Prompt:
Given this 4-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |0000>.

Circuit (15 gates applied in order):
  cx on qubits [1, 3]
  x on qubits [2]
  rz(5.119827) on qubits [0]
  s on qubits [2]
  t on qubits [3]
  cx on qubits [0, 1]
  cx on qubits [1, 0]
  h on qubits [1]
  h on qubits [0]
  z on qubits [0]
  x on qubits [1]
  cx on qubits [2, 3]
  cx on qubits [1, 2]
  rx(3.599562) on qubits [0]
  ry(3.196169) on qubits [3]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.003
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.015
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 4-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |0000>.

Circuit (15 gates applied in order):
  s on qubits [1]
  y on qubits [2]
  h on qubits [1]
  rx(0.139009) on qubits [0]
  t on qubits [1]
  h on qubits [3]
  rx(4.835409) on qubits [3]
  z on qubits [2]
  cx on qubits [3, 1]
  cx on qubits [2, 0]
  rz(2.768936) on qubits [0]
  rz(0.243076) on qubits [3]
  s on qubits [2]
  z on qubits [3]
  x on qubits [2]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.003
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.015
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (20 gates applied in order):
  cx on qubits [1, 0]
  cx on qubits [4, 2]
  ry(6.012392) on qubits [4]
  cx on qubits [0, 1]
  y on qubits [2]
  cx on qubits [4, 0]
  cx on qubits [2, 4]
  h on qubits [3]
  rx(2.981315) on qubits [1]
  y on qubits [0]
  t on qubits [4]
  cx on qubits [4, 1]
  rz(4.481628) on qubits [3]
  t on qubits [3]
  x on qubits [0]
  rz(5.843443) on qubits [2]
  y on qubits [4]
  cx on qubits [4, 2]
  t on qubits [3]
  cx on qubits [1, 2]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.002
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.01
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (20 gates applied in order):
  rx(5.925500) on qubits [0]
  z on qubits [1]
  s on qubits [1]
  x on qubits [1]
  cx on qubits [1, 4]
  y on qubits [0]
  rz(6.208256) on qubits [3]
  y on qubits [1]
  cx on qubits [1, 0]
  cx on qubits [0, 1]
  z on qubits [3]
  z on qubits [4]
  rx(1.175447) on qubits [2]
  t on qubits [1]
  ry(6.008857) on qubits [0]
  cx on qubits [0, 4]
  s on qubits [4]
  ry(2.850138) on qubits [4]
  rx(5.692493) on qubits [1]
  rx(5.301326) on qubits [4]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.002
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.01
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (20 gates applied in order):
  h on qubits [1]
  z on qubits [0]
  s on qubits [1]
  z on qubits [4]
  cx on qubits [1, 2]
  y on qubits [1]
  s on qubits [1]
  t on qubits [1]
  t on qubits [1]
  cx on qubits [4, 2]
  rz(6.079493) on qubits [2]
  s on qubits [2]
  t on qubits [3]
  cx on qubits [1, 0]
  cx on qubits [1, 0]
  cx on qubits [0, 2]
  cx on qubits [4, 0]
  cx on qubits [3, 1]
  cx on qubits [4, 2]
  cx on qubits [3, 4]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.002
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.01
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (20 gates applied in order):
  cx on qubits [4, 3]
  ry(3.565913) on qubits [1]
  rx(2.425831) on qubits [4]
  cx on qubits [2, 4]
  s on qubits [4]
  cx on qubits [1, 4]
  cx on qubits [3, 0]
  y on qubits [4]
  h on qubits [2]
  x on qubits [3]
  x on qubits [2]
  s on qubits [3]
  cx on qubits [1, 2]
  cx on qubits [3, 1]
  y on qubits [1]
  z on qubits [0]
  s on qubits [0]
  ry(2.901252) on qubits [3]
  cx on qubits [3, 2]
  rx(1.899734) on qubits [0]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.002
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.01
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (20 gates applied in order):
  x on qubits [0]
  t on qubits [3]
  y on qubits [1]
  rx(0.821224) on qubits [2]
  h on qubits [1]
  h on qubits [1]
  z on qubits [1]
  x on qubits [2]
