The phase-flip code and Shor’s nine-qubit code

Seen in the H basis, a phase flip turns into a bit flip. Use this to correct phase flips, and meet Shor’s code, the first quantum code that corrects any single-qubit error by nesting two codes.

In the H basis a phase flip is a bit flip

The H gate swaps the Z basis and the X basis: H·Z·H = X. So a phase flip behaves like a bit flip on qubits with H applied. To catch phase flips it is enough to encode the data as |+++⟩ and |−−−⟩ instead of |000⟩ and |111⟩.

Before measuring the syndrome, H is applied to the three qubits again; the phase flip turns back into a bit flip, and the same syndrome circuit as in the repetition code finds where it is.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Operator, Statevector

hzh = QuantumCircuit(1)
hzh.h(0)
hzh.z(0)
hzh.h(0)
x = QuantumCircuit(1)
x.x(0)
print(Operator(hzh).equiv(Operator(x)))   # True: H·Z·H = X

for wrong in [None, 0, 1, 2]:
    qc = QuantumCircuit(5)
    qc.cx(0, 1)
    qc.cx(0, 2)
    qc.h([0, 1, 2])              # |000⟩ → |+++⟩
    if wrong is not None:
        qc.z(wrong)              # a single phase flip
    qc.h([0, 1, 2])              # go back: the phase flip became a bit flip
    qc.cx(0, 3)
    qc.cx(1, 3)
    qc.cx(1, 4)
    qc.cx(2, 4)
    probs = Statevector(qc).probabilities_dict()
    print(wrong, max(probs, key=probs.get)[:2])
# True
# None 00
# 0 01
# 1 11
# 2 10

Shor’s code

In 1995 Peter Shor nested the two codes: the logical qubit is first spread over three qubits with the phase-flip code, then each of those three is spread over three qubits with the bit-flip code. The result is one logical qubit stored in nine physical qubits, which corrects a single X error, Z error or a combination of the two on any qubit.

This was the first proof that quantum error correction is possible. Until then many physicists believed that the decay of continuous-valued amplitudes could not be corrected. The fact that errors can be broken down into X and Z components removed this obstacle.

More efficient codes

After Shor’s code more efficient codes were found: Steane’s 7-qubit code and the 5-qubit code, the smallest code that can correct a single qubit error. Today the code with the highest hopes is the surface code, which only needs operations between neighbouring qubits on a two-dimensional grid. It is the subject of the next lesson.

Try it yourself

The H, Z, H circuit is the core of the single-qubit phase-flip code: the Z error acts like an X between the H gates and makes the result 1. Put an X in place of the Z: this time the X becomes invisible in the H basis and the result stays 0. X and Z errors are each other’s mirror image in the H basis.

Open the circuit in your browser

The circuit playground is in Turkish. Button guide: “1000 kez ölç” = measure 1000 times, “Durum vektörünü göster” = show the state vector, “Bu devrenin Qiskit kodu” = this circuit’s Qiskit code, “Geri al” = undo, “Temizle” = clear. Click a gate in the circuit to remove it.

Key points

H·Z·H = X: in the H basis a phase flip turns into a bit flip.
The phase-flip code encodes data as |+++⟩ and |−−−⟩ and uses the same syndrome circuit.
Shor’s 9-qubit code (1995) is the first quantum code that corrects any single-qubit error.
Later the 7- and 5-qubit codes and the surface code were found.

Test yourself

In how many physical qubits does Shor’s code store one logical qubit?

Show the answer

9. Each of the three qubits of the phase-flip code is spread over three with the bit-flip code: 3 × 3 = 9.

← The repetition code: finding an error without looking at the data · The surface code and the threshold: the road to fault-tolerant quantum computers →