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:

# 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.

# 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