Quantum Program Semantic Verification

---
name: quantum-program-semantic-verification
description: "Semantics-based verification methodology for quantum programs implementing number-theoretic algorithms. Covers oracle specification, refinement-style verification, and semantic auditing for Shor-style quantum algorithms. Based on arXiv:2605.01008."
---

# Quantum Program Semantic Verification

## Description

Semantics-first verification methodology for quantum programs, particularly those implementing number-theoretic algorithms like Shor's algorithm for ECDLP. Provides a framework for specifying oracles at the program semantics level, deriving verification obligations, and performing semantic auditing to ensure correctness before deployment.

Based on: *Semantics-Based Verification of an Implemented Shor Oracle for ECDLP in Qrisp* (arXiv:2605.01008, 2026-05-01)

## Activation Keywords
- quantum program verification
- shor oracle verification
- quantum semantic auditing
- ECDLP quantum implementation
- quantum oracle specification
- refinement verification quantum
- Qrisp verification
- 量子程序验证
- 量子语义审计
- quantum semantics

## Tools Used
- exec: Run quantum program verification tools, Qrisp compiler
- read: Analyze quantum program source code, verification obligations
- write: Create verification specifications, test cases

## Core Methodology

### Step 1: Oracle Specification at Semantic Level

Define the quantum oracle at the level of program semantics rather than gate-level implementation:

```python
# Specification: What the oracle should compute
# For ECDLP: O|x⟩|y⟩ = |x⟩|y ⊕ f(x)⟩
# where f(x) = x*P (elliptic curve point multiplication)

def oracle_specification():
    return {
        "input_state": "|x⟩|y⟩ where x ∈ Z_n, y ∈ E(F_p)",
        "output_state": "|x⟩|y ⊕ (x*P)⟩",
        "constraints": [
            "x must be well-formed (valid field element)",
            "P must be a valid point on the curve",
            "Controlled execution must preserve superposition semantics"
        ]
    }
```

### Step 2: Derive Refinement-Style Verification Obligations

Break down the oracle into components and derive verification conditions:

1. **Point-update primitive**: Verify against classical reference implementation
2. **Controlled execution**: Verify control law preservation under superposition
3. **Resource estimation**: Verify gate count and qubit requirements
4. **Modular arithmetic**: Verify correctness of finite field operations

### Step 3: Semantic Auditing

Key insight from the paper: **Controlled execution may violate expected control law even when trivial sanity checks pass.**

```python
# Audit checklist:
audit_items = [
    "Core primitive agrees with classical reference on well-formed inputs",
    "Controlled execution preserves superposition semantics",
    "No unintended phase kickback in control qubits",
    "Modular inverse operations are correctly implemented",
    "Point-at-infinity edge cases handled",
    "Quantum resource estimation matches theoretical bounds"
]
```

### Step 4: Complexity Argument

Provide high-level complexity analysis for the oracle family:
- Gate complexity: O(log³ n) for n-bit field elements
- Qubit requirements: 2n + ancilla qubits
- Depth: O(log² n) with parallel arithmetic

## Key Findings from arXiv:2605.01008

1. **Semantic sensitivity**: Shor-style quantum algorithms for ECDLP are highly sensitive to exact semantics of group-operation oracles
2. **Minor implementation choices** can invalidate the mathematical model
3. **Trivial control sanity checks are insufficient** - semantic auditing is required
4. **Controlled execution violations** may occur under the evaluated toolchain despite passing basic tests
5. **Semantic auditing is a practical prerequisite** for trustworthy ECDLP-oriented quantum software

## Verification Pattern for Number-Theoretic Quantum Algorithms

### For Shor's Algorithm (Factoring):
- Specify modular exponentiation oracle: U|a⟩|b⟩ = |a⟩|b·aˣ mod N⟩
- Verify: periodicity is preserved under quantum superposition
- Check: controlled-U operations don't introduce phase errors

### For ECDLP (Elliptic Curve):
- Specify point addition oracle: U|P⟩|Q⟩ = |P⟩|P+Q⟩
- Verify: group law is preserved for all valid inputs
- Check: point-at-infinity and identity element handling

## Error Handling

### Verification Fails on Controlled Execution
- Check: toolchain-specific controlled gate implementations
- Verify: no implicit phase factors in controlled operations
- Test: with known eigenstates to detect phase errors

### Oracle Disagrees with Classical Reference
- Check: input encoding conventions (endianness, representation)
- Verify: modular arithmetic implementation matches specification
- Test: boundary cases (0, 1, N-1, etc.)

## Related Skills
- quantum-error-correction-methods
- quantum-program-reliability
- quantum-ml-certification

## Resources
- arXiv:2605.01008 - Semantics-Based Verification of Shor Oracle for ECDLP
- Qrisp framework: https://qrisp.eu/
- Quantum program verification literature