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In Stim, after performing syndrome measurements circuit using stabilizer codes like surface codes, how can we know the expected value of errors occurring on each data qubit? Even if there are no errors in the circuit, measuring each data qubit transversally after the syndrome measurement will result in the measurement outcome flipping non-deterministically (up to the stabilizer). This means that simply sampling the results of the transversal measurement of each data qubit to obtain the expected value won't yield the correct expectation value. I'd like to know if there's another way.

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Since stim v1.12.0, you can use stim.FlipSimulator to determine the rate at which data qubits are flipped (or whether, in a specific shot, it has been flipped a particular way).

Truncate the circuit to just before the data measurements, run the circuit in the flip simulator (with stabilizer randomization disabled), and count how often you see each Pauli.

import numpy as np
import stim

# Get a surface code circuit.
circuit = stim.Circuit.generated(
    "surface_code:rotated_memory_x",
    distance=5,
    rounds=15,
    after_clifford_depolarization=1e-3,
    before_round_data_depolarization=1e-3,
    before_measure_flip_probability=1e-3,
    after_reset_flip_probability=1e-3,
)

# Truncate the circuit so it stops just before the data measurements.
last_measurement_layer = len(circuit) - 1
while circuit[last_measurement_layer].name != 'MR':
    last_measurement_layer -= 1
circuit = circuit[:last_measurement_layer]


# Collect stats over a million+ shots, in batches of 1024 (takes ~10 seconds).
i_hits = np.zeros(shape=circuit.num_qubits, dtype=np.uint64)
x_hits = np.zeros(shape=circuit.num_qubits, dtype=np.uint64)
y_hits = np.zeros(shape=circuit.num_qubits, dtype=np.uint64)
z_hits = np.zeros(shape=circuit.num_qubits, dtype=np.uint64)
for _ in range(1000):
    sim = stim.FlipSimulator(
        batch_size=1024,
        disable_stabilizer_randomization=True,
    )
    sim.do(circuit)

    # Count number of times each Pauli occurred on each qubit.
    instance_paulis: stim.PauliString
    for instance_paulis in sim.peek_pauli_flips():
        xs, zs = instance_paulis.to_numpy(bit_packed=False)
        i_hits += ~xs & ~zs
        x_hits += xs & ~zs
        y_hits += xs & zs
        z_hits += ~xs & zs


# Print results.
qubit_coords = circuit.get_final_qubit_coordinates()
for q, coords in qubit_coords.items():
    i, x, y, z = i_hits[q], x_hits[q], y_hits[q], z_hits[q]
    t = i + x + y + z
    print(f"qubit at {str(coords):>14}: X={x/t:<16} Y={y/t:<16} Z={z/t:<16}")

Example results (note this includes data and measure qubits):