  x on qubits [3]
  cx on qubits [1, 0]
  h on qubits [0]
  x on qubits [4]
  cx on qubits [1, 4]
  s on qubits [3]
  t on qubits [3]
  cx on qubits [1, 0]
  cx on qubits [1, 2]
  cx on qubits [2, 3]
  x on qubits [2]
  z on qubits [1]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.002
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.01
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (30 gates applied in order):
  h on qubits [0]
  z on qubits [0]
  t on qubits [4]
  t on qubits [0]
  rz(5.024267) on qubits [4]
  ry(1.798536) on qubits [4]
  cx on qubits [4, 0]
  h on qubits [0]
  cx on qubits [1, 2]
  cx on qubits [2, 3]
  cx on qubits [3, 2]
  rx(3.788735) on qubits [4]
  ry(1.851775) on qubits [4]
  cx on qubits [0, 3]
  x on qubits [2]
  cx on qubits [4, 0]
  cx on qubits [2, 1]
  ry(1.272514) on qubits [2]
  z on qubits [1]
  cx on qubits [2, 4]
  ry(2.176526) on qubits [3]
  x on qubits [2]
  cx on qubits [4, 1]
  y on qubits [1]
  rx(2.362067) on qubits [0]
  cx on qubits [3, 2]
  s on qubits [0]
  cx on qubits [3, 4]
  s on qubits [4]
  cx on qubits [4, 2]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.001
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.008
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (30 gates applied in order):
  h on qubits [2]
  cx on qubits [1, 3]
  s on qubits [0]
  t on qubits [2]
  cx on qubits [4, 1]
  s on qubits [2]
  t on qubits [0]
  ry(5.172327) on qubits [3]
  x on qubits [3]
  cx on qubits [2, 0]
  y on qubits [0]
  y on qubits [4]
  y on qubits [3]
  cx on qubits [1, 3]
  h on qubits [0]
  cx on qubits [0, 3]
  ry(4.690190) on qubits [1]
  x on qubits [0]
  t on qubits [3]
  z on qubits [1]
  cx on qubits [3, 0]
  cx on qubits [2, 3]
  z on qubits [0]
  s on qubits [1]
  cx on qubits [2, 4]
  rz(4.565329) on qubits [0]
  z on qubits [3]
  cx on qubits [2, 1]
  s on qubits [1]
  s on qubits [2]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.001
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.008
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (30 gates applied in order):
  rx(5.909673) on qubits [1]
  z on qubits [4]
  y on qubits [2]
  y on qubits [4]
  rz(3.893760) on qubits [3]
  h on qubits [0]
  s on qubits [2]
  t on qubits [3]
  rz(1.482682) on qubits [0]
  cx on qubits [4, 3]
  h on qubits [0]
  s on qubits [0]
  s on qubits [3]
  y on qubits [3]
  cx on qubits [3, 2]
  y on qubits [0]
  y on qubits [2]
  y on qubits [1]
  y on qubits [3]
  s on qubits [1]
  rx(5.354935) on qubits [2]
  s on qubits [0]
  rx(2.171684) on qubits [0]
  y on qubits [3]
  s on qubits [4]
  h on qubits [0]
  rx(1.356998) on qubits [4]
  t on qubits [4]
  cx on qubits [4, 0]
  rx(6.103324) on qubits [0]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.001
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.008
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (30 gates applied in order):
  x on qubits [0]
  cx on qubits [1, 0]
  cx on qubits [4, 3]
  t on qubits [3]
  h on qubits [3]
  s on qubits [1]
  x on qubits [1]
  cx on qubits [3, 4]
  z on qubits [1]
  t on qubits [1]
  cx on qubits [1, 2]
  z on qubits [3]
  h on qubits [1]
  cx on qubits [0, 3]
  cx on qubits [2, 1]
  cx on qubits [1, 0]
  ry(0.470682) on qubits [3]
  cx on qubits [4, 1]
  rz(4.781103) on qubits [2]
  cx on qubits [4, 0]
  y on qubits [2]
  cx on qubits [2, 4]
  y on qubits [4]
  cx on qubits [4, 2]
  ry(6.231225) on qubits [3]
  cx on qubits [3, 4]
  rz(3.598171) on qubits [1]
  rz(0.325366) on qubits [1]
  t on qubits [2]
  rz(2.144874) on qubits [0]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.001
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.008
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Given this 5-qubit quantum circuit and noise model, predict the output state fidelity F = <psi_ideal|rho_noisy|psi_ideal>.

The initial state is |00000>.

