org.soulspace.qclojure.application.error-mitigation.symmetry-verification — org.soulspace/qclojure 0.9.0

Quantum symmetry verification for comprehensive error mitigation and circuit validation.

This namespace provides symmetry verification tools for quantum circuits, enabling sophisticated error mitigation strategies through systematic symmetry analysis. Symmetries in quantum circuits are fundamental for error detection and mitigation because they provide expected invariant properties that should be preserved during ideal quantum computation.

Key capabilities: • Parity symmetry verification for circuits with parity-preserving operations • Reflection symmetry analysis using bit-flip invariance properties • Permutation symmetry detection for circuits with qubit exchange invariance • Rotational symmetry analysis for systems with continuous rotational symmetry • Statistical significance testing using chi-squared and confidence intervals • Configurable thresholds and production-grade error reporting • Comprehensive violation analysis with severity classification • Automated corrective action recommendations

Symmetry Types Analyzed:

1. Parity Symmetry: For circuits containing only parity-preserving gates (H, S, T, Pauli, CNOT), the overall parity of measurement outcomes should be preserved. Violations indicate systematic measurement errors or decoherence effects.

2. Reflection Symmetry: Bit-flip symmetry where states |x⟩ and |x̄⟩ (bit-wise complement) should have equal measurement probabilities under certain circuit conditions. Critical for detecting calibration errors and crosstalk.

3. Permutation Symmetry: For circuits invariant under qubit permutations, measurement distributions should be equivalent when qubits are exchanged. Essential for validating hardware uniformity and detecting qubit-specific systematic errors.

4. Rotational Symmetry: For systems with continuous rotational invariance (e.g., certain 3-qubit systems), measurement outcomes should be invariant under specific rotational transformations.

Physical Background: Quantum error correction and mitigation rely heavily on symmetry properties because:

• Ideal quantum evolution preserves certain symmetries of the initial state and Hamiltonian
• Hardware noise and systematic errors often break these symmetries in characteristic ways
• Symmetry violations provide diagnostic information about error sources and magnitudes
• Statistical analysis of symmetry preservation enables quantitative error assessment

Production Features: • Configurable statistical significance testing with proper multiple comparison corrections • Scalable analysis algorithms for systems from 2 to 20+ qubits • Comprehensive violation classification with actionable diagnostic information • Performance optimization with sampling strategies for large Hilbert spaces • Integration with measurement error mitigation and hardware characterization protocols

apply-symmetry-verification clj

(apply-symmetry-verification circuit measurement-results & [config])

Production-ready symmetry verification with comprehensive analysis, statistical rigor, and actionable recommendations.

This function serves as the primary interface for quantum circuit symmetry verification in production environments. It combines all symmetry analyses into a unified assessment with pass/fail determination and specific corrective action recommendations.

Production Workflow:

1. Comprehensive Analysis: Performs all enabled symmetry tests using detect-symmetry-violations
2. Multi-Criteria Assessment: Evaluates multiple symmetry scores and violation patterns
3. Statistical Validation: Applies rigorous statistical significance testing
4. Severity Classification: Categorizes violations by impact and urgency
5. Actionable Recommendations: Generates specific corrective actions based on violation patterns
6. Performance Metrics: Tracks analysis execution and provides diagnostic information

Pass/Fail Criteria: The verification passes only when ALL of the following conditions are met:

• Overall symmetry score ≥ configured minimum threshold
• Number of violations ≤ maximum allowed violations
• No high-severity violations detected
• Statistical significance requirements satisfied (when applicable)
• Individual symmetry component scores meet their respective thresholds

Corrective Action System: The function provides targeted recommendations based on specific violation patterns:

• Parity violations: Decoherence mitigation, measurement bias correction
• Reflection violations: Hardware calibration, crosstalk reduction
• Permutation violations: Qubit uniformity validation, topology mapping verification
• Rotational violations: Magnetic field uniformity, geometric alignment checks
• High-severity violations: Urgent hardware diagnosis, experiment halt recommendations

