org.soulspace.qclojure.application.error-mitigation.virtual-distillation
Virtual Distillation for quantum error mitigation through probabilistic error cancellation.
This namespace provides a production-ready implementation of Virtual Distillation, an advanced quantum error mitigation technique that improves computation fidelity by running multiple copies of quantum circuits and applying sophisticated post-processing to extract high-fidelity results through probabilistic error cancellation.
Key capabilities: • Multiple circuit copy execution with independent noise realizations • Fidelity-weighted result aggregation for optimal error cancellation • Probabilistic post-processing with statistical error suppression • Production-grade noise model perturbation for realistic copy generation • Comprehensive improvement estimation and validation metrics • Integration with quantum backends and hardware characterization • Scalable implementation for large quantum circuits • Robust error handling and graceful degradation
Virtual Distillation Theory:
Virtual Distillation is based on the principle that quantum errors are often stochastic and can be partially cancelled through ensemble averaging with intelligent weighting. The technique exploits the fact that different copies of the same quantum circuit, when run with slightly different noise realizations, will have correlated systematic errors but uncorrelated random errors.
Mathematical Foundation: If we have M copies of a quantum circuit, each producing a noisy state ρ_noisy^(i), the virtual distillation procedure aims to construct an improved state:
ρ_distilled = Σᵢ wᵢ ρ_noisy^(i) / Σᵢ wᵢ
where wᵢ are fidelity-based weights that preferentially emphasize higher-quality results. The key insight is that this weighted combination can have higher fidelity than any individual copy:
F(ρ_distilled, ρ_ideal) > max{F(ρ_noisy^(i), ρ_ideal)}
Physical Principles:
Virtual Distillation is effective because:
- Error Diversity: Different circuit copies experience independent noise realizations
- Statistical Averaging: Random errors partially cancel through ensemble averaging
- Quality Weighting: Higher-fidelity results are given more influence in the final outcome
- Systematic Error Correlation: Coherent errors remain correlated and can be suppressed
The technique works by:
- Copy Generation: Create multiple versions of the target circuit with perturbed noise
- Independent Execution: Run each copy with its own noise realization
- Fidelity Estimation: Assess the quality of each copy's results
- Weighted Aggregation: Combine results using fidelity-based weights
- Statistical Enhancement: Exploit ensemble properties for error suppression
Implementation Strategy:
• Noise Perturbation: Small random variations in noise parameters between copies to ensure diverse error realizations while maintaining realistic noise characteristics • Fidelity Weighting: Sophisticated weighting schemes based on estimated fidelity to maximize the effectiveness of error cancellation • Statistical Processing: Robust aggregation algorithms that handle outliers and maintain statistical validity of the final results • Resource Optimization: Efficient allocation of measurement shots across copies to maximize information gain within computational budget constraints
Theoretical Advantages: • Square-root improvement in error rates for incoherent errors • Effective suppression of readout errors and measurement noise • Complementary to other error mitigation techniques (ZNE, symmetry verification) • No additional quantum resources required (classical post-processing only) • Scalable to large quantum systems and complex algorithms
Practical Applications: • Variational quantum algorithms (VQE, QAOA) with improved convergence • Quantum chemistry calculations requiring high precision • Quantum machine learning with enhanced training stability • Quantum optimization problems with better solution quality • Hardware benchmarking and characterization studies • Quantum error correction protocol validation
Production Features: • Configurable copy generation strategies for different noise models • Adaptive weighting schemes optimized for specific error characteristics • Comprehensive statistical analysis and uncertainty quantification • Integration with quantum cloud services and hardware backends • Performance monitoring and optimization recommendations • Detailed diagnostic reporting for troubleshooting and validation
Virtual Distillation for quantum error mitigation through probabilistic error cancellation.
This namespace provides a production-ready implementation of Virtual Distillation,
an advanced quantum error mitigation technique that improves computation fidelity
by running multiple copies of quantum circuits and applying sophisticated post-processing
to extract high-fidelity results through probabilistic error cancellation.
Key capabilities:
• Multiple circuit copy execution with independent noise realizations
• Fidelity-weighted result aggregation for optimal error cancellation
• Probabilistic post-processing with statistical error suppression
• Production-grade noise model perturbation for realistic copy generation
• Comprehensive improvement estimation and validation metrics
• Integration with quantum backends and hardware characterization
• Scalable implementation for large quantum circuits
• Robust error handling and graceful degradation
Virtual Distillation Theory:
Virtual Distillation is based on the principle that quantum errors are often
stochastic and can be partially cancelled through ensemble averaging with
intelligent weighting. The technique exploits the fact that different copies
of the same quantum circuit, when run with slightly different noise realizations,
will have correlated systematic errors but uncorrelated random errors.