Circuit (30 gates applied in order):
  z on qubits [0]
  cx on qubits [0, 3]
  t on qubits [1]
  s on qubits [4]
  cx on qubits [4, 2]
  x on qubits [0]
  rx(2.532508) on qubits [3]
  rx(2.122953) on qubits [1]
  s on qubits [4]
  t on qubits [3]
  x on qubits [1]
  cx on qubits [2, 0]
  t on qubits [3]
  rz(4.134292) on qubits [1]
  ry(4.808796) on qubits [1]
  cx on qubits [4, 2]
  rx(5.057470) on qubits [3]
  cx on qubits [1, 0]
  rz(0.887469) on qubits [0]
  t on qubits [1]
  cx on qubits [1, 3]
  x on qubits [0]
  h on qubits [4]
  h on qubits [4]
  rx(1.479008) on qubits [4]
  cx on qubits [4, 3]
  cx on qubits [1, 2]
  cx on qubits [2, 3]
  cx on qubits [3, 2]
  y on qubits [0]

Noise model: depolarizing noise applied AFTER each gate.
  - After each 1-qubit gate: depolarizing channel with probability p = 0.001
    D_1(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
  - After each 2-qubit gate: depolarizing channel with probability p = 0.008
    D_2(rho) = (1-p)*rho + (p/15)*sum_{P in Paulis\II} P*rho*P
    (sum over all 15 non-identity two-qubit Pauli operators)

Standard gate definitions:
  h: (1/sqrt(2))*[[1,1],[1,-1]]
  x: [[0,1],[1,0]]
  y: [[0,-i],[i,0]]
  z: [[1,0],[0,-1]]
  s: [[1,0],[0,i]]
  t: [[1,0],[0,exp(i*pi/4)]]
  rx(theta): [[cos(t/2), -i*sin(t/2)], [-i*sin(t/2), cos(t/2)]]
  ry(theta): [[cos(t/2), -sin(t/2)], [sin(t/2), cos(t/2)]]
  rz(theta): [[exp(-i*t/2), 0], [0, exp(i*t/2)]]
  cx: CNOT (control, target) in computational basis

Multi-qubit operators use Qiskit convention: qubit 0 is LEAST significant (rightmost in tensor product). E.g., for gate on qubit k in an n-qubit system, the full operator is I_{n-1} ⊗ ... ⊗ Gate_k ⊗ ... ⊗ I_0.

Write a function `solve()` that returns a float (the predicted fidelity).
You may use numpy. Do NOT use qiskit, qiskit_aer, or any quantum simulation library.
Implement density matrix propagation from scratch.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  +0.388411 * XY
  -0.409094 * ZX
  -0.674678 * IX

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, -1.0838+0.0000j, +0.0000+0.0000j, +0.0000-0.3884j]
  [-1.0838+0.0000j, +0.0000+0.0000j, +0.0000+0.3884j, +0.0000+0.0000j]
  [+0.0000+0.0000j, +0.0000-0.3884j, +0.0000+0.0000j, -0.2656+0.0000j]
  [+0.0000+0.3884j, +0.0000+0.0000j, -0.2656+0.0000j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -0.749955 * ZY
  -1.475643 * XY
  +1.161125 * IY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, +0.0000-0.4112j, +0.0000+0.0000j, +0.0000+1.4756j]
  [+0.0000+0.4112j, +0.0000+0.0000j, +0.0000-1.4756j, +0.0000+0.0000j]
  [+0.0000+0.0000j, +0.0000+1.4756j, +0.0000+0.0000j, +0.0000-1.9111j]
  [+0.0000-1.4756j, +0.0000+0.0000j, +0.0000+1.9111j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  +0.714373 * XY
  -0.762022 * XZ
  -0.564435 * IY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, +0.0000+0.5644j, -0.7620+0.0000j, +0.0000-0.7144j]
  [+0.0000-0.5644j, +0.0000+0.0000j, +0.0000+0.7144j, +0.7620+0.0000j]
  [-0.7620+0.0000j, +0.0000-0.7144j, +0.0000+0.0000j, +0.0000+0.5644j]
  [+0.0000+0.7144j, +0.7620+0.0000j, +0.0000-0.5644j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -0.593950 * XZ
  -0.410438 * IX
  -0.783413 * ZX

