The three-qubit phase-flip code protects one logical qubit against single phase-flip…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch The stabilizer formalism

Problem

The three-qubit phase-flip code protects one logical qubit against single phase-flip (

Z

) errors using the stabilizer generators

S_1 = X_1X_2

and

S_2 = X_2X_3

on three physical qubits. An arbitrary logical input

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

is encoded by the circuit: a CNOT from qubit

1

(control) to qubit

2

(target), then a CNOT from qubit

1

(control) to qubit

3

(target), then a Hadamard gate on each of the three qubits, applied to the initial state

|\psi\rangle_1|0\rangle_2|0\rangle_3

. All gates are ideal and noiseless. ((a)) Check that

S_1

and

S_2

commute and are independent, and that neither

S_1

S_2

nor

S_1S_2

equals

-I

. State the dimension of the space they fix, using

\dim V_{\mathcal S} = 2^{n-r}

for

n = 3

physical qubits and

r

independent generators. ((b)) Represent the uncoded input by the operators

Z_1

and

X_1

(the input qubit's own logical Pauli operators, which do not fix

|\psi\rangle

), together with

Z_2

and

Z_3

(which fix the ancillas

|0\rangle_2|0\rangle_3

). Using the conjugation rules

\mathrm{CNOT}: X_{\rm c} \to X_{\rm c}X_{\rm t}

Z_{\rm t} \to Z_{\rm c}Z_{\rm t}

, with

X_{\rm t}

and

Z_{\rm c}

unchanged (c = control, t = target), and

H: X \leftrightarrow Z

on the qubit it acts on, update all four operators through the three gates of the encoding circuit in turn. Give the four Pauli strings that result. ((c)) Identify which two of the four resulting strings generate the same group as

S_1, S_2

, and which two are the logical operators

\bar Z

and

\bar X

of the encoded qubit. ((d)) By multiplying

\bar Z

\bar X

by elements of

\mathcal S = \{I, S_1, S_2, S_1S_2\}

, find the minimum-weight nontrivial element of the normaliser outside

\mathcal S

, and give the code's distance

d

. State in one sentence what kind of single-qubit error this reveals as undetectable. ((e)) Count the number of two-qubit gates used in the encoding circuit.

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