challenging for large systems (e.g. 100+ qubits).

• Approach Two: Randomized Pauli measurements.

  • Requires exponential measurements for most target states, works for low-entangled states.

• Cross Entropy Benchmarking:

  • Measures in Z basis, but may not capture off-diagonal errors.

New Theorems Presented

Theorem 1

• Certification for almost all ENCB states with O(n²) single qubit measurements.
• Works without assumptions about errors in experimental setup.

Theorem 2

• For a chosen basis that induces a relaxation time, certification can also be done with O(t²) single qubit measurements.

Protocol for Certification

• Simple protocol involves:
  • Randomly select one qubit to measure in a different basis.
  • Measure all other qubits in a fixed basis.
  • Repeat for O(n²) times to get results.

Post-Processing Measurement Data

• Use the measurement outcomes to estimate fidelity through shadow overlap, a bias estimator linking measurement outcomes to fidelity.
• Shadow overlap tracks true fidelity closely, especially for certification.

Applications of Certification

1. Benchmarking:

   • Numerical experiments comparing shadow overlap vs. traditional methods under various noise conditions.

2. Machine Learning for Quantum State Tomography:

   • Using shadow overlap to certify neural network models representing quantum states.
   • Allows for accurate predictions by confirming the model against experimental states.

3. Quantum State Preparation Optimization:

   • Maximize shadow overlap to improve circuit design for generating desired states.

Conclusion

• Certification of almost all quantum states with few single qubit measurements is possible.
• Open questions about specific states that may defy certification and the computational complexities involved.

Questions and Discussion

• Audience questions addressed, including the relation of shadow overlap to the efficiency of the protocol and computational challenges in measuring fidelity.