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What is Surface code?

A code for quantum error correction in which qubits are arranged on a two-dimensional grid and errors are detected using only operations between neighboring qubits. Its threshold is high, roughly 1%, and it suits superconducting qubit chips well. The rotated version with code distance d uses 2d² − 1 physical qubits per logical qubit.

Japanese name: 表面符号

Surface code appears in 2 practice questions across 1 topic, including Quantum Computing Hardware.

Field: Quantum Computing Hardware

Try it: questions about Surface code

A rotated surface code of code distance d uses, for one logical qubit, d² physical data qubits and d² − 1 physical measurement qubits that check for errors. For d = 15, how many physical qubits does one logical qubit need?

  1. A29
  2. B225
  3. C449
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Answer: C. 449

There are 15² = 225 data qubits and 225 − 1 = 224 measurement qubits, a total of 2d² − 1 = 449. A code of distance d can correct up to (d − 1)/2 errors, so d = 15 can handle up to 7 errors. Estimates suggest that with physical error rates of about 0.1%, reaching the logical error rates needed for practical computations requires d of roughly 15 to 25, which means several hundred to about 1,500 physical qubits per logical qubit. Because the surface code works with operations only between neighboring qubits on a two-dimensional grid, it suits superconducting qubit chips well.

Suppose the logical error rate p_L of a surface code can be approximated as p_L ∝ (p/p_th)^((d+1)/2), using the physical error rate p, the threshold p_th and the code distance d. With p = 0.1% and p_th = 1%, which is closest to the ratio of logical error rates when d is increased from 5 to 9 (the value at d = 5 divided by the value at d = 9)?

  1. AAbout 1.8
  2. BAbout 10
  3. CAbout 100
Show answer and explanation

Answer: C. About 100

The exponent (d+1)/2 is 3 for d = 5 and 5 for d = 9, so the ratio is (p_th/p) squared = 10² = 100, about 100. If the physical error rate is well below the threshold, each increase of the code distance by 2 reduces the logical error rate by a factor of p_th/p (10 here), so errors fall exponentially as more qubits are used. Conversely, if p is above the threshold, a larger d only adds more chances for errors and makes the logical error rate worse. The surface code threshold depends on the error model but is roughly 1%, and keeping the physical error rate well below the threshold is a prerequisite for error correction.

Surface code in the explanations

Questions that mention Surface code (2)

  1. A rotated surface code of code distance d uses, for one logical qubit, d² physical data qubits and d² − 1 physical measurement qubits that check for errors. For d = 15, how many physical qubits does one logical qubit need?
  2. Suppose the logical error rate p_L of a surface code can be approximated as p_L ∝ (p/p_th)^((d+1)/2), using the physical error rate p, the threshold p_th and the code distance d. With p = 0.1% and p_th = 1%, which is closest to the ratio of logical error rates when d is increased from 5 to 9 (the value at d = 5 divided by the value at d = 9)?

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Related terms

Qubit Decoherence Dilution refrigerator Ion trap Josephson junction NISQ Post-quantum cryptography Quantum entanglement Quantum error correction Quantum gate

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