Mystifying!!

 I went back to my 10-qubit Hadamard number run on ibm_sherbrooke and had

the computer output every one of the 1024 possible binary outputs (2^10) as a list.

I then put the results in groups of 64, for 16 bins. Below...

                                                          

It's a bit mystifying: it appears the Hadamard flip the most to 0 in the initial 64 and -

if not - tend to make it to the end...

This was a qasm run for the computer, so it is bare-boned with respect to error

mitigation. I asked Gemini to explain the info from the run site,,,

                                                                   

OPENQASM 2.0;

include "qelib1.inc";

gate rzx(param0) q0,q1 { h q1; cx q0,q1; rz(param0) q1; cx q0,q1; h q1; }

gate ecr q0,q1 { rzx(pi/4) q0,q1; x q0; rzx(-pi/4) q0,q1; }

qreg q[127];

creg meas[10];

rz(-pi) q[0];

sx q[0];

rz(-pi/2) q[1];

sx q[1];

rz(-pi) q[1];

ecr q[1],q[0];

rz(-pi/2) q[0];

sx q[0];

rz(pi/2) q[0];

rz(pi/2) q[1];

sx q[1];

rz(-pi) q[2];

sx q[2];

rz(-pi) q[2];

ecr q[1],q[2];

x q[1];

sx q[2];

rz(-pi/2) q[2];

rz(-pi/2) q[3];

sx q[3];

rz(-pi) q[3];

ecr q[3],q[2];

rz(-pi/2) q[2];

sx q[2];

rz(pi/2) q[2];

rz(-pi) q[3];

sx q[3];

rz(-pi/2) q[4];

sx q[4];

rz(-pi) q[4];

ecr q[4],q[3];

rz(-pi/2) q[3];

sx q[3];

rz(pi/2) q[3];

rz(-pi) q[4];

sx q[4];

rz(-pi/2) q[5];

sx q[5];

rz(-pi) q[5];

ecr q[5],q[4];

rz(-pi/2) q[4];

sx q[4];

rz(pi/2) q[4];

rz(-pi) q[5];

sx q[5];

rz(-pi/2) q[6];

sx q[6];

rz(-pi) q[6];

ecr q[6],q[5];

rz(-pi/2) q[5];

sx q[5];

rz(pi/2) q[5];

rz(-pi) q[6];

sx q[6];

rz(-pi/2) q[7];

sx q[7];

rz(-pi) q[7];

ecr q[7],q[6];

rz(-pi/2) q[6];

sx q[6];

rz(pi/2) q[6];

rz(pi/2) q[7];

sx q[7];

rz(-pi) q[8];

sx q[8];

rz(-pi) q[8];

ecr q[7],q[8];

x q[7];

rz(-pi/2) q[8];

rz(-pi) q[9];

sx q[9];

rz(-pi) q[9];

ecr q[8],q[9];

x q[8];

barrier q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8],q[9];

measure q[0] -> meas[0];

measure q[1] -> meas[1];

measure q[2] -> meas[2];

measure q[3] -> meas[3];

measure q[4] -> meas[4];

measure q[5] -> meas[5];

measure q[6] -> meas[6];

measure q[7] -> meas[7];

measure q[8] -> meas[8];

measure q[9] -> meas[9];

Circuit Analysis and Error Mitigation

The provided QASM code defines a quantum circuit consisting of multiple parts. Let's break it down and see if there's any evidence of error mitigation techniques:

Circuit Breakdown:

1. Qubit Initialization:

   • The circuit starts by initializing 11 qubits (q[0] to q[10]) in the |0> state using rz(-pi) and applying a Pauli-X (sx) gate.

2. Entanglement and Error Correction (ECR) Gates:

   • The circuit then applies a series of gates that create entanglement between qubit pairs (q[0,1], q[1,2], etc.) followed by an ecr gate.
     • The ecr gate (defined using the rzx gate) is a type of error correction code specifically designed for single-qubit bit-flip errors. It helps to mitigate these errors by introducing redundancy in the quantum state.

3. Additional Gates:

   • The circuit continues with further gates like rz (rotations around the Z-axis) and sx (Pauli-X) on various qubits. These gates likely implement the specific quantum algorithm you're running.

4. Barrier and Measurement:

   • A barrier synchronizes all qubits, ensuring they complete their operations before measurement.
   • Finally, the circuit measures qubits q[0] to q[9] and stores the results in classical bits (meas[0] to meas[9]).

Error Mitigation Techniques:

Based on the code, the circuit utilizes the ecr gate for error correction. This is a positive sign, as it actively tries to counteract bit-flip errors during the execution.

However, it's important to note that:

• The ecr gate can only handle bit-flip errors. Other types of errors, like phase-flip errors or multi-qubit errors, are not addressed by this specific code.
• The effectiveness of error correction codes like ecr depends on various factors, including the code itself, the number of qubits involved, and the inherent error rates of the quantum hardware.

Additional Considerations:

• It's possible that additional error mitigation techniques might have been applied during the execution on the quantum computer. These might involve software techniques or hardware-specific optimizations. Unfortunately, the provided code doesn't reveal those details.
• To get a more comprehensive picture of the error mitigation used, you could consult the documentation or reports associated with the specific quantum computer you used.

Overall:

The circuit demonstrates an awareness of potential errors by incorporating the ecr gate. However, it's likely just one part of a larger error mitigation strategy employed by the quantum hardware and software stack.