# Day 670: Month 24 Review - Error Channels and Quantum Error Correction ## Week 96: Semester 1B Review | Month 24: Quantum Channels & Error Introduction --- ## Review Scope **Month 24: Quantum Channels & Error Introduction (Days 645-672)** - Week 93: Channel Representations (Kraus, Choi, Stinespring) - Week 94: Quantum Error Types (Bit-flip, Phase-flip, Depolarizing, Amplitude Damping) - Week 95: Error Detection/Correction Introduction --- ## Core Concepts: Error Channels ### 1. Error Channel Classification | Channel | Kraus Operators | Effect | |---------|-----------------|--------| | Bit-flip | $K_0 = \sqrt{1-p}I$, $K_1 = \sqrt{p}X$ | Flips $\|0\rangle \leftrightarrow \|1\rangle$ | | Phase-flip | $K_0 = \sqrt{1-p}I$, $K_1 = \sqrt{p}Z$ | Applies relative phase | | Bit-phase flip | $K_0 = \sqrt{1-p}I$, $K_1 = \sqrt{p}Y$ | Both X and Z | | Depolarizing | Symmetric Pauli | Contracts Bloch sphere | | Amplitude damping | Energy decay | Non-unital, T1 process | ### 2. Pauli Channel General Form $$\mathcal{E}(\rho) = (1-p_x-p_y-p_z)\rho + p_x X\rho X + p_y Y\rho Y + p_z Z\rho Z$$ **Bloch sphere effect:** $$\vec{r} \to \begin{pmatrix}1-2(p_y+p_z) & 0 & 0\\0 & 1-2(p_x+p_z) & 0\\0 & 0 & 1-2(p_x+p_y)\end{pmatrix}\vec{r}$$ ### 3. Depolarizing Channel $$\mathcal{E}_{dep}(\rho) = (1-p)\rho + p\frac{I}{2} = \left(1-\frac{3p}{4}\right)\rho + \frac{p}{4}(X\rho X + Y\rho Y + Z\rho Z)$$ **Properties:** - Isotropic contraction: $\vec{r} \to (1-p)\vec{r}$ - Unital: $\mathcal{E}(I) = I$ - "Worst-case" noise model ### 4. Amplitude Damping $$K_0 = \begin{pmatrix}1 & 0\\0 & \sqrt{1-\gamma}\end{pmatrix}, \quad K_1 = \begin{pmatrix}0 & \sqrt{\gamma}\\0 & 0\end{pmatrix}$$ **Properties:** - Non-unital: $\mathcal{E}(I) \neq I$ - Fixed point: $|0\rangle$ - Models T1 relaxation: $\gamma = 1 - e^{-t/T_1}$ ### 5. T1/T2 Parameters $$\boxed{T_2 \leq 2T_1}$$ | Process | Rate | Channel | |---------|------|---------| | T1 (relaxation) | $\gamma = 1/T_1$ | Amplitude damping | | T2 (dephasing) | $\gamma_\phi$ | Phase damping | | Pure dephasing | $1/T_\phi$ | $1/T_2 = 1/(2T_1) + 1/T_\phi$ | --- ## Core Concepts: Error Correction ### 6. Classical vs Quantum EC | Challenge | Classical | Quantum Solution | |-----------|-----------|------------------| | Copy data | Repeat bits | Entanglement encoding | | Check errors | Measure directly | Syndrome measurement | | Error types | Bit flip only | X, Y, Z, continuous | | Discretization | N/A | Pauli projection | ### 7. Knill-Laflamme Conditions A code with projector $P$ corrects errors $\{E_a\}$ iff: $$\boxed{PE_a^\dagger E_b P = \alpha_{ab} P}$$ **Equivalent:** $\langle i_L|E_a^\dagger E_b|j_L\rangle = \alpha_{ab}\delta_{ij}$ **Interpretation:** - Different errors → distinguishable syndromes - Same error effect on all logical states ### 8. Three-Qubit Bit-Flip Code **Encoding:** $$|0_L\rangle = |000\rangle, \quad |1_L\rangle = |111\rangle$$ **Syndromes:** $Z_1Z_2$, $Z_2Z_3$ | Error | Syndrome | Correction | |-------|----------|------------| | None | (0,0) | None | | $X_1$ | (1,0) | $X_1$ | | $X_2$ | (1,1) | $X_2$ | | $X_3$ | (0,1) | $X_3$ | **Limitation:** Cannot correct Z errors ### 9. Three-Qubit Phase-Flip Code **Encoding:** $$|0_L\rangle = |+++\rangle, \quad |1_L\rangle = |---\rangle$$ **Syndromes:** $X_1X_2$, $X_2X_3$ **Duality:** Hadamard transform of bit-flip code **Limitation:** Cannot correct X errors ### 10. Shor [[9,1,3]] Code **Concatenation:** Phase-flip code over bit-flip code $$|0_L\rangle = \frac{1}{2\sqrt{2}}(|000\rangle + |111\rangle)^{\otimes 3}$$ $$|1_L\rangle = \frac{1}{2\sqrt{2}}(|000\rangle - |111\rangle)^{\otimes 3}$$ **Properties:** - Corrects ANY single-qubit error - 8 syndrome bits (6 bit-flip + 2 phase-flip) - First complete QEC code (Shor, 1995) ### 11. Stabilizer Formalism Preview **Stabilizer group:** Abelian subgroup $\mathcal{S} \subset \mathcal{P}_n$ **Code space:** Simultaneous +1 eigenspace of all stabilizers **Parameters:** $[[n, k, d]]$ where $k = n - r$ (r = number of generators) **Syndrome:** Commutation pattern with stabilizers --- ## Integration: The QEC Pipeline ``` Physical Noise (T1, T2, gate errors) │ ▼ Error Channels (Kraus representation) │ ▼ Error Discretization (Pauli basis) │ ▼ Syndrome Measurement │ ▼ Correction Operation │ ▼ Recovered Quantum State ``` --- ## Practice Problems ### Problem 1: Error Identification A qubit prepared in $|+\rangle$ is measured after some noise and found to have density matrix: $$\rho = \begin{pmatrix}0.5 & 0.4\\0.4 & 0.5\end{pmatrix}$$ What type of noise occurred? Estimate the parameter. **Solution:** For $|+\rangle$: $\rho_0 = \begin{pmatrix}0.5 & 0.5\\0.5 & 0.5\end{pmatrix}$ Coherence decayed: $0.5 \to 0.4$ (factor of 0.8) This is consistent with: - Dephasing with $(1-2p) = 0.8 \Rightarrow p = 0.1$ - Or amplitude damping with $\sqrt{1-\gamma} = 0.8 \Rightarrow \gamma = 0.36$ Population unchanged → pure dephasing (p = 0.1) ### Problem 2: Syndrome Calculation For the bit-flip code, calculate the syndrome when error $X_1X_2$ occurs. **Solution:** Syndromes: $Z_1Z_2$, $Z_2Z_3$ For $X_1X_2$: - $Z_1Z_2$ anti-commutes with $X_1$ and $X_2$ → commutes overall → syndrome 0 - $Z_2Z_3$ anti-commutes with $X_2$ only → anti-commutes → syndrome 1 Syndrome: (0, 1) → Interpreted as $X_3$ error! **Conclusion:** Two-qubit error misdiagnosed as single-qubit → correction fails. ### Problem 3: Channel Composition A qubit experiences amplitude damping ($\gamma = 0.1$) followed by dephasing ($p = 0.05$). Find the effective coherence decay. **Solution:** After amplitude damping: coherence $\to \sqrt{1-\gamma} = \sqrt{0.9} \approx 0.949$ After dephasing: coherence $\to (1-2p) = 0.9$ Total: $0.949 \times 0.9 \approx 0.854$ Compare to pure T2 process with same final coherence. ### Problem 4: Code Distance The Shor code has $d = 3$. How many errors can it: a) Detect? b) Correct? **Solution:** a) Can detect up to $d-1 = 2$ errors b) Can correct up to $\lfloor(d-1)/2\rfloor = 1$ error --- ## Computational Lab ```python """Day 670: Month 24 Review - Error Channels and QEC""" import numpy as np # Pauli matrices I = np.eye(2, dtype=complex) X = np.array([[0, 1], [1, 0]], dtype=complex) Y = np.array([[0, -1j], [1j, 0]], dtype=complex) Z = np.array([[1, 0], [0, -1]], dtype=complex) def tensor(*matrices): result = matrices[0] for m in matrices[1:]: result = np.kron(result, m) return result print("Month 24 Review: Error Channels and QEC") print("=" * 60) # ============================================ # Part 1: Error Channel Comparison # ============================================ print("\nPART 1: Error Channel Comparison") print("-" * 40) def apply_channel(rho, kraus_ops): result = np.zeros_like(rho) for K in kraus_ops: result += K @ rho @ K.conj().T return result # Initial state |+⟩ rho_plus = 0.5 * np.array([[1, 1], [1, 1]], dtype=complex) # Error channels p = 0.1 gamma = 0.1 channels = { "Bit-flip": [np.sqrt(1-p) * I, np.sqrt(p) * X], "Phase-flip": [np.sqrt(1-p) * I, np.sqrt(p) * Z], "Depolarizing": [np.sqrt(1-3*p/4) * I, np.sqrt(p/4) * X, np.sqrt(p/4) * Y, np.sqrt(p/4) * Z], "Amp. damping": [np.array([[1, 0], [0, np.sqrt(1-gamma)]]), np.array([[0, np.sqrt(gamma)], [0, 0]])] } print(f"Initial state |+⟩, error parameter = {p}") print(f"\n{'Channel':<15} {'Purity':>10} {'Coherence':>12} {'P(|1⟩)':>10}") print("-" * 50) for name, K in channels.items(): rho_out = apply_channel(rho_plus, K) purity = np.real(np.trace(rho_out @ rho_out)) coherence = np.abs(rho_out[0, 1]) p1 = np.real(rho_out[1, 1]) print(f"{name:<15} {purity:>10.4f} {coherence:>12.4f} {p1:>10.4f}") # ============================================ # Part 2: Error Correction Simulation # ============================================ print("\n" + "=" * 60) print("PART 2: Error Correction Simulation") print("-" * 40) # Basis states ket_0 = np.array([[1], [0]], dtype=complex) ket_1 = np.array([[0], [1]], dtype=complex) # Bit-flip code logical states ket_0L = tensor(ket_0, ket_0, ket_0) ket_1L = tensor(ket_1, ket_1, ket_1) # Syndrome operators Z1Z2 = tensor(Z, Z, I) Z2Z3 = tensor(I, Z, Z) def get_syndrome(state, syndromes): """Get syndrome bits.""" result = [] for S in syndromes: ev = np.real((state.conj().T @ S @ state)[0, 0]) result.append(0 if ev > 0 else 1) return tuple(result) def correct_bitflip(state, syndrome): """Apply correction for bit-flip code.""" corrections = { (0, 0): tensor(I, I, I), (1, 0): tensor(X, I, I), (1, 1): tensor(I, X, I), (0, 1): tensor(I, I, X) } return corrections[syndrome] @ state # Encoded |+_L⟩ psi_L = (ket_0L + ket_1L) / np.sqrt(2) # Test single-qubit errors errors = { "None": tensor(I, I, I), "X1": tensor(X, I, I), "X2": tensor(I, X, I), "X3": tensor(I, I, X), "Z1": tensor(Z, I, I), "X1X2": tensor(X, X, I) } print("Bit-flip code error correction:") print(f"{'Error':<8} {'Syndrome':<12} {'Corrected':>12} {'Fidelity':>10}") print("-" * 45) for name, E in errors.items(): psi_err = E @ psi_L syn = get_syndrome(psi_err, [Z1Z2, Z2Z3]) psi_corr = correct_bitflip(psi_err, syn) fidelity = np.abs((psi_L.conj().T @ psi_corr)[0, 0])**2 success = "Yes" if fidelity > 0.99 else "No" print(f"{name:<8} {str(syn):<12} {success:>12} {fidelity:>10.4f}") # ============================================ # Part 3: Knill-Laflamme Verification # ============================================ print("\n" + "=" * 60) print("PART 3: Knill-Laflamme Verification") print("-" * 40) errors_to_check = [tensor(I, I, I), tensor(X, I, I), tensor(I, X, I), tensor(I, I, X)] error_names = ["I", "X1", "X2", "X3"] print("Matrix ⟨0L|Ea†Eb|0L⟩:") print(" ", end="") for name in error_names: print(f"{name:>8}", end="") print() for i, (Ea, name_a) in enumerate(zip(errors_to_check, error_names)): print(f"{name_a:>4}", end="") for j, (Eb, name_b) in enumerate(zip(errors_to_check, error_names)): val = (ket_0L.conj().T @ Ea.conj().T @ Eb @ ket_0L)[0, 0] print(f"{np.real(val):>8.2f}", end="") print() print("\nMatrix ⟨0L|Ea†Eb|1L⟩ (should be all zeros):") print(" ", end="") for name in error_names: print(f"{name:>8}", end="") print() for i, (Ea, name_a) in enumerate(zip(errors_to_check, error_names)): print(f"{name_a:>4}", end="") for j, (Eb, name_b) in enumerate(zip(errors_to_check, error_names)): val = (ket_0L.conj().T @ Ea.conj().T @ Eb @ ket_1L)[0, 0] print(f"{np.abs(val):>8.2f}", end="") print() print("\nKnill-Laflamme satisfied: errors produce orthogonal subspaces ✓") # ============================================ # Part 4: Circuit Error Accumulation # ============================================ print("\n" + "=" * 60) print("PART 4: Circuit Error Accumulation") print("-" * 40) def circuit_fidelity(n_gates, gate_error): """Estimate circuit fidelity with independent gate errors.""" return (1 - gate_error) ** n_gates gate_errors = [0.001, 0.01, 0.1] depths = [10, 100, 1000] print(f"{'Depth':<10}", end="") for err in gate_errors: print(f"ε={err:<8}", end="") print() print("-" * 40) for d in depths: print(f"{d:<10}", end="") for err in gate_errors: fid = circuit_fidelity(d, err) print(f"{fid:<10.4f}", end="") print() print("\nConclusion: Need error correction for deep circuits!") # ============================================ # Part 5: Threshold Estimate # ============================================ print("\n" + "=" * 60) print("PART 5: Error Threshold Concept") print("-" * 40) print(""" Error Correction Threshold: - If physical error rate ε < ε_th, arbitrary accuracy achievable - Typical threshold: ε_th ~ 1% for surface code - Overhead: O(log(1/ε_target)) code distance Current technology: - Superconducting qubits: ε ~ 0.1-1% - Trapped ions: ε ~ 0.01-0.1% - At or near threshold → Error correction essential! """) print("=" * 60) print("Review Complete!") ``` --- ## Summary ### Error Channels | Channel | Corrects | Effect | |---------|----------|--------| | Bit-flip | With bit-flip code | X errors | | Phase-flip | With phase-flip code | Z errors | | Depolarizing | With Shor/CSS codes | All Pauli | | Amplitude damping | Requires special codes | T1 decay | ### Error Correction Codes | Code | Parameters | Corrects | |------|------------|----------| | Bit-flip | [[3,1,1]] for X | Single X | | Phase-flip | [[3,1,1]] for Z | Single Z | | Shor | [[9,1,3]] | Any single-qubit | ### Key Principles 1. **Discretization:** Continuous errors → Pauli errors via measurement 2. **Syndrome:** Extract error info without disturbing data 3. **Concatenation:** Combine codes for more error types 4. **Threshold:** Below critical error rate → scalable QC possible --- ## Preview: Day 671 Tomorrow: **Comprehensive Problems** - Integration across all Semester 1B topics!

Day 670: Month 24 Review - Error Channels and Quantum Error Correction

Today's Schedule (7 hours)

Day 670: Month 24 Review - Error Channels and Quantum Error Correction

Today's Schedule (7 hours)