3-Qubit Phase-Flip Code

Overview

The 3-qubit phase-flip code protects against single Z (phase-flip) errors by encoding one logical qubit into three physical qubits in the Hadamard basis. It is the exact dual of the bit-flip code: where the bit-flip code protects |0>/|1> against X errors, the phase-flip code protects |+>/|-> against Z errors.

Duality with Bit-Flip Code

The phase-flip code is obtained by conjugating the entire bit-flip code with Hadamard gates. This duality is one of the most elegant ideas in quantum error correction:

Property	Bit-Flip Code	Phase-Flip Code
Protects against	X errors (bit-flip)	Z errors (phase-flip)
Encoding basis	Computational (Z)	Hadamard (X)
\|0>_L	\|000>	\|+++>
\|1>_L	\|111>	\|--->
Syndrome checks	Z-basis parity	X-basis parity
Correction gate	X	Z
Transform	Identity	H before/after syndrome

Key insight: A Z error flips |+> to |-> (and vice versa), which is exactly a "bit-flip in the X basis." So we apply H to all qubits, use the standard parity check, then apply H again.

Encoding

Logical State	Physical