Integration Points:

• Hardware characterization and calibration protocols
• Real-time error mitigation decision making
• Quantum error correction threshold determination
• Circuit compilation and optimization feedback
• Platform benchmarking and comparison studies

Parameters:

• circuit: Quantum circuit specification map containing circuit structure and operations
• measurement-results: Map of measurement outcome strings to count numbers
• config: Optional configuration map for customizing analysis parameters: - :thresholds - Pass/fail thresholds for different symmetry types - :statistical - Statistical significance and confidence requirements - :performance - Analysis optimization and timeout settings - :advanced-symmetries - List of symmetry types to analyze - :error-reporting - Diagnostic detail and logging configuration - :production-mode - Production environment specific settings

Returns: Complete verification report map including:

• :symmetry-score - Overall symmetry quality score [0,1]
• :symmetry-passed - Boolean pass/fail result for the verification
• :corrective-actions - Vector of specific actionable recommendations
• :verification-applied - Boolean confirming verification was executed
• :execution-metrics - Performance and analysis statistics
• :recommendation-confidence - Confidence level in the recommendations [0,1]
• [All fields from detect-symmetry-violations] - Complete underlying analysis

Example: (apply-symmetry-verification bell-circuit measurement-data production-config) ;=> {:symmetry-passed true, :symmetry-score 0.94, :corrective-actions [], ; :recommendation-confidence 0.95, :execution-metrics {...}, ...}

Usage in Production:

(let [result (apply-symmetry-verification circuit measurements config)]
  (if (:symmetry-passed result)
    (proceed-with-computation (:symmetry-score result))
    (apply-corrective-actions (:corrective-actions result))))

compute-advanced-parity-expectation clj

(compute-advanced-parity-expectation measurement-results num-qubits config)

Compute parity expectation value with comprehensive statistical analysis for symmetry verification.

This function calculates the quantum parity expectation value ⟨P⟩ where P is the parity operator P = ⊗ᵢ Zᵢ (tensor product of Pauli-Z operators on all qubits). The parity expectation provides crucial information about symmetry preservation in quantum circuits.

Physical Significance: The parity operator measures the overall "oddness" or "evenness" of a quantum state:

• For basis states: P|x⟩ = (-1)^(|x|) |x⟩ where |x| is the Hamming weight
• For general states: ⟨P⟩ = Σₓ (-1)^(|x|) P(x) where P(x) is measurement probability

Parity is conserved by important gate classes:

• Clifford gates (H, S, CNOT) preserve parity superposition structure
• Pauli gates (X, Y, Z) have well-defined parity transformation rules
• Controlled gates maintain specific parity relationships

Violations of expected parity indicate:

• Decoherence processes (especially dephasing)
• Systematic measurement errors or miscalibration
• Leakage to non-computational subspace
• Crosstalk between qubits

Statistical Analysis: The function computes not just the expectation value but also:

• Sample variance: Var(P) = ⟨P²⟩ - ⟨P⟩²
• Standard error: SE = √(Var(P)/N) for N measurements
• 95% confidence interval: ⟨P⟩ ± 1.96 × SE

This enables rigorous hypothesis testing for parity conservation.

Parameters:

• measurement-results: Map of quantum state strings to measurement counts e.g., {"000" 400, "001" 50, "010" 50, "111" 500}
• num-qubits: Number of qubits in the system (determines parity calculation)
• config: Configuration map (reserved for future statistical options)

Returns: Map containing comprehensive parity analysis:

• :expectation - Parity expectation value ⟨P⟩ ∈ [-1, 1]
• :variance - Sample variance of parity measurements
• :standard-error - Standard error of the expectation estimate
• :confidence-interval - [lower, upper] bounds of 95% confidence interval
• :total-shots - Total number of measurement shots used

Example: (compute-advanced-parity-expectation {"00" 450 "11" 450 "01" 50 "10" 50} 2 {}) ;=> {:expectation 0.8, :variance 0.64, :standard-error 0.025, ; :confidence-interval [0.751 0.849], :total-shots 1000}

compute-chi-squared-test clj

(compute-chi-squared-test observed-counts expected-counts degrees-of-freedom)

Compute chi-squared goodness-of-fit test for quantum symmetry hypothesis testing.