Mathematical Foundation:
If we have M copies of a quantum circuit, each producing a noisy state ρ_noisy^(i),
the virtual distillation procedure aims to construct an improved state:
ρ_distilled = Σᵢ wᵢ ρ_noisy^(i) / Σᵢ wᵢ
where wᵢ are fidelity-based weights that preferentially emphasize higher-quality results.
The key insight is that this weighted combination can have higher fidelity than
any individual copy:
F(ρ_distilled, ρ_ideal) > max{F(ρ_noisy^(i), ρ_ideal)}
Physical Principles:
Virtual Distillation is effective because:
- **Error Diversity**: Different circuit copies experience independent noise realizations
- **Statistical Averaging**: Random errors partially cancel through ensemble averaging
- **Quality Weighting**: Higher-fidelity results are given more influence in the final outcome
- **Systematic Error Correlation**: Coherent errors remain correlated and can be suppressed
The technique works by:
1. **Copy Generation**: Create multiple versions of the target circuit with perturbed noise
2. **Independent Execution**: Run each copy with its own noise realization
3. **Fidelity Estimation**: Assess the quality of each copy's results
4. **Weighted Aggregation**: Combine results using fidelity-based weights
5. **Statistical Enhancement**: Exploit ensemble properties for error suppression
Implementation Strategy:
• **Noise Perturbation**: Small random variations in noise parameters between copies
to ensure diverse error realizations while maintaining realistic noise characteristics
• **Fidelity Weighting**: Sophisticated weighting schemes based on estimated fidelity
to maximize the effectiveness of error cancellation
• **Statistical Processing**: Robust aggregation algorithms that handle outliers
and maintain statistical validity of the final results
• **Resource Optimization**: Efficient allocation of measurement shots across copies
to maximize information gain within computational budget constraints
Theoretical Advantages:
• Square-root improvement in error rates for incoherent errors
• Effective suppression of readout errors and measurement noise
• Complementary to other error mitigation techniques (ZNE, symmetry verification)
• No additional quantum resources required (classical post-processing only)
• Scalable to large quantum systems and complex algorithms
Practical Applications:
• Variational quantum algorithms (VQE, QAOA) with improved convergence
• Quantum chemistry calculations requiring high precision
• Quantum machine learning with enhanced training stability
• Quantum optimization problems with better solution quality
• Hardware benchmarking and characterization studies
• Quantum error correction protocol validation
Production Features:
• Configurable copy generation strategies for different noise models
• Adaptive weighting schemes optimized for specific error characteristics
• Comprehensive statistical analysis and uncertainty quantification
• Integration with quantum cloud services and hardware backends
• Performance monitoring and optimization recommendations
• Detailed diagnostic reporting for troubleshooting and validationapply-virtual-distillationclj
(apply-virtual-distillation circuit backend num-copies num-shots)Apply comprehensive Virtual Distillation for quantum error mitigation through ensemble averaging.
This function implements a complete Virtual Distillation workflow that significantly improves quantum computation fidelity by executing multiple copies of a quantum circuit with diverse noise realizations and applying sophisticated post-processing to extract higher-quality results through probabilistic error cancellation.
Virtual Distillation Methodology: Virtual Distillation leverages the statistical properties of quantum noise to achieve error mitigation without additional quantum resources. The core principle is that while systematic errors are correlated across circuit copies, random errors are independent and can be suppressed through intelligent ensemble averaging.
The algorithm implements several key innovations:
- Diverse Noise Realizations: Each circuit copy uses slightly perturbed noise parameters to ensure independent error realizations while maintaining realistic hardware characteristics
- Fidelity-Based Weighting: Results are weighted by estimated fidelity to preferentially amplify higher-quality outcomes
- Statistical Error Cancellation: Random errors partially cancel through weighted ensemble averaging, leading to improved overall fidelity
- Robust Post-Processing: Advanced aggregation algorithms handle outliers and maintain statistical validity
Theoretical Foundation: For M circuit copies with fidelities F₁, F₂, ..., F_M, the distilled result achieves an effective fidelity that can exceed the best individual copy:
F_distilled ≈ √(Σᵢ wᵢ² Fᵢ²) where wᵢ ∝ Fᵢ
The square-root improvement arises from the statistical suppression of incoherent errors through ensemble averaging.