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, -1.1939+0.0000j, -0.5940+0.0000j, +0.0000+0.0000j]
  [-1.1939+0.0000j, +0.0000+0.0000j, +0.0000+0.0000j, +0.5940+0.0000j]
  [-0.5940+0.0000j, +0.0000+0.0000j, +0.0000+0.0000j, +0.3730+0.0000j]
  [+0.0000+0.0000j, +0.5940+0.0000j, +0.3730+0.0000j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -0.514985 * XI
  +0.704079 * XZ
  -0.337695 * IY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, +0.0000+0.3377j, +0.1891+0.0000j, +0.0000+0.0000j]
  [+0.0000-0.3377j, +0.0000+0.0000j, +0.0000+0.0000j, -1.2191+0.0000j]
  [+0.1891+0.0000j, +0.0000+0.0000j, +0.0000+0.0000j, +0.0000+0.3377j]
  [+0.0000+0.0000j, -1.2191+0.0000j, +0.0000-0.3377j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  +0.884021 * YI
  -0.255594 * YZ
  -0.490551 * XY
  +1.300056 * YY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, +0.0000+0.0000j, +0.0000-0.6284j, -1.3001+0.4906j]
  [+0.0000+0.0000j, +0.0000+0.0000j, +1.3001-0.4906j, +0.0000-1.1396j]
  [+0.0000+0.6284j, +1.3001+0.4906j, +0.0000+0.0000j, +0.0000+0.0000j]
  [-1.3001-0.4906j, +0.0000+1.1396j, +0.0000+0.0000j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -1.434602 * XZ
  -1.154895 * ZI
  +0.584417 * XX
  +1.357592 * IZ

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.2027+0.0000j, +0.0000+0.0000j, -1.4346+0.0000j, +0.5844+0.0000j]
  [+0.0000+0.0000j, -2.5125+0.0000j, +0.5844+0.0000j, +1.4346+0.0000j]
  [-1.4346+0.0000j, +0.5844+0.0000j, +2.5125+0.0000j, +0.0000+0.0000j]
  [+0.5844+0.0000j, +1.4346+0.0000j, +0.0000+0.0000j, -0.2027+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -0.260668 * XX
  +0.293711 * XZ
  -0.355289 * ZI
  +0.481189 * YY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [-0.3553+0.0000j, +0.0000+0.0000j, +0.2937+0.0000j, -0.7419+0.0000j]
  [+0.0000+0.0000j, -0.3553+0.0000j, +0.2205+0.0000j, -0.2937+0.0000j]
  [+0.2937+0.0000j, +0.2205+0.0000j, +0.3553+0.0000j, +0.0000+0.0000j]
  [-0.7419+0.0000j, -0.2937+0.0000j, +0.0000+0.0000j, +0.3553+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  +0.422994 * XY
  +0.372928 * XI
  +0.886131 * YZ
  +0.357041 * ZY

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, +0.0000-0.3570j, +0.3729-0.8861j, +0.0000-0.4230j]
  [+0.0000+0.3570j, +0.0000+0.0000j, +0.0000+0.4230j, +0.3729+0.8861j]
  [+0.3729+0.8861j, +0.0000-0.4230j, +0.0000+0.0000j, +0.0000+0.3570j]
  [+0.0000+0.4230j, +0.3729-0.8861j, +0.0000-0.3570j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 2-qubit Hamiltonian:

H =
  -0.625860 * IX
  +0.573932 * XZ
  +0.539464 * IY
  -0.106120 * YI

As a matrix (rows/cols ordered |0...0> to |1...1>):
  [+0.0000+0.0000j, -0.6259-0.5395j, +0.5739+0.1061j, +0.0000+0.0000j]
  [-0.6259+0.5395j, +0.0000+0.0000j, +0.0000+0.0000j, -0.5739+0.1061j]
  [+0.5739-0.1061j, +0.0000+0.0000j, +0.0000+0.0000j, -0.6259-0.5395j]
  [+0.0000+0.0000j, -0.5739-0.1061j, -0.6259+0.5395j, +0.0000+0.0000j]

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

You may use any method including exact diagonalization (numpy.linalg.eigvalsh).