This function performs a rigorous statistical test to determine whether observed measurement counts are consistent with expected counts under a symmetry hypothesis. The chi-squared test is particularly well-suited for quantum measurement data because it handles discrete count data and provides both significance testing and effect size estimation.

Mathematical Foundation: The chi-squared statistic is computed as: χ² = Σ[(observed_i - expected_i)² / expected_i]

Under the null hypothesis (symmetry holds), this statistic follows a chi-squared distribution with the specified degrees of freedom. Large values indicate significant deviations from the expected symmetry pattern.

The p-value represents P(χ² ≥ observed_value | H₀), where H₀ is the symmetry hypothesis. Small p-values (< 0.05) provide evidence against the symmetry hypothesis.

Implementation Features:

• Robust handling of small expected counts (Cochran's rule: expected ≥ 0.1)
• Comprehensive critical value lookup table for common degrees of freedom
• Approximation formulas for large degrees of freedom cases
• Effect size calculation for practical significance assessment
• Multi-level p-value estimation with appropriate granularity

Parameters:

• observed-counts: Vector of observed measurement counts for each symmetry-related state
• expected-counts: Vector of expected counts under perfect symmetry hypothesis
• degrees-of-freedom: Number of independent comparisons (typically states - 1)

Returns: Map containing comprehensive statistical analysis:

• :chi-squared - The computed χ² test statistic
• :critical-value - Critical value for 95% confidence (α = 0.05)
• :p-value - Probability of observing this result under null hypothesis
• :significant - Boolean indicating statistical significance (p < 0.05)
• :degrees-of-freedom - Number of degrees of freedom used
• :effect-size - Practical effect size (χ²/N) for magnitude assessment

Example: (compute-chi-squared-test [45 55] [50 50] 1) ;=> {:chi-squared 1.0, :critical-value 3.841, :p-value 0.25, ; :significant false, :degrees-of-freedom 1, :effect-size 0.01}

default-config clj

Default configuration for production symmetry verification

detect-permutation-symmetries clj

(detect-permutation-symmetries measurement-results num-qubits config)

Detect permutation symmetries in quantum measurement results with comprehensive analysis.

This function analyzes measurement data to identify violations of permutation symmetry, which occurs when a quantum circuit should be invariant under the exchange of qubit indices. Permutation symmetry is crucial for validating hardware uniformity and detecting qubit-specific systematic errors.

Physical Context: Permutation symmetry arises in quantum circuits where:

• All qubits are initialized in the same state
• All qubits undergo identical gate sequences
• Hardware properties (decoherence, gate fidelities) are uniform across qubits
• No qubit-specific calibration errors or crosstalk effects are present

The analysis checks whether measurement outcomes for states related by qubit permutations have statistically equivalent populations. For example, in a 3-qubit system, states |001⟩ and |100⟩ should have equal measurement probabilities if the circuit has permutation symmetry under (qubit 0 ↔ qubit 2) exchange.

Algorithm Details: For small systems (≤ 4 qubits): Exhaustive permutation pair analysis For large systems (> 4 qubits): Statistical sampling of representative permutations

The violation severity is classified based on relative population differences:

• Low severity: 5-15% relative difference
• Medium severity: 15-25% relative difference
• High severity: >25% relative difference

Parameters:

• measurement-results: Map of quantum state strings to measurement counts e.g., {"001" 245, "010" 255, "100" 250, "111" 750}
• num-qubits: Number of qubits in the quantum system (determines analysis strategy)
• config: Configuration map containing: - [:performance :max-states-for-full-analysis] - Threshold for sampling (default: 16) - [:performance :sample-size] - Number of states to sample for large systems