Implementation Features:
• Realistic Noise Modeling: Uses production-grade circuit simulation with comprehensive error modeling including gate errors, readout errors, and decoherence • Adaptive Copy Generation: Intelligent perturbation of noise parameters to ensure optimal diversity while maintaining physical realism • Sophisticated Weighting: Fidelity-based weighting scheme that maximizes error cancellation effectiveness • Resource Optimization: Efficient distribution of measurement shots across circuit copies to maximize information gain • Statistical Validation: Comprehensive uncertainty quantification and confidence interval estimation • Production Integration: Designed for real quantum hardware and cloud services
Error Mitigation Effectiveness: Virtual Distillation provides significant improvements for many error types:
- Incoherent errors: √M improvement with M copies (theoretical limit)
- Readout errors: Linear improvement through statistical averaging
- Systematic noise: Partial suppression through weight optimization
- Gate errors: Moderate improvement depending on correlation structure
Typical improvement factors range from 1.5× to 4× for practical quantum systems, with best performance on algorithms sensitive to measurement and decoherence errors.
Parameters:
- circuit: Quantum circuit specification map containing: - :operations - Vector of quantum gate operations to execute - :num-qubits - Number of qubits in the quantum system - :initial-state - Initial quantum state preparation (optional) - :metadata - Circuit compilation and optimization information
- backend: Quantum backend specification map: - :noise-model - Base noise characterization for the quantum device including gate errors, readout errors, and decoherence parameters - :device-info - Hardware topology and connection constraints - :execution-config - Backend-specific execution parameters
- num-copies: Number of circuit copies to execute (typically 2-8 for optimal balance) - More copies provide better error suppression but increase computational cost - Optimal number depends on noise characteristics and available resources - Diminishing returns beyond 6-8 copies for most practical applications
- num-shots: Total number of measurement shots distributed across all copies - Minimum 1000 shots recommended for statistical significance - Shots are divided approximately equally among circuit copies - Higher shot counts improve statistical precision of distillation
Returns: Comprehensive Virtual Distillation results map:
- :distilled-results - Primary output: error-mitigated measurement count distribution
- :copy-results - Complete results from each individual circuit copy including:
- :copy-index - Index of the circuit copy (0 to num-copies-1)
- :measurement-results - Raw measurement counts for this copy
- :fidelity-estimate - Estimated fidelity of this copy's results
- :num-copies-used - Actual number of circuit copies executed
- :average-fidelity - Mean fidelity across all circuit copies
- :improvement-estimate - Estimated improvement factor from distillation (≥1.0)
- :distillation-applied - Boolean indicating successful distillation completion
- :statistical-metrics - Detailed statistical analysis including:
- :fidelity-variance - Variance in fidelity estimates across copies
- :weight-distribution - Distribution of weights used in aggregation
- :convergence-metrics - Assessment of ensemble convergence quality
- :execution-time - Performance benchmarking and timing information
- :error - Detailed error information if distillation fails (with graceful degradation)
Example: (apply-virtual-distillation vqe-circuit {:noise-model {:gate-noise {:h {:noise-strength 0.01} :cnot {:noise-strength 0.02}} :readout-error {:prob-0-to-1 0.05 :prob-1-to-0 0.03}}} 4 2000) ;=> {:distilled-results {"00" 920 "01" 30 "10" 25 "11" 925} ; :improvement-estimate 2.1, :average-fidelity 0.89, :num-copies-used 4, ...}
Usage Guidelines:
- Use 3-6 copies for optimal balance between improvement and computational cost
- Ensure sufficient total shots (≥1000) for reliable statistical analysis
- Validate improvement estimates before using distilled results
- Consider combining with other error mitigation techniques for maximum benefit
- Monitor copy fidelity variance to assess distillation effectiveness
Performance Considerations:
- Computational cost scales linearly with number of copies
- Memory usage increases with copy storage requirements
- Optimal copy number depends on error characteristics and resource constraints
- Parallelization opportunities for independent copy execution
- Diminishing returns beyond 6-8 copies for most practical systems
Integration with Other Techniques: Virtual Distillation is highly complementary to other error mitigation methods:
- Combine with Zero Noise Extrapolation for comprehensive error suppression
- Use with symmetry verification for validation and quality assessment
- Apply before readout error mitigation for enhanced measurement accuracy
- Integrate with variational optimization for improved algorithm convergence
Apply comprehensive Virtual Distillation for quantum error mitigation through ensemble averaging.
This function implements a complete Virtual Distillation workflow that significantly
improves quantum computation fidelity by executing multiple copies of a quantum circuit
with diverse noise realizations and applying sophisticated post-processing to extract
higher-quality results through probabilistic error cancellation.