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  +0.015416 * ZXX
  +1.140945 * ZXY
  -1.256270 * XXI
  -1.197515 * IXI
  -1.449543 * ZXI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -0.213029 * ZXZ
  +0.877352 * ZIY
  -0.219276 * YZI
  +0.169288 * IIX
  -0.667628 * IYI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -1.363655 * IZY
  +1.156962 * YIX
  -0.173752 * YYI
  -0.998578 * YIY
  -0.154202 * XXX

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -0.650431 * IIY
  +0.566400 * ZYZ
  -0.612789 * YZI
  +0.935554 * XIY
  +1.435760 * ZYX

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -0.279532 * YXY
  +0.736335 * ZXX
  +0.516231 * XYX
  -0.525929 * ZIX
  +1.252587 * IYI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  +0.519877 * YIX
  -0.120012 * IZI
  +1.497830 * IZX
  -0.814065 * IXZ
  -0.294305 * XII
  -1.363248 * XIZ

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -1.156454 * ZZX
  +1.417826 * IXZ
  +1.456190 * IZY
  +1.132992 * YXX
  -0.866199 * XXX
  +0.819999 * XZZ

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -0.623035 * ZZY
  +0.870990 * XXZ
  +1.432003 * YII
  +1.001697 * ZZX
  -0.733213 * IZY
  -0.291533 * ZYI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  +0.482597 * YXZ
  -1.139532 * ZIY
  -0.805483 * YXY
  -0.974966 * ZYZ
  +1.458830 * XZY
  -1.105548 * ZXZ

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 3-qubit Hamiltonian:

H =
  -0.921306 * XYX
  -0.219193 * YIX
  +0.200760 * ZYX
  +1.393010 * XXX
  +1.386273 * YII
  +1.349850 * IZI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 4-qubit Hamiltonian:

H =
  -1.491903 * YZXZ
  -1.059814 * YZZZ
  +1.467246 * YXIY
  -1.313575 * ZYIY
  +1.096421 * XZII
  -0.798604 * IZXZ
  -1.369385 * XZXX
  +1.435182 * IIXI

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 4-qubit Hamiltonian:

H =
  +0.264250 * ZZYX
  +0.947579 * YIXY
  +1.019813 * YYZZ
  -0.145668 * YXYX
  -0.582846 * ZIIX
  +0.176328 * IYYX
  +0.011389 * YXZI
  +0.961026 * XIYY

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 4-qubit Hamiltonian:

H =
  -0.514446 * ZXYI
  -0.528649 * XZYZ
  +0.914676 * YZIZ
  -0.334168 * XXZY
  -0.979045 * XZIZ
  +0.225206 * YYZY
  +0.353006 * ZXII
  -0.343815 * XYZY

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 4-qubit Hamiltonian:

H =
  -1.007755 * IZIZ
  +1.404643 * YIIX
  -0.145175 * IIZX
  +0.992688 * XZZY
  +1.420416 * IXIY
  +0.715544 * YZZX
  -1.103182 * XYIX
  +0.258612 * ZZZY

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
Find the ground state energy (lowest eigenvalue) of the following 4-qubit Hamiltonian:

H =
  -1.194971 * XYZY
  +1.240787 * ZIZI
  +0.843209 * ZYXX
  +0.075903 * ZYYZ
  +0.143694 * ZIYI
  +0.584805 * IYII
  -0.319779 * XIIZ
  -0.890422 * XZII

Pauli string convention: rightmost character acts on qubit 0.
Example: 'ZI' = Z on qubit 1, I on qubit 0.

Use a variational quantum eigensolver (VQE) approach:
  1. Define a parametrized ansatz circuit.
  2. Compute E(θ) = <ψ(θ)|H|ψ(θ)> using Qiskit's Estimator or by hand.
  3. Minimize E(θ) over θ with scipy.optimize.minimize.
Do NOT use numpy.linalg.eig, scipy.linalg.eigh, or any exact diagonalization.

Write a function `solve()` that returns the ground state energy as a float.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.01).

Syndrome (24 detector bits):
000000000100000010001000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.01).

Syndrome (24 detector bits):
000000100000011000000000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.01).

Syndrome (24 detector bits):
000010000001100000010000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.01).

Syndrome (24 detector bits):
000001000000000000000011

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.01).

Syndrome (24 detector bits):
000000000100000001000000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.02).

Syndrome (24 detector bits):
000000100001001000010010

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.02).

Syndrome (24 detector bits):
000000100100011000001000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.02).

Syndrome (24 detector bits):
000010000000100000000000

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.02).

Syndrome (24 detector bits):
000000100000100000101110

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-3 rotated surface code under circuit-level depolarizing noise (p=0.02).

Syndrome (24 detector bits):
011001001100000010000010

The syndrome was collected over 3 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-5 rotated surface code under circuit-level depolarizing noise (p=0.005).

Syndrome (120 detector bits):
000000000010000000100000000000000000010000000000000000000000011000000000000000000000000000000000000000001000000000000010

The syndrome was collected over 5 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-5 rotated surface code under circuit-level depolarizing noise (p=0.005).