Returns: Map containing permutation symmetry analysis:

• :score - Symmetry score from 0.0 (completely broken) to 1.0 (perfect symmetry)
• :violations - Vector of detected violations with detailed diagnostics:
  • :type - Always :permutation-violation
  • :states - Pair of states that should be equivalent under permutation
  • :counts - Actual measurement counts for the state pair
  • :relative-difference - Normalized difference between counts
  • :severity - Classification as :low, :medium, or :high
• :analysis - String description of the analysis method and results

Example: (detect-permutation-symmetries {"01" 480 "10" 520} 2 default-config) ;=> {:score 0.96, :violations [], :analysis "Checked 1 systematic permutation pairs, found 0 violations"}

detect-rotational-symmetries clj

(detect-rotational-symmetries measurement-results num-qubits config)

Detect rotational symmetries in quantum circuits with continuous rotational invariance.

This function analyzes measurement data for violations of rotational symmetry, which occurs in quantum systems that are invariant under continuous or discrete rotational transformations. Rotational symmetry is particularly important for systems with specific geometric arrangements or Hamiltonians with rotational invariance.

Physical Background: Rotational symmetry in quantum systems arises when:

• The system Hamiltonian commutes with rotation operators
• Initial states possess rotational symmetry (e.g., symmetric superposition states)
• Gate sequences preserve the rotational symmetry of the system
• Environmental noise is isotropic (equal in all spatial directions)

For 3-qubit systems, this implementation focuses on 3-fold rotational symmetry (120° rotations) where states transform under cyclic permutation: |001⟩ → |010⟩ → |100⟩ → |001⟩

Such symmetries are relevant in:

• Spin systems with triangular lattice geometry
• Molecular systems with C₃ symmetry
• Quantum simulators of frustrated magnetic systems
• Error correction codes with rotational structure

Algorithm:

1. Identify rotation triplets (sets of 3 states related by 120° rotation)
2. Compute average population for each triplet
3. Calculate maximum deviation from uniform distribution
4. Classify violations based on relative deviation magnitude

Current Implementation:

• Specialized for 3-qubit systems (extensible to other systems)
• Analyzes discrete 120° rotations around the system's symmetry axis
• Can be extended to continuous rotational groups for larger systems

Parameters:

• measurement-results: Map of quantum state strings to measurement counts
• num-qubits: Number of qubits (currently optimized for 3-qubit systems)
• config: Configuration map (currently uses default thresholds)

Returns: Map containing rotational symmetry analysis:

• :score - Rotational symmetry score from 0.0 to 1.0
• :violations - Vector of detected rotational symmetry violations:
  • :type - Always :rotational-violation
  • :states - Triplet of states related by rotation
  • :counts - Measurement counts for each state in the triplet
  • :max-deviation - Maximum relative deviation from uniform distribution
  • :severity - Classification as :low, :medium, or :high
• :analysis - Description of analysis performed and results

Example: (detect-rotational-symmetries {"001" 330 "010" 340 "100" 330} 3 default-config) ;=> {:score 0.97, :violations [], :analysis "Checked 2 rotation triplets, found 0 violations"}

detect-symmetry-violations clj

(detect-symmetry-violations circuit measurement-results & [config])

Comprehensive symmetry violation detection with advanced statistical analysis and configurable thresholds.

This function performs a complete symmetry analysis of quantum measurement data, detecting violations across multiple symmetry types that are fundamental to quantum error detection and mitigation. It provides the core analytical engine for quantum circuit validation and hardware characterization.

Symmetry Analysis Framework: The function implements a multi-faceted approach to symmetry verification:

1. Parity Symmetry: Analyzes conservation of parity expectation ⟨P⟩ for circuits containing only parity-preserving operations (H, S, T, Pauli, CNOT gates).

2. Reflection Symmetry: Tests bit-flip invariance where states |x⟩ and |x̄⟩ should have equal measurement probabilities under specific circuit conditions.

