Quantum Bayesian Inference

Quantum Bayesian Inference: Experimental Validation and Extensions · NCHU Department of Applied Mathematics

NSTC Undergraduate Research Project Project No. 112-2813-C-005-038-E · Advisor: Prof. Yu-Tsung Tai · Period: 2023/07 – 2024/02

Overview

This thesis investigated whether quantum computer outputs conform to Bayesian probability laws. Using IBM’s Qiskit framework on real IBMQ hardware and simulators, we tested the law of large numbers for quantum bit measurements and applied Bayesian updating to analyze qubit measurement outcomes.

The key finding: real quantum computers and simulators behave differently — the quantum computer’s cumulative probability showed a systematic drift away from the theoretical 0.5, raising fundamental questions about objective randomness in quantum states.

Methodology

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flowchart TD
    A["Phase 1: Law of Large Numbers     "] --> B["Build H-gate Quantum Circuit     "]
    B --> C["Run on Simulator (20,000 shots)     "]
    B --> D["Run on IBMQ Hardware (20,000 shots)     "]
    C --> E["Cumulative P(0) Visualization     "]
    D --> E
    E --> F["Compare Convergence Behavior     "]
    F --> G["Phase 2: Bayesian Inference     "]
    G --> H["Collect Measurement Results as CSV     "]
    H --> I["Compute Prior from First 50 Observations     "]
    I --> J["Iterative Bayesian Updating (20,000 rounds)     "]
    J --> K["Find MAP Estimate of P(0)     "]

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    style D fill:#FFF3E0,stroke:#FFCC80
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Phase 1: Law of Large Numbers Verification

We applied a Hadamard gate (H-gate) to a single qubit, putting it into superposition state \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\), then measured it 20,000 times. If quantum measurements follow classical probability, the cumulative \(P(|0\rangle)\) should converge to 0.5.

Setup Backend Shots Expected \(P(0)\)
H-gate \(\rightarrow\) Measure ibmq_qasm_simulator 20,000 0.5
H-gate \(\rightarrow\) Measure ibm_osaka (real hardware) 20,000 0.5

Simulator Result

The simulator’s cumulative probability converges smoothly to 0.5, perfectly following the law of large numbers:

Simulator: Cumulative P(0) over 20,000 measurements — converges to 0.5

Real Quantum Computer Result

The real quantum computer (IBM Osaka) shows a systematic downward drift — the cumulative probability does NOT converge to 0.5:

IBM Osaka: Cumulative P(0) over 20,000 measurements — drifts below 0.5

Key Finding: The simulator perfectly follows the law of large numbers, but the real quantum computer exhibits a persistent bias. This divergence suggests that either: (1) hardware noise introduces systematic error, (2) the sample size is insufficient, or (3) quantum superposition may not produce truly objective randomness.

Zoomed-in Comparison (1,000 shots)

A closer look at individual runs with 1,000 measurements each:

Simulator: 1,000 shots — tight convergence to 0.5

Simulator: Rapid, clean convergence to the theoretical 0.5 line.

Quantum Computer: 1,000 shots — wider oscillation around 0.5

IBMQ: Noticeably wider variance and slower convergence, with persistent oscillation.

Phase 2: Bayesian Inference on Quantum Outputs

In the second phase, we treated the quantum measurement outcomes as observed data and applied Bayesian updating to infer the most likely probability parameter \(\theta = P(|0\rangle)\).

Steps:

  1. Collected 20,000 measurement results from IBMQ as a binary sequence (0/1)
  2. Used the first 50 observations to construct the prior distribution over \(\theta\)
  3. Iteratively applied Bayes’ theorem for 19,950 rounds of updating
  4. Identified the MAP (Maximum A Posteriori) estimate of \(\theta\)

Bayesian Convergence

Bayesian updating: cumulative probability (blue) vs Bayesian posterior mode (red) — both converge, but the posterior MAP drifts toward ~0.483 instead of 0.5

Bayesian Result: With prior density \(n = 50\), the MAP estimate converged to \(\theta \approx 0.483\) with posterior probability 0.955. This confirms the quantum hardware’s systematic bias — Bayesian inference correctly identifies that the true \(P(|0\rangle)\) on real hardware is NOT exactly 0.5.

Conclusions

Aspect Simulator Real Quantum Computer
Law of Large Numbers Perfectly satisfied Violated (systematic drift)
Convergence to 0.5 Yes (smooth) No (biased toward ~0.49)
Bayesian MAP estimate \(\theta \approx 0.500\) \(\theta \approx 0.483\)
Variance Low High (noisy)

Three possible explanations for the divergence:

  1. Insufficient sample size — 20,000 shots may not be enough for quantum hardware (limited by IBMQ queue constraints)
  2. Hardware noise and error — current IBMQ devices have gate errors, decoherence, and readout errors that introduce systematic bias
  3. Quantum randomness is not classical randomness — the superposition state may follow measurement-theoretic probability models (e.g., Gleason’s theorem) that don’t fully align with Kolmogorov’s axioms

Tech Stack: Python, Qiskit, IBMQ (ibm_osaka), matplotlib, csv · Grant: NSTC 112-2813-C-005-038-E

Source Code

The Qiskit programs developed for this research are available on GitHub: