CUDA-Q QEC - Quantum Error Correction Library

Overview

The cudaq-qec library provides a comprehensive framework for quantum error correction research and development. It leverages GPU acceleration for efficient syndrome decoding and error correction simulations (coming soon).

Core Components

cudaq-qec is composed of two main interfaces - the cudaq::qec::code and cudaq::qec::decoder types. These types are meant to be extended by developers to provide new error correcting codes and new decoding strategies.

QEC Code Framework cudaq::qec::code

The cudaq::qec::code class serves as the base class for all quantum error correcting codes in CUDA-Q QEC. It provides a flexible extension point for implementing new codes and defines the core interface that all QEC codes must support.

The core abstraction here is that of a mapping or dictionary of logical operations to their corresponding physical implementation in the error correcting code as CUDA-Q quantum kernels.

Class Structure

The code base class provides:

1. Operation Enumeration: Defines supported logical operations

   enum class operation {
       x,     // Logical X gate
       y,     // Logical Y gate
       z,     // Logical Z gate
       h,     // Logical Hadamard gate
       s,     // Logical S gate
       cx,    // Logical CNOT gate
       cy,    // Logical CY gate
       cz,    // Logical CZ gate
       stabilizer_round,  // Stabilizer measurement round
       prep0, // Prepare |0⟩ state
       prep1, // Prepare |1⟩ state
       prepp, // Prepare |+⟩ state
       prepm  // Prepare |-⟩ state
   };
   

2. Patch Type: Defines the structure of a logical qubit patch

   struct patch {
       cudaq::qview<> data;  // View of data qubits
       cudaq::qview<> ancx;  // View of X stabilizer ancilla qubits
       cudaq::qview<> ancz;  // View of Z stabilizer ancilla qubits
   };
   

   The patch type represents a logical qubit in quantum error correction codes. It contains: - data: A view of the data qubits in the patch - ancx: A view of the ancilla qubits used for X stabilizer measurements - ancz: A view of the ancilla qubits used for Z stabilizer measurements

   This structure is designed for use within CUDA-Q kernel code and provides a convenient way to access different qubit subsets within a logical qubit patch.

3. Kernel Type Aliases: Defines quantum kernel signatures

   using one_qubit_encoding = cudaq::qkernel<void(patch)>;
   using two_qubit_encoding = cudaq::qkernel<void(patch, patch)>;
   using stabilizer_round = cudaq::qkernel<std::vector<cudaq::measure_result>(
       patch, const std::vector<std::size_t>&, const std::vector<std::size_t>&)>;
   

4. Protected Members:

   • operation_encodings: Maps operations to their quantum kernel implementations. The key is the operation enum and the value is a variant on the above kernel type aliases.

   • m_stabilizers: Stores the code’s stabilizer generators

Implementing a New Code

To implement a new quantum error correcting code:

1. Create a New Class:

   class my_code : public qec::code {
   protected:
       // Implement required virtual methods
   public:
       my_code(const heterogeneous_map& options);
   };
   

2. Implement Required Virtual Methods:

   // Number of physical data qubits
   std::size_t get_num_data_qubits() const override;
   
   // Total number of ancilla qubits
   std::size_t get_num_ancilla_qubits() const override;
   
   // Number of X-type ancilla qubits
   std::size_t get_num_ancilla_x_qubits() const override;
   
   // Number of Z-type ancilla qubits
   std::size_t get_num_ancilla_z_qubits() const override;
   

3. Define Quantum Kernels:

   Create CUDA-Q kernels for each logical operation:

   __qpu__ void x(patch p) {
       // Implement logical X
   }
   
   __qpu__ std::vector<cudaq::measure_result> stabilizer(patch p,
       const std::vector<std::size_t>& x_stabs,
       const std::vector<std::size_t>& z_stabs) {
       // Implement stabilizer measurements
   }
   

4. Register Operations:

   In the constructor, register quantum kernels for each operation:

   my_code::my_code(const heterogeneous_map& options) : code() {
       // Register operations
       operation_encodings.insert(
          std::make_pair(operation::x, x));
       operation_encodings.insert(
          std::make_pair(operation::stabilizer_round, stabilizer));
   
       // Define stabilizer generators
       m_stabilizers = qec::stabilizers({"XXXX", "ZZZZ"});
   }
   

   Note that in your constructor, you have access to user-provided options. For example, if your code depends on an integer paramter called distance, you can retrieve that from the user via

   my_code::my_code(const heterogeneous_map& options) : code() {
       // ... fill the map and stabilizers ...
   
       // Get the user-provided distance, or just
       // set to 3 if user did not provide one
       this->distance = options.get<int>("distance", /*defaultValue*/ 3);
   }
   

5. Register Extension Point:

   Add extension point registration:

   CUDAQ_EXTENSION_CUSTOM_CREATOR_FUNCTION(
       my_code,
       static std::unique_ptr<qec::code> create(
           const heterogeneous_map &options) {
           return std::make_unique<my_code>(options);
       }
   )
   
   CUDAQ_REGISTER_TYPE(my_code)
   

Example: Steane Code

The Steane [[7,1,3]] code provides a complete example implementation:

1. Header Definition:

   • Declares quantum kernels for all logical operations

   • Defines the code class with required virtual methods

   • Specifies 7 data qubits and 6 ancilla qubits (3 X-type, 3 Z-type)

2. Implementation:

   steane::steane(const heterogeneous_map &options) : code() {
       // Register all logical operations
       operation_encodings.insert(
           std::make_pair(operation::x, x));
       // ... register other operations ...
   
       // Define stabilizer generators
       m_stabilizers = qec::stabilizers({
           "XXXXIII", "IXXIXXI", "IIXXIXX",
           "ZZZZIII", "IZZIZZI", "IIZZIZZ"
       });
   }
   

3. Quantum Kernels:

   Implements fault-tolerant logical operations:

   __qpu__ void x(patch logicalQubit) {
       // Apply logical X to specific data qubits
       x(logicalQubit.data[4], logicalQubit.data[5],
         logicalQubit.data[6]);
   }
   
   __qpu__ std::vector<cudaq::measure_result> stabilizer(patch logicalQubit,
       const std::vector<std::size_t>& x_stabilizers,
       const std::vector<std::size_t>& z_stabilizers) {
       // Measure X stabilizers
       h(logicalQubit.ancx);
       // ... apply controlled-X gates ...
       h(logicalQubit.ancx);
   
       // Measure Z stabilizers
       // ... apply controlled-X gates ...
   
       // Return measurement results
       return mz(logicalQubit.ancz, logicalQubit.ancx);
   }
   

Implementing a New Code in Python

CUDA-Q QEC supports implementing quantum error correction codes in Python using the @qec.code decorator. This provides a more accessible way to prototype and develop new codes.

1. Create a New Python File:

   Create a new file (e.g., my_steane.py) with your code implementation:

   import cudaq
   import cudaq_qec as qec
   from cudaq_qec import patch
   

2. Define Quantum Kernels:

   Implement the required quantum kernels using the @cudaq.kernel decorator:

   @cudaq.kernel
   def prep0(logicalQubit: patch):
       h(logicalQubit.data[0], logicalQubit.data[4], logicalQubit.data[6])
       x.ctrl(logicalQubit.data[0], logicalQubit.data[1])
       x.ctrl(logicalQubit.data[4], logicalQubit.data[5])
       # ... additional initialization gates ...
   
   @cudaq.kernel
   def stabilizer(logicalQubit: patch,
                 x_stabilizers: list[int],
                 z_stabilizers: list[int]) -> list[bool]:
       # Measure X stabilizers
       h(logicalQubit.ancx)
       for xi in range(len(logicalQubit.ancx)):
           for di in range(len(logicalQubit.data)):
               if x_stabilizers[xi * len(logicalQubit.data) + di] == 1:
                   x.ctrl(logicalQubit.ancx[xi], logicalQubit.data[di])
       h(logicalQubit.ancx)
   
       # Measure Z stabilizers
       for zi in range(len(logicalQubit.ancx)):
           for di in range(len(logicalQubit.data)):
               if z_stabilizers[zi * len(logicalQubit.data) + di] == 1:
                   x.ctrl(logicalQubit.data[di], logicalQubit.ancz[zi])
   
       # Get and reset ancillas
       results = mz(logicalQubit.ancz, logicalQubit.ancx)
       reset(logicalQubit.ancx)
       reset(logicalQubit.ancz)
       return results
   

3. Implement the Code Class:

   Create a class decorated with @qec.code that implements the required interface:

   @qec.code('py-steane-example')
   class MySteaneCodeImpl:
       def __init__(self, **kwargs):
           qec.Code.__init__(self, **kwargs)
   
           # Define stabilizer generators
           self.stabilizers = qec.Stabilizers([
               "XXXXIII", "IXXIXXI", "IIXXIXX",
               "ZZZZIII", "IZZIZZI", "IIZZIZZ"
           ])
   
           # Register quantum kernels
           self.operation_encodings = {
               qec.operation.prep0: prep0,
               qec.operation.stabilizer_round: stabilizer
           }
   
       def get_num_data_qubits(self):
           return 7
   
       def get_num_ancilla_x_qubits(self):
           return 3
   
       def get_num_ancilla_z_qubits(self):
           return 3
   
       def get_num_ancilla_qubits(self):
           return 6
   

4. Install the Code:

   Install your Python-implemented code using cudaqx-config:

   cudaqx-config --install-code my_steane.py
   

5. Using the Code:

   The code can now be used like any other CUDA-Q QEC code:

   import cudaq_qec as qec
   
   # Create instance of your code
   code = qec.get_code('py-steane-example')
   
   # Use the code for various numerical experiments
   

Key Points

• The @qec.code decorator takes the name of the code as an argument

• Operation encodings are registered via the operation_encodings dictionary

• Stabilizer generators are defined using the qec.Stabilizers class

• The code must implement all required methods from the base class interface

Using the Code Framework

To use an implemented code:

Python

# Create a code instance
code = qec.get_code("steane")

# Access stabilizer information
stabilizers = code.get_stabilizers()
parity = code.get_parity()

# The code can now be used for various numerical
# experiments - see section below.

C++

// Create a code instance
auto code = cudaq::qec::get_code("steane");

// Access stabilizer information
auto stabilizers = code->get_stabilizers();
auto parity = code->get_parity();

// The code can now be used for various numerical
// experiments - see section below.

Pre-built Quantum Error Correction Codes

CUDA-Q QEC provides several well-studied quantum error correction codes out of the box. Here’s a detailed overview of each:

Steane Code

The Steane code is a [[7,1,3]] CSS (Calderbank-Shor-Steane) code that encodes one logical qubit into seven physical qubits with a code distance of 3.

Key Properties:

• Data qubits: 7

• Encoded qubits: 1

• Code distance: 3

• Ancilla qubits: 6 (3 for X stabilizers, 3 for Z stabilizers)

Stabilizer Generators:

• X-type: ["XXXXIII", "IXXIXXI", "IIXXIXX"]

• Z-type: ["ZZZZIII", "IZZIZZI", "IIZZIZZ"]

The Steane code can correct any single-qubit error and detect up to two errors. It is particularly notable for being the smallest CSS code that can implement a universal set of transversal gates.

Usage:

Python

import cudaq_qec as qec

# Create Steane code instance
steane = qec.get_code("steane")

C++

auto steane = cudaq::qec::get_code("steane");

Repetition Code

The repetition code is a simple [[n,1,n]] code that protects against bit-flip (X) errors by encoding one logical qubit into n physical qubits, where n is the code distance.

Key Properties:

• Data qubits: n (distance)

• Encoded qubits: 1

• Code distance: n

• Ancilla qubits: n-1 (all for Z stabilizers)

Stabilizer Generators:

• For distance 3: ["ZZI", "IZZ"]

• For distance 5: ["ZZIII", "IZZII", "IIZZI", "IIIZZ"]

The repetition code is primarily educational as it can only correct X errors. However, it serves as an excellent introduction to QEC concepts.

Usage:

Python

import cudaq_qec as qec

# Create distance-3 repetition code
code = qec.get_code('repetition', distance=3})

# Access stabilizers
stabilizers = code.get_stabilizers()  # Returns ["ZZI", "IZZ"]

C++

auto code = qec::get_code("repetition", {{"distance", 3}});

// Access stabilizers
auto stabilizers = code->get_stabilizers();

Decoder Framework cudaq::qec::decoder

The CUDA-Q QEC decoder framework provides an extensible system for implementing quantum error correction decoders through the cudaq::qec::decoder base class.

Class Structure

The decoder base class defines the core interface for syndrome decoding:

class decoder {
protected:
    std::size_t block_size;     // For [n,k] code, this is n
    std::size_t syndrome_size;   // For [n,k] code, this is n-k
    tensor<uint8_t> H;          // Parity check matrix

public:
    struct decoder_result {
        bool converged;                  // Decoder convergence status
        std::vector<float_t> result;     // Soft error probabilities
    };

    virtual decoder_result decode(
        const std::vector<float_t>& syndrome) = 0;

    virtual std::vector<decoder_result> decode_multi(
        const std::vector<std::vector<float_t>>& syndrome);
};

Key Components:

• Parity Check Matrix: Defines the code structure via H

• Block Size: Number of physical qubits in the code

• Syndrome Size: Number of stabilizer measurements

• Decoder Result: Contains convergence status and error probabilities

• Multiple Decoding Modes: Single syndrome or batch processing

Implementing a New Decoder in C++

To implement a new decoder:

1. Create Decoder Class:

class my_decoder : public qec::decoder {
private:
    // Decoder-specific members

public:
    my_decoder(const tensor<uint8_t>& H,
              const heterogeneous_map& params)
        : decoder(H) {
        // Initialize decoder
    }

    decoder_result decode(
        const std::vector<float_t>& syndrome) override {
        // Implement decoding logic
    }
};

2. Register Extension Point:

CUDAQ_EXTENSION_CUSTOM_CREATOR_FUNCTION(
    my_decoder,
    static std::unique_ptr<decoder> create(
        const tensor<uint8_t>& H,
        const heterogeneous_map& params) {
        return std::make_unique<my_decoder>(H, params);
    }
)

CUDAQ_REGISTER_TYPE(my_decoder)

Example: Lookup Table Decoder

Here’s a simple lookup table decoder for the Steane code:

class single_error_lut : public decoder {
private:
    std::map<std::string, std::size_t> single_qubit_err_signatures;

public:
    single_error_lut(const tensor<uint8_t>& H,
                      const heterogeneous_map& params)
        : decoder(H) {
        // Build lookup table for single-qubit errors
        for (std::size_t qErr = 0; qErr < block_size; qErr++) {
            std::string err_sig(syndrome_size, '0');
            for (std::size_t r = 0; r < syndrome_size; r++) {
                bool syndrome = 0;
                for (std::size_t c = 0; c < block_size; c++)
                    syndrome ^= (c != qErr) && H.at({r, c});
                err_sig[r] = syndrome ? '1' : '0';
            }
            single_qubit_err_signatures.insert({err_sig, qErr});
        }
    }

    decoder_result decode(
        const std::vector<float_t>& syndrome) override {
        decoder_result result{false,
            std::vector<float_t>(block_size, 0.0)};

        // Convert syndrome to string
        std::string syndrome_str(syndrome_size, '0');
        for (std::size_t i = 0; i < syndrome_size; i++)
            syndrome_str[i] = (syndrome[i] >= 0.5) ? '1' : '0';

        // Lookup error location
        auto it = single_qubit_err_signatures.find(syndrome_str);
        if (it != single_qubit_err_signatures.end()) {
            result.converged = true;
            result.result[it->second] = 1.0;
        }

        return result;
    }
};

Implementing a Decoder in Python

CUDA-Q QEC supports implementing decoders in Python using the @qec.decoder decorator:

1. Create Decoder Class:

@qec.decoder("my_decoder")
class MyDecoder:
    def __init__(self, H, **kwargs):
        qec.Decoder.__init__(self, H)
        self.H = H
        # Initialize with optional kwargs

    def decode(self, syndrome):
        # Create result object
        result = qec.DecoderResult()

        # Implement decoding logic
        # ...

        # Set results
        result.converged = True
        result.result = [0.0] * self.block_size

        return result

2. Using Custom Parameters:

# Create decoder with custom parameters
decoder = qec.get_decoder("my_decoder",
                        H=parity_check_matrix,
                        custom_param=42)

Key Features

• Soft Decision Decoding: Results are probabilities in [0,1]

• Batch Processing: Support for decoding multiple syndromes

• Asynchronous Decoding: Optional async interface for parallel processing

• Custom Parameters: Flexible configuration via heterogeneous_map

• Python Integration: First-class support for Python implementations

Usage Example

Python

import cudaq_qec as qec

# Get a code instance
code = qec.get_code('steane')

# Create decoder with code's parity matrix
decoder = qec.get_decoder('single_error_lut',
                        H=code.get_parity())

# Run stabilizer measurements
syndromes, dataQubitResults = qec.sample_memory_circuit(steane, numShots, numRounds)

# Decode syndrome
result = decoder.decode(syndromes[0])
if result.converged:
    print("Error locations:",
        [i for i,p in enumerate(result.result) if p > 0.5])

C++

using namespace cudaq;

// Get a code instance
auto code = qec::get_code("steane");

// Create decoder with code's parity matrix
auto decoder = qec::get_decoder("single_error_lut",
                        code->get_parity());

// Run stabilizer measurements
auto [syndromes, dataQubitResults] = qec::sample_memory_circuit(*code, /*numShots*/numShots, /*numRounds*/ 1);

// Decode syndrome
auto result = decoder->decode(syndromes[0]);

Numerical Experiments

CUDA-Q QEC provides utilities for running numerical experiments with quantum error correction codes.

Conventions

To address vectors of qubits (cudaq::qvector), CUDAQ indexing starts from 0, and 0 corresponds to the leftmost position when working with pauli strings (cudaq::spin_op). For example, applying a pauli X operator to qubit 1 out of 7 would be X_1 = IXIIIII.

While implementing your own codes and decoders, you are free to follow any convention that is convenient to you. However, to interact with the pre-built QEC codes and decoders within this library, the following conventions are used. All of these codes are CSS codes, and so we separate \(X\)-type and \(Z\)-type errors. For example, an error vector for 3 qubits will have 6 entries, 3 bits representing the presence of a bit-flip on each qubit, and 3 bits representing a phase-flip on each qubit. An error vector representing a bit-flip on qubit 0, and a phase-flip on qubit 1 would look like E = 100010. This means that this error vector is just two error vectors (E_X, E_Z) concatenated together (E = E_X | E_Z).

These errors are detected by stabilizers. \(Z\)-stabilizers detect \(X\)-type errors and vice versa. Thus we write our CSS parity check matrices as

\[\begin{split}H_{CSS} = \begin{pmatrix} H_Z & 0 \\ 0 & H_X \end{pmatrix},\end{split}\]

so that when we generate a syndrome vector by multiplying the parity check matrix by an error vector we get

\[\begin{split}\begin{align} S &= H \cdot E\\ S_X &= H_Z \cdot E_x\\ S_Z &= H_X \cdot E_Z. \end{align}\end{split}\]

This means that for the concatenated syndrome vector S = S_X | S_Z, the first part, S_X, are syndrome bits triggered by Z stabilizers detecting X errors. This is because the Z stabilizers like ZZI and IZZ anti-commute with X errors like IXI.

The decoder prediction as to what error happened is D = D_X | D_Z. A successful error decoding does not require that D = E, but that D + E is not a logical operator. There are a couple ways to check this. For bitflip errors, we check that the residual error R = D_X + E_X is not L_X. Since X anticommutes with Z, we can check that L_Z(D_X + E_X) = 0. This is because we just need to check if they have mutual support on an even or odd number of qubits. We could also check that R is not a stabilizer.

Similar to the parity check matrix, the logical obvervables are also stored in a matrix as

\[\begin{split}L = \begin{pmatrix} L_Z & 0 \\ 0 & L_X \end{pmatrix},\end{split}\]

so that when determining logical errors, we can do matrix multiplication

\[\begin{split}\begin{align} P &= L \cdot R\\ P_X &= L_Z \cdot R_x\\ P_Z &= L_X \cdot R_Z. \end{align}\end{split}\]

Here we’re using P as this can be stored in a Pauli frame tracker to track observable flips.

Each logical qubit has logical observables associated with it. Depending on what basis the data qubits are measured in, either the X or Z logical observables can be measured. The data qubits which support the logical observable is contained the qec::code class as well.

To do a logical Z(X) measurement, measure out all of the data qubits in the Z(X) basis. Then check support on the appropriate Z(x) observable.

Memory Circuit Experiments

Memory circuit experiments test a QEC code’s ability to preserve quantum information over time by:

1. Preparing an initial logical state

2. Performing multiple rounds of stabilizer measurements

3. Measuring data qubits to verify state preservation

4. Optionally applying noise during the process

Function Variants

Python

import cudaq_qec as qec

# Basic memory circuit with |0⟩ state
syndromes, measurements = qec.sample_memory_circuit(
    code,           # QEC code instance
    numShots=1000, # Number of circuit executions
    numRounds=1    # Number of stabilizer rounds
)

# Memory circuit with custom initial state
syndromes, measurements = qec.sample_memory_circuit(
    code,                       # QEC code instance
    state_prep=qec.operation.prep1,  # Initial state
    numShots=1000,            # Number of shots
    numRounds=1               # Number of rounds
)

# Memory circuit with noise model
noise = cudaq.noise_model()
noise.add_channel(...)  # Configure noise
syndromes, measurements = qec.sample_memory_circuit(
    code,           # QEC code instance
    numShots=1000, # Number of shots
    numRounds=1,   # Number of rounds
    noise=noise     # Noise model
)

C++

// Basic memory circuit with |0⟩ state
auto [syndromes, measurements] = qec::sample_memory_circuit(
    code,           // QEC code instance
    numShots,       // Number of circuit executions
    numRounds       // Number of stabilizer rounds
);

// Memory circuit with custom initial state
auto [syndromes, measurements] = qec::sample_memory_circuit(
    code,                    // QEC code instance
    operation::prep1,  // Initial state preparation
    numShots,                // Number of circuit executions
    numRounds               // Number of stabilizer rounds
);

// Memory circuit with noise model
auto noise_model = cudaq::noise_model();
noise_model.add_channel(...);  // Configure noise
auto [syndromes, measurements] = qec::sample_memory_circuit(
    code,           // QEC code instance
    numShots,       // Number of circuit executions
    numRounds,      // Number of stabilizer rounds
    noise_model     // Noise model to apply
);

Return Values

The functions return a tuple containing:

1. Syndrome Measurements (tensor<uint8_t>):

   • Shape: (num_shots, (num_rounds-1) * syndrome_size)

   • Contains stabilizer measurement results

   • Values are 0 or 1 representing measurement outcomes

2. Data Measurements (tensor<uint8_t>):

   • Shape: (num_shots, block_size)

   • Contains final data qubit measurements

   • Used to verify logical state preservation

Example Usage

Here’s a complete example of running a memory experiment:

Python

import cudaq
import cudaq_qec as qec

# Create code and decoder
code = qec.get_code('steane')
decoder = qec.get_decoder('single_error_lut',
                          code.get_parity())

# Configure noise
noise = cudaq.noise_model()
noise.add_channel('x', depolarizing=0.001)

# Run memory experiment
syndromes, measurements = qec.sample_memory_circuit(
    code,
    state_prep=qec.operation.prep0,
    num_shots=1000,
    num_rounds=10,
    noise=noise
)

# Analyze results
for shot in range(1000):
    # Get syndrome for this shot
    syndrome = syndromes[shot].tolist()

    # Decode syndrome
    result = decoder.decode(syndrome)
    if result.converged:
        # Process correction
        pass

C++

// Top of file
#include "cudaq/qec/experiments.h"

// Create a Steane code instance
auto code = cudaq::qec::get_code("steane");

// Configure noise model
auto noise = cudaq::noise_model();
noise.add_all_qubit_channel("x", cudaq::qec::two_qubit_depolarization(0.1),
                    /*num_controls=*/1);

// Run memory experiment
auto [syndromes, measurements] = qec::sample_memory_circuit(
    code,                    // Code instance
    operation::prep0,  // Prepare |0⟩ state
    1000,                    // 1000 shots
    10,                      // 10 rounds
    noise                    // Apply noise
);

// Analyze results
auto decoder = qec::get_decoder("single_error_lut", code->get_parity());
for (std::size_t shot = 0; shot < 1000; shot++) {
    // Get syndrome for this shot
    std::vector<float> syndrome(code->get_syndrome_size());
    for (std::size_t i = 0; i < syndrome.size(); i++)
        syndrome[i] = syndromes.at({shot, i});

    // Decode syndrome
    auto result = decoder->decode(syndrome);
    // Process correction
    // ...
}

Additional Noise Models:

Python

noise = cudaq.NoiseModel()

# Add multiple error channels
noise.add_all_qubit_channel('h', cudaq.BitFlipChannel(0.001))

# Specify two qubit errors
noise.add_all_qubit_channel("x", qec.TwoQubitDepolarization(p), 1)

C++

cudaq::noise_model noise;

# Add multiple error channels
noise.add_all_qubit_channel(
    "x", cudaq::BitFlipChannel(/*probability*/ 0.01));

# Specify two qubit errors
noise.add_all_qubit_channel(
    "x", cudaq::qec::two_qubit_depolarization(/*probability*/ 0.01),
    /*numControls*/ 1);