Chapter 35 Quiz: Quantum Error Correction

Multiple Choice

Q1. Which of the following is NOT a reason why quantum error correction is harder than classical error correction?

(a) The no-cloning theorem prevents copying quantum states (b) Measurement collapses quantum superpositions (c) Quantum errors are continuous rather than discrete (d) Classical bits can store more information than qubits

Q2. The three types of single-qubit Pauli errors are:

(a) $\hat{X}$ (bit-flip), $\hat{Z}$ (phase-flip), $\hat{Y}$ (combined bit-and-phase-flip) (b) $\hat{X}$ (phase-flip), $\hat{Z}$ (bit-flip), $\hat{Y}$ (amplitude damping) (c) $\hat{H}$ (Hadamard error), $\hat{S}$ (phase error), $\hat{T}$ (rotation error) (d) $\hat{X}$, $\hat{Y}$, $\hat{Z}$ (position, momentum, and energy errors)

Q3. The 3-qubit bit-flip code encodes $|0\rangle_L = |000\rangle$ and $|1\rangle_L = |111\rangle$. This encoding is:

(a) A violation of the no-cloning theorem (b) Three independent copies of the original qubit (c) An entangled state that encodes the logical information without copying it (d) Only valid for $|0\rangle$ and $|1\rangle$ states, not superpositions

Q4. Syndrome measurement in quantum error correction:

(a) Measures each physical qubit individually to check for errors (b) Measures parity operators that reveal error information without disturbing the encoded state (c) Requires destroying the quantum state and re-encoding it (d) Can only detect errors, not identify which qubit was affected

Q5. The 3-qubit bit-flip code can correct:

(a) Any single-qubit error (b) Only single bit-flip ($\hat{X}$) errors (c) Only single phase-flip ($\hat{Z}$) errors (d) Both bit-flip and phase-flip errors

Q6. The Hadamard gate $\hat{H}$ interchanges $\hat{X}$ and $\hat{Z}$: $\hat{H}\hat{X}\hat{H} = \hat{Z}$. This fact is used in:

(a) The encoding circuit for the Shor code (b) Converting the bit-flip code into the phase-flip code (c) Proving the no-cloning theorem (d) Implementing the threshold theorem

Q7. Shor's 9-qubit code uses:

(a) 9 copies of the original qubit (b) A phase-flip code applied to each qubit of a bit-flip code (c) A bit-flip code applied to each qubit of a phase-flip code, or equivalently the reverse (d) Three independent 3-qubit bit-flip codes

Q8. The Steane [[7,1,3]] code is built from:

(a) The classical [7,4,3] Hamming code (b) The classical [7,3,5] BCH code (c) A random classical code (d) No classical code — it was designed from scratch

Q9. In the stabilizer formalism, the code space is defined as:

(a) The subspace annihilated by all stabilizer generators (b) The simultaneous $+1$ eigenspace of all stabilizer generators (c) The simultaneous $-1$ eigenspace of all stabilizer generators (d) The full Hilbert space of $n$ qubits

Q10. The threshold theorem states that fault-tolerant quantum computation is possible if:

(a) The number of qubits exceeds a critical threshold (b) The error rate per physical gate is below a critical threshold $p_{\text{th}}$ (c) The temperature is below a critical threshold (d) The computation length is below a critical threshold

Q11. For the surface code, the threshold error rate is approximately:

(a) $10^{-6}$ (b) $10^{-4}$ (c) $10^{-2}$ (1%) (d) $10^{-1}$ (10%)

Q12. Current estimates for breaking RSA-2048 encryption using a fault-tolerant quantum computer require approximately:

(a) $\sim 100$ physical qubits (b) $\sim 10{,}000$ physical qubits (c) $\sim 1{,}000{,}000$ physical qubits (d) $\sim 10{,}000{,}000$ physical qubits

True or False

Q13. True or False: Quantum error correction works by making copies of the quantum state and using majority voting, just like classical error correction.

Q14. True or False: A quantum error-correcting code that can correct $\hat{X}$, $\hat{Y}$, and $\hat{Z}$ errors individually can also correct any continuous single-qubit error.

Q15. True or False: The Eastin-Knill theorem states that no quantum error-correcting code can implement a universal gate set using only transversal gates.

Q16. True or False: The physical error rates of current quantum computers (as of 2024) are already well below the surface code threshold, so fault-tolerant quantum computing is straightforward.

Short Answer

Q17. Explain in 2-3 sentences why syndrome measurement does not destroy the encoded quantum information, even though measurement generally collapses quantum states.

