Self-Assessment Quiz: Surface Codes and Fault-Tolerant Computation

Twenty questions on the surface code, thresholds, decoding, magic states, and the path from noisy qubits to reliable computation. Aim for 16+.

Question 1

The surface code arranges qubits:

A) On a 2D lattice with data and measure qubits alternating B) In a 1D chain C) All-to-all D) In a 3D crystal

Question 2

Its stabilizers are:

A) Weight-4 $X$-type and $Z$-type plaquette/vertex operators B) Weight-2 only C) Global operators D) Non-Pauli

Question 3

A distance-$d$ surface code uses approximately:

A) $d$ physical qubits B) $2d^2$ C) $d^3$ D) $2^d$

Question 4

The surface code's threshold is approximately:

A) $10^{-4}$ B) $10^{-3}$ C) $10^{-2}$ D) $10^{-1}$

Question 5

Below threshold, logical error rate scales as:

A) $\sim(p/p_{th})^{(d+1)/2}$ — exponential suppression in $d$ B) Linearly in $d$ C) Constant D) Increasing with $d$

Question 6

Increasing $d$ from 11 to 13 at $p = 10^{-3}$ changes logical error by roughly:

A) No change B) One order of magnitude C) Doubling D) Two orders

Question 7

A logical error occurs when:

A) A chain of physical errors spans the lattice, forming a logical operator B) Any single qubit errs C) A syndrome is measured D) An ancilla fails

Question 8

The standard surface-code decoder is:

A) Minimum-weight perfect matching B) Gaussian elimination C) Brute-force search D) Belief propagation only

Question 9

Syndrome extraction must be repeated because:

A) Measurements themselves are faulty, so a time-series is needed B) It is faster C) The code degrades D) It reduces qubit count

Question 10

Repeating $d$ rounds makes the decoding problem:

A) 3D — two spatial dimensions plus time B) 1D C) Unnecessary D) Classical only

Question 11

Logical operations on surface codes are typically done via:

A) Lattice surgery — merging and splitting patches B) Physical SWAPs C) Direct transversal application of all gates D) Measurement only

Question 12

Which gates are relatively easy on the surface code?

A) Clifford gates B) $T$ gates C) Toffoli D) All equally

Question 13

$T$ gates require:

A) Magic-state distillation and gate teleportation B) Nothing special C) More rounds of syndrome extraction only D) A different code

Question 14

In a large fault-tolerant computation, magic-state factories typically occupy:

A) A negligible fraction of the chip B) A majority of the physical qubits C) Exactly half D) None

Question 15

"Below threshold" experimentally demonstrated means:

A) Increasing code distance reduced the logical error rate B) A single qubit worked C) Gates were fast D) The decoder was fast

Question 16

True or false: The surface code can correct any error pattern.

Question 17

True or false: Real-time decoding is needed for fault-tolerant computation.

Question 18

True or false: The surface code has a good rate.

Question 19

Short answer. Explain why the decoding problem is three-dimensional and what that costs.

Question 20

Short answer. Why is the surface code the dominant choice despite its poor rate?

Answer Key

Q Ans Note
1 A A checkerboard of data qubits and measure qubits.
2 A Weight-4 checks, local on the lattice — the property that makes it hardware-friendly.
3 B $d^2$ data plus $d^2-1$ measure qubits.
4 C ~1%, among the highest known, which is why it is practical.
5 A Each two units of distance buys roughly another order of magnitude of suppression.
6 B The exponent increases by 1, multiplying suppression by $p/p_{th} = 0.1$.
7 A Errors must form a connected chain spanning the lattice — which is why distance is a geometric length.
8 A MWPM, near-linear time and highly accurate for the surface code's matching-graph structure.
9 A A single round's syndrome may itself be wrong; repetition distinguishes measurement errors from data errors.
10 A Space × time, decoded jointly.
11 A Lattice surgery: merge and split patches to realize logical CNOTs and measurements.
12 A Cliffords via lattice surgery and transversal operations.
13 A Eastin–Knill forbids transversal universality, so $T$ comes from distilled magic states.
14 B Typically over half, sometimes far more, in $T$-heavy algorithms.
15 A Google's Willow result: $d=5$ outperformed $d=3$, the first clear demonstration of the scaling working.
16 False It corrects errors up to weight $\lfloor(d-1)/2\rfloor$ and fails on spanning chains. It is highly degenerate, so many higher-weight patterns are also corrected, but not all.
17 True Syndromes must be decoded fast enough to keep up with extraction rounds; a backlog grows exponentially. This forces dedicated classical hardware near the cryostat, with latency budgets in microseconds.
18 False Rate is $1/d^2$, among the worst of any useful code family — the price paid for locality, threshold, and decodability.
19 — Each syndrome extraction round yields a 2D snapshot, but measurement errors mean a single round cannot be trusted. Repeating over $d$ rounds gives a 2D × time volume, and the decoder must find the most likely error chain through that 3D structure — chains can move in space (data errors) or in time (measurement errors). Cost: $d$ times more measurement rounds, a decoding problem of size $O(d^3)$, and a hard real-time latency requirement, since decoding must keep pace with extraction.
20 — Because rate is only one criterion of four, and the surface code wins the other three. Its stabilizers are weight-4 and geometrically local, matching 2D superconducting and neutral-atom hardware. Its ~1% threshold is the highest among practical codes, and current hardware sits just below it — a code with better rate but a 0.1% threshold would be unusable. Its matching decoder runs in near-linear time with real-time-capable implementations. And lattice surgery gives a complete, well-understood set of logical operations. A code must be implementable, correctable, decodable, and computable-with before rate becomes relevant.