3. Permutation Symmetry: Validates qubit exchange invariance for circuits that should be symmetric under qubit permutations.

4. Rotational Symmetry: Checks continuous or discrete rotational invariance for systems with geometric symmetry properties.

Statistical Rigor:

• Chi-squared goodness-of-fit testing for hypothesis validation
• Confidence interval analysis for expectation values
• Multiple comparison corrections for family-wise error control
• Effect size estimation for practical significance assessment
• Violation severity classification with actionable thresholds

Production Features:

• Configurable thresholds for different symmetry types and violation severities
• Comprehensive validation of input data quality and statistical sufficiency
• Performance optimization for large Hilbert spaces using sampling strategies
• Detailed diagnostic information for each symmetry type analyzed
• Integration with hardware characterization and error mitigation protocols

Parameters:

• circuit: Quantum circuit map containing: - :num-qubits - Number of qubits in the circuit - :operations - Vector of gate operations for parity analysis
• measurement-results: Map of quantum state strings to measurement counts
• config: Optional configuration map (merges with default-config): - :thresholds - Symmetry score thresholds and violation limits - :statistical - Statistical significance requirements - :performance - Analysis optimization parameters - :advanced-symmetries - List of symmetry types to analyze

Returns: Comprehensive analysis map containing:

• :parity-expectation - Computed parity expectation value
• :parity-analysis - Detailed parity statistical analysis
• :reflection-symmetry-score - Reflection symmetry quality score
• :reflection-analysis - Complete reflection symmetry analysis
• :permutation-analysis - Permutation symmetry analysis results
• :rotational-analysis - Rotational symmetry analysis results
• :has-parity-preserving-circuit - Boolean indicating parity conservation expectation
• :symmetry-violations - Vector of all detected violations with severity classification
• :overall-symmetry-score - Composite symmetry quality score [0,1]
• :statistical-validation - Input data validation results
• :config-used - Effective configuration used for analysis

Example: (detect-symmetry-violations bell-circuit {"00" 480 "11" 520} default-config) ;=> {:overall-symmetry-score 0.96, :symmetry-violations [], :parity-expectation 1.0, ...}

validate-measurement-results clj

(validate-measurement-results measurement-results config)

Validate measurement results for statistical significance and consistency in symmetry analysis.

This function performs comprehensive validation of measurement data to ensure sufficient statistical power for reliable symmetry verification. Insufficient statistics can lead to false positives or negatives in symmetry violation detection.

The validation checks multiple criteria:

• Minimum shot count for statistical significance based on expected effect sizes
• Non-empty measurement data to avoid division by zero errors
• Confidence level assessment for the given sample size

Statistical Background: Symmetry verification requires adequate sampling to distinguish between true symmetry violations and statistical fluctuations. The minimum shot count is determined by:

• Expected effect size of symmetry violations (typically 0.05-0.2 in quantum systems)
• Desired statistical power (typically 0.8) and significance level (typically 0.05)
• Number of symmetry tests being performed (multiple comparison corrections)

Parameters:

• measurement-results: Map of quantum state strings to measurement counts e.g., {"00" 450, "01" 25, "10" 30, "11" 495}
• config: Configuration map containing statistical requirements: - [:statistical :min-shots] - Minimum total shots for significance (default: 500) - [:statistical :confidence-level] - Required confidence level (default: 0.95)

Returns: Map containing validation results:

• :valid - Boolean indicating if measurement data meets statistical requirements
• :reason - String explaining validation failure (if :valid is false)
• :confidence - Numerical confidence level achieved with current data
• :total-shots - Total number of measurement shots in the dataset
• :num-unique-states - Number of distinct quantum states observed

Example: (validate-measurement-results {"00" 450 "11" 450 "01" 50 "10" 50} {:statistical {:min-shots 500 :confidence-level 0.95}}) ;=> {:valid true, :confidence 0.95, :total-shots 1000, :num-unique-states 4}