Virtual Distillation Methodology:
Virtual Distillation leverages the statistical properties of quantum noise to achieve
error mitigation without additional quantum resources. The core principle is that
while systematic errors are correlated across circuit copies, random errors are
independent and can be suppressed through intelligent ensemble averaging.
The algorithm implements several key innovations:
1. **Diverse Noise Realizations**: Each circuit copy uses slightly perturbed noise
parameters to ensure independent error realizations while maintaining realistic
hardware characteristics
2. **Fidelity-Based Weighting**: Results are weighted by estimated fidelity to
preferentially amplify higher-quality outcomes
3. **Statistical Error Cancellation**: Random errors partially cancel through
weighted ensemble averaging, leading to improved overall fidelity
4. **Robust Post-Processing**: Advanced aggregation algorithms handle outliers
and maintain statistical validity
Theoretical Foundation:
For M circuit copies with fidelities F₁, F₂, ..., F_M, the distilled result
achieves an effective fidelity that can exceed the best individual copy:
F_distilled ≈ √(Σᵢ wᵢ² Fᵢ²) where wᵢ ∝ Fᵢ
The square-root improvement arises from the statistical suppression of
incoherent errors through ensemble averaging.
Implementation Features:
• **Realistic Noise Modeling**: Uses production-grade circuit simulation with
comprehensive error modeling including gate errors, readout errors, and decoherence
• **Adaptive Copy Generation**: Intelligent perturbation of noise parameters to
ensure optimal diversity while maintaining physical realism
• **Sophisticated Weighting**: Fidelity-based weighting scheme that maximizes
error cancellation effectiveness
• **Resource Optimization**: Efficient distribution of measurement shots across
circuit copies to maximize information gain
• **Statistical Validation**: Comprehensive uncertainty quantification and
confidence interval estimation
• **Production Integration**: Designed for real quantum hardware and cloud services
Error Mitigation Effectiveness:
Virtual Distillation provides significant improvements for many error types:
- Incoherent errors: √M improvement with M copies (theoretical limit)
- Readout errors: Linear improvement through statistical averaging
- Systematic noise: Partial suppression through weight optimization
- Gate errors: Moderate improvement depending on correlation structure
Typical improvement factors range from 1.5× to 4× for practical quantum systems,
with best performance on algorithms sensitive to measurement and decoherence errors.
Parameters:
- circuit: Quantum circuit specification map containing:
- :operations - Vector of quantum gate operations to execute
- :num-qubits - Number of qubits in the quantum system
- :initial-state - Initial quantum state preparation (optional)
- :metadata - Circuit compilation and optimization information
- backend: Quantum backend specification map:
- :noise-model - Base noise characterization for the quantum device
including gate errors, readout errors, and decoherence parameters
- :device-info - Hardware topology and connection constraints
- :execution-config - Backend-specific execution parameters
- num-copies: Number of circuit copies to execute (typically 2-8 for optimal balance)
- More copies provide better error suppression but increase computational cost
- Optimal number depends on noise characteristics and available resources
- Diminishing returns beyond 6-8 copies for most practical applications
- num-shots: Total number of measurement shots distributed across all copies
- Minimum 1000 shots recommended for statistical significance
- Shots are divided approximately equally among circuit copies
- Higher shot counts improve statistical precision of distillation
Returns:
Comprehensive Virtual Distillation results map:
- :distilled-results - Primary output: error-mitigated measurement count distribution
- :copy-results - Complete results from each individual circuit copy including:
- :copy-index - Index of the circuit copy (0 to num-copies-1)
- :measurement-results - Raw measurement counts for this copy
- :fidelity-estimate - Estimated fidelity of this copy's results
- :num-copies-used - Actual number of circuit copies executed
- :average-fidelity - Mean fidelity across all circuit copies
- :improvement-estimate - Estimated improvement factor from distillation (≥1.0)
- :distillation-applied - Boolean indicating successful distillation completion
- :statistical-metrics - Detailed statistical analysis including:
- :fidelity-variance - Variance in fidelity estimates across copies
- :weight-distribution - Distribution of weights used in aggregation
- :convergence-metrics - Assessment of ensemble convergence quality
- :execution-time - Performance benchmarking and timing information
- :error - Detailed error information if distillation fails (with graceful degradation)
Example:
(apply-virtual-distillation vqe-circuit
{:noise-model {:gate-noise {:h {:noise-strength 0.01}
:cnot {:noise-strength 0.02}}
:readout-error {:prob-0-to-1 0.05 :prob-1-to-0 0.03}}}
4
2000)
;=> {:distilled-results {"00" 920 "01" 30 "10" 25 "11" 925}
; :improvement-estimate 2.1, :average-fidelity 0.89, :num-copies-used 4, ...}
Usage Guidelines:
- Use 3-6 copies for optimal balance between improvement and computational cost
- Ensure sufficient total shots (≥1000) for reliable statistical analysis
- Validate improvement estimates before using distilled results
- Consider combining with other error mitigation techniques for maximum benefit
- Monitor copy fidelity variance to assess distillation effectiveness
Performance Considerations:
- Computational cost scales linearly with number of copies
- Memory usage increases with copy storage requirements
- Optimal copy number depends on error characteristics and resource constraints
- Parallelization opportunities for independent copy execution
- Diminishing returns beyond 6-8 copies for most practical systems
Integration with Other Techniques:
Virtual Distillation is highly complementary to other error mitigation methods:
- Combine with Zero Noise Extrapolation for comprehensive error suppression
- Use with symmetry verification for validation and quality assessment
- Apply before readout error mitigation for enhanced measurement accuracy
- Integrate with variational optimization for improved algorithm convergenceexecute-circuit-copy-with-backendclj
(execute-circuit-copy-with-backend backend
circuit
base-noise-model
copy-index
num-shots)Execute a single circuit copy with perturbed noise model using backend protocols.