Q18. What is the difference between the code distance $d$ and the number of correctable errors?

Q19. Why does concatenation of quantum error-correcting codes lead to a doubly exponential decrease in the logical error rate?

Q20. State one fundamental advantage of the Steane code over the Shor code, and one advantage that both codes share.

Answer Key

Q1: (d) — Classical bits do NOT store more information than qubits (in fact, a qubit can encode more information in some contexts). The other three options are genuine obstacles to quantum error correction.

Q2: (a) — $\hat{X}$ flips $|0\rangle \leftrightarrow |1\rangle$ (bit-flip), $\hat{Z}$ adds a relative minus sign (phase-flip), and $\hat{Y} = i\hat{X}\hat{Z}$ is a combined error.

Q3: (c) — The state $\alpha|000\rangle + \beta|111\rangle$ is an entangled state, not three copies. The no-cloning theorem is not violated because $\alpha$ and $\beta$ appear only once.

Q4: (b) — Syndrome measurements are parity measurements ($\hat{Z}_i\hat{Z}_j$ or $\hat{X}_i\hat{X}_j$) that determine whether two qubits agree or disagree without measuring either qubit individually.

Q5: (b) — The 3-qubit bit-flip code only corrects $\hat{X}$ errors. It cannot detect or correct phase-flip ($\hat{Z}$) errors.

Q6: (b) — The Hadamard transformation converts $\hat{X}$ errors to $\hat{Z}$ errors and vice versa, so the phase-flip code is obtained by applying $\hat{H}$ to each qubit of the bit-flip code.

Q7: (c) — Shor's code applies the bit-flip code to each qubit of the phase-flip code (or equivalently, the phase-flip code to each qubit of the bit-flip code), yielding $3 \times 3 = 9$ qubits.

Q8: (a) — The Steane code is a CSS code built from the classical [7,4,3] Hamming code, using it for both $\hat{X}$ and $\hat{Z}$ error correction.

Q9: (b) — The code space is the simultaneous $+1$ eigenspace. Errors move the state out of this eigenspace, producing $-1$ eigenvalues (the syndrome).

Q10: (b) — The threshold theorem requires the physical error rate per gate to be below $p_{\text{th}}$.

Q11: (c) — The surface code has a threshold of approximately 1%, the highest among known codes and the reason it is the leading candidate for near-term fault tolerance.

Q12: (d) — Estimates range from $\sim 4$ million to $\sim 20$ million physical qubits, roughly $10^7$.

Q13: False — Quantum error correction does NOT copy the state (that would violate the no-cloning theorem). It encodes the information in entangled states and uses syndrome measurements to detect errors without measuring the encoded information.

Q14: True — This is the discretization of errors. Syndrome measurement projects any continuous error onto one of the discrete Pauli errors, which can then be corrected.

Q15: True — The Eastin-Knill theorem proves that transversal gates alone cannot be universal for any code that detects all single-qubit errors. This is why magic state distillation is needed.

Q16: False — Current error rates ($\sim 10^{-3}$ for two-qubit gates) are near the surface code threshold ($\sim 10^{-2}$), but not "well below" it. Achieving reliable fault tolerance requires further improvement in error rates and significant qubit overhead.

Q17: Syndrome measurements measure parity operators (e.g., $\hat{Z}_i\hat{Z}_j$) that determine whether two qubits are the same or different, without measuring either qubit's individual value. These parity operators commute with the logical operators, so they project the state within the code space (collapsing the error) without distinguishing between the logical $|0\rangle_L$ and $|1\rangle_L$ states.

Q18: The code distance $d$ is the minimum weight of any uncorrectable (undetectable) error — equivalently, the minimum number of physical qubit errors needed to cause a logical error. A code with distance $d$ can correct up to $\lfloor(d-1)/2\rfloor$ errors, because any pattern of fewer than $d/2$ errors produces a unique syndrome.

Q19: At each level of concatenation, the logical error rate squares (in appropriate units): $p_k \sim (cp)^{2^k}/c$. The exponent $2^k$ grows exponentially with the concatenation level $k$, so the logical error rate decreases doubly exponentially: exponentially fast in the number of concatenation levels.

Q20: The Steane code uses 7 physical qubits per logical qubit, compared to 9 for the Shor code (efficiency advantage). Additionally, the Steane code supports many transversal logical gates (e.g., transversal Hadamard), making fault-tolerant gate implementation simpler. Both codes share the property of correcting any single-qubit error (distance 3).