qubit at     [1.0, 1.0]: X=0.00206640625    Y=0.0008642578125  Z=0.0033662109375 
qubit at     [2.0, 0.0]: X=0.006416015625   Y=0.0011796875     Z=0.0012509765625 
qubit at     [3.0, 1.0]: X=0.0041142578125  Y=0.0010810546875  Z=0.0027060546875 
qubit at     [5.0, 1.0]: X=0.004029296875   Y=0.0011611328125  Z=0.003640625     
qubit at     [6.0, 0.0]: X=0.006451171875   Y=0.001201171875   Z=0.0011552734375 
qubit at     [7.0, 1.0]: X=0.0041064453125  Y=0.001087890625   Z=0.00265625      
qubit at     [9.0, 1.0]: X=0.00259765625    Y=0.00084765625    Z=0.0024365234375 
qubit at     [1.0, 3.0]: X=0.0034638671875  Y=0.0011806640625  Z=0.003703125     
qubit at     [2.0, 2.0]: X=0.007291015625   Y=0.00102734375    Z=0.0010830078125 
qubit at     [3.0, 3.0]: X=0.004375         Y=0.0014482421875  Z=0.0039501953125 
qubit at     [4.0, 2.0]: X=0.0123115234375  Y=0.0018193359375  Z=0.0017705078125 
qubit at     [5.0, 3.0]: X=0.0043759765625  Y=0.0014013671875  Z=0.0039853515625 
qubit at     [6.0, 2.0]: X=0.0089931640625  Y=0.001046875      Z=0.0010712890625 
qubit at     [7.0, 3.0]: X=0.004392578125   Y=0.0014306640625  Z=0.0039423828125 
qubit at     [8.0, 2.0]: X=0.0118388671875  Y=0.00170703125    Z=0.0017861328125 
qubit at     [9.0, 3.0]: X=0.0018125        Y=0.0011689453125  Z=0.002677734375  
qubit at    [10.0, 2.0]: X=0.0059580078125  Y=0.000498046875   Z=0.00056640625   
qubit at     [0.0, 4.0]: X=0.003765625      Y=0.000537109375   Z=0.0005283203125 
qubit at     [1.0, 5.0]: X=0.002279296875   Y=0.0011962890625  Z=0.0036416015625 
qubit at     [2.0, 4.0]: X=0.0132919921875  Y=0.0018017578125  Z=0.0017685546875 
qubit at     [3.0, 5.0]: X=0.004310546875   Y=0.001388671875   Z=0.00394140625   
qubit at     [4.0, 4.0]: X=0.0094189453125  Y=0.00109765625    Z=0.0010927734375 
qubit at     [5.0, 5.0]: X=0.004279296875   Y=0.001435546875   Z=0.0039130859375 
qubit at     [6.0, 4.0]: X=0.01394921875    Y=0.0017841796875  Z=0.001677734375  
qubit at     [7.0, 5.0]: X=0.0043056640625  Y=0.001427734375   Z=0.0039638671875 
qubit at     [8.0, 4.0]: X=0.0096455078125  Y=0.0010634765625  Z=0.0010810546875 
qubit at     [9.0, 5.0]: X=0.0028818359375  Y=0.0011396484375  Z=0.002625        
qubit at     [1.0, 7.0]: X=0.003470703125   Y=0.0011318359375  Z=0.0036708984375 
qubit at     [2.0, 6.0]: X=0.00767578125    Y=0.0010361328125  Z=0.0010830078125 
qubit at     [3.0, 7.0]: X=0.0042568359375  Y=0.00137109375    Z=0.0040126953125 
qubit at     [4.0, 6.0]: X=0.0139052734375  Y=0.0017578125     Z=0.0016982421875 
qubit at     [5.0, 7.0]: X=0.004298828125   Y=0.001373046875   Z=0.0039921875    
qubit at     [6.0, 6.0]: X=0.009423828125   Y=0.001046875      Z=0.00103515625   
qubit at     [7.0, 7.0]: X=0.0043837890625  Y=0.001388671875   Z=0.003896484375  
qubit at     [8.0, 6.0]: X=0.013353515625   Y=0.001693359375   Z=0.0017841796875 
qubit at     [9.0, 7.0]: X=0.001828125      Y=0.00116796875    Z=0.0025703125    
qubit at    [10.0, 6.0]: X=0.0066552734375  Y=0.0005341796875  Z=0.000509765625  
qubit at     [0.0, 8.0]: X=0.003779296875   Y=0.0005263671875  Z=0.00051953125   
qubit at     [1.0, 9.0]: X=0.0020869140625  Y=0.00085546875    Z=0.001837890625  
qubit at     [2.0, 8.0]: X=0.0133505859375  Y=0.0017529296875  Z=0.0017900390625 
qubit at     [3.0, 9.0]: X=0.0029619140625  Y=0.0011484375     Z=0.00323828125   
qubit at     [4.0, 8.0]: X=0.008765625      Y=0.001076171875   Z=0.0010576171875 
qubit at     [5.0, 9.0]: X=0.0030146484375  Y=0.001171875      Z=0.0021728515625 
qubit at     [6.0, 8.0]: X=0.0140751953125  Y=0.0018154296875  Z=0.00168359375   
qubit at     [7.0, 9.0]: X=0.0030009765625  Y=0.001162109375   Z=0.0032080078125 
qubit at     [8.0, 8.0]: X=0.0088955078125  Y=0.00108984375    Z=0.0010390625    
qubit at     [9.0, 9.0]: X=0.0015361328125  Y=0.0008759765625  Z=0.001861328125  
qubit at    [4.0, 10.0]: X=0.009263671875   Y=0.0011455078125  Z=0.0011689453125 
qubit at    [8.0, 10.0]: X=0.00867578125    Y=0.0012060546875  Z=0.001201171875  
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