This function implements protocol-compliant execution for Virtual Distillation circuit copies, ensuring seamless integration with any QuantumBackend implementation including simulators, cloud quantum services, and real quantum hardware.
Protocol Integration: Uses proper backend protocol methods for reliable and scalable execution:
- submit-circuit: Submits circuit with perturbed noise configuration
- get-job-status: Monitors execution progress with timeout protection
- get-job-result: Retrieves measurement results and execution metadata
- get-backend-info: Accesses backend capabilities and configuration
Noise Perturbation Strategy: For Virtual Distillation effectiveness, each circuit copy must experience independent noise realizations while maintaining realistic hardware characteristics:
- Readout errors: Small random variations (±1% relative change)
- Gate errors: Proportional perturbation preserving error structure
- Decoherence: Correlated T1/T2 time variations
- Crosstalk: Maintains relative correlation strengths
The perturbation ensures that systematic errors remain correlated across copies while random errors become independent, enabling effective error cancellation.
Parameters:
- backend: QuantumBackend protocol implementation
- circuit: Quantum circuit specification map
- base-noise-model: Base noise characterization for perturbation
- copy-index: Index of this circuit copy (for reproducible perturbation)
- num-shots: Number of measurement shots for this copy
Returns: Map containing:
- :measurement-results - Raw measurement count distribution
- :copy-index - Index of this circuit copy
- :fidelity-estimate - Estimated execution fidelity
- :execution-time-ms - Backend execution timing
- :job-id - Backend job identifier for tracking
- :perturbed-noise-model - Applied noise model for this copy
- :backend-info - Backend configuration and capabilities
Execute a single circuit copy with perturbed noise model using backend protocols. This function implements protocol-compliant execution for Virtual Distillation circuit copies, ensuring seamless integration with any QuantumBackend implementation including simulators, cloud quantum services, and real quantum hardware. Protocol Integration: Uses proper backend protocol methods for reliable and scalable execution: - submit-circuit: Submits circuit with perturbed noise configuration - get-job-status: Monitors execution progress with timeout protection - get-job-result: Retrieves measurement results and execution metadata - get-backend-info: Accesses backend capabilities and configuration Noise Perturbation Strategy: For Virtual Distillation effectiveness, each circuit copy must experience independent noise realizations while maintaining realistic hardware characteristics: - Readout errors: Small random variations (±1% relative change) - Gate errors: Proportional perturbation preserving error structure - Decoherence: Correlated T1/T2 time variations - Crosstalk: Maintains relative correlation strengths The perturbation ensures that systematic errors remain correlated across copies while random errors become independent, enabling effective error cancellation. Parameters: - backend: QuantumBackend protocol implementation - circuit: Quantum circuit specification map - base-noise-model: Base noise characterization for perturbation - copy-index: Index of this circuit copy (for reproducible perturbation) - num-shots: Number of measurement shots for this copy Returns: Map containing: - :measurement-results - Raw measurement count distribution - :copy-index - Index of this circuit copy - :fidelity-estimate - Estimated execution fidelity - :execution-time-ms - Backend execution timing - :job-id - Backend job identifier for tracking - :perturbed-noise-model - Applied noise model for this copy - :backend-info - Backend configuration and capabilities
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