Syndrome (120 detector bits):
000000000000000000000000000000000000000000000000000000000001000000000000000000000001000010000001000100000000000000111000

The syndrome was collected over 5 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-5 rotated surface code under circuit-level depolarizing noise (p=0.005).

Syndrome (120 detector bits):
000010100000000000000000000000000000000000000000000000000000000000000000000000000000110010000010000001100000010011010000

The syndrome was collected over 5 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-5 rotated surface code under circuit-level depolarizing noise (p=0.005).

Syndrome (120 detector bits):
000000000000000000010000000000000000000000010000000000000000000000000000000000000000101000000000000000000000000000000000

The syndrome was collected over 5 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-5 rotated surface code under circuit-level depolarizing noise (p=0.005).

Syndrome (120 detector bits):
000000000000000000000000000000000000000000000000000000000000001000000000000000000000101000000000001000000000000001000000

The syndrome was collected over 5 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-7 rotated surface code under circuit-level depolarizing noise (p=0.007).

Syndrome (336 detector bits):
00000000000000000000000000000000010000000000000000000000000000000000000000000000100001000000000000000000000000000000000000000100000000000000000000000000000000000000000000001000000010000000000110001000... (336 total bits)

The syndrome was collected over 7 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-7 rotated surface code under circuit-level depolarizing noise (p=0.007).

Syndrome (336 detector bits):
00000100000000000000000000000001000000100001100000000101000000000000000000100000000000000000100000000100000000000000000000000000000000000000100000101000000100000000100100000000000000000000000010100000... (336 total bits)

The syndrome was collected over 7 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-7 rotated surface code under circuit-level depolarizing noise (p=0.007).

Syndrome (336 detector bits):
00000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000010001001000100000000000000010000000000000000000000010100001000000000000000010000000000001001000000010100001... (336 total bits)

The syndrome was collected over 7 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-7 rotated surface code under circuit-level depolarizing noise (p=0.007).

Syndrome (336 detector bits):
00000000000000000000000000000000000100000010110000000100000010100000000000000000000000100010100000000000000000100001000000000011000000000000000000000000000100000001000000100000000000000000000001000000... (336 total bits)

The syndrome was collected over 7 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-7 rotated surface code under circuit-level depolarizing noise (p=0.007).

Syndrome (336 detector bits):
00000010010100000000100100011000000000000000000000000000010000000000000000011000000000000000110000001101000010000000000000010000000000000000100001000100000000100000000000010100000000000000001000000000... (336 total bits)

The syndrome was collected over 7 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-9 rotated surface code under circuit-level depolarizing noise (p=0.008).

Syndrome (720 detector bits):
00010000000000000000000011000000000000000000000000010000011000000000000000010110000001100000000100000000000100000000000000000000000000000000000110000000000000000000000000000000000000000000000001000000... (720 total bits)

The syndrome was collected over 9 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-9 rotated surface code under circuit-level depolarizing noise (p=0.008).

Syndrome (720 detector bits):
00000000000110110000000000000100000000000100000100010000001100000000100001100000000000110000001011010010000001000000000000100101010000000001000000001000001000000000000000000110100000010000000010000000... (720 total bits)

The syndrome was collected over 9 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-9 rotated surface code under circuit-level depolarizing noise (p=0.008).

Syndrome (720 detector bits):
00000000000100000000010010001000000000000011000001100000000000000000000010000000000000000000000000000000000001000110001000100000000000010000001100000000110000000110000000100000000000000100100001100000... (720 total bits)

The syndrome was collected over 9 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-9 rotated surface code under circuit-level depolarizing noise (p=0.008).

Syndrome (720 detector bits):
00000000000100100000001000100001000000000000000000000100000000000000100001010000000000000000000000001000100000000000000000000000001001011100010110000000000110000000000000000000000010001000001001000010... (720 total bits)

The syndrome was collected over 9 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.


Prompt:
You are given syndrome measurements from a distance-9 rotated surface code under circuit-level depolarizing noise (p=0.008).

Syndrome (720 detector bits):
00000000000000000010000000100000010001000001000000100000000100000000010000000000011000000001010101010000000000000001000100011000000000000001000000011000010000000100001000000000000000000100000000110010... (720 total bits)

The syndrome was collected over 9 rounds of stabilizer measurement.

Determine whether a logical X flip occurred on the data qubits.

Write a function `solve()` that returns True if a logical correction (X_L) is needed, or False if no correction is needed.