Error Correction over time

Question

I am in year 12 and am a Science Extension student. My research question was "To what extent does an error-correcting algorithm reduce the inaccuracy of a quantum computation over time?".

To conduct this experiment I utilised Shors 3-qubit bit flip error correcting algorithm within my circuit as shown below. I have utilised a loop within my circuit to increase the amount of time the that the computation takes thus leaving more time for error to occur. I also ran a circuit without the error-correcting algorithm and just the loops to measure the fidelity of the quantum computation. I was expecting the circuit with the error-correcting algorithm to have a higher fidelity however after running both circuits through IBM's quantum computer ibmq_belem the circuit without the error-correcting algorithm turned out to be significantly more accurate. Can someone please tell me why this is?

num_repeats = 1000 #This number represents how many times the certain part of the circuit will be repeated. 
pause_time = gate_length * num_repeats


qreg_q = QuantumRegister(4, 'q')
creg_c = ClassicalRegister(4, 'c')
circuit = QuantumCircuit(qreg_q, creg_c)

circuit.h(qreg_q[0])
circuit.cx(qreg_q[0], qreg_q[1])
circuit.cx(qreg_q[0], qreg_q[2])
circuit.h(qreg_q[0])
circuit.h(qreg_q[1])
circuit.h(qreg_q[2])

for blah in range(num_repeats):
    circuit.barrier(qreg_q[0], qreg_q[1], qreg_q[2], qreg_q[3])
    circuit.x(qreg_q[3])
    
circuit.barrier(qreg_q[0], qreg_q[1], qreg_q[2], qreg_q[3])
    
circuit.h(qreg_q[0])
circuit.h(qreg_q[1])
circuit.h(qreg_q[2])
circuit.cx(qreg_q[0], qreg_q[1])
circuit.cx(qreg_q[0], qreg_q[2])
circuit.ccx(qreg_q[2], qreg_q[1], qreg_q[0])
circuit.h(qreg_q[0])
circuit.measure(qreg_q[0], creg_c[0])
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Answer 1

This is because error correcting codes only help if the physical noise rate is below some threshold value. Available quantum computers are still too noisy, so error correction experiments are still error worsening experiments. The Shor code has a particularly demanding threshold, if I recall correctly, so I'm not surprised your experiment didn't work on real hardware. It could only work in a simulation with lower noise than current hardware.

Keep in mind there are still very few experiments ever that have shown more error correction helping instead of hurting, and the ones that do (like https://arxiv.org/abs/2107.07505 and https://arxiv.org/abs/2207.06431) tend to only show it helping in one way while hurting in another or show it not helping very much. A major experimental goal of the field is to make machines with lower noise, so that error correction helps instead of hurts. If we can make the machines ten times less noisy, then error correction will start working (and in fact start working well) and it will become conceivable to run computations with billions of steps instead of only thousands... given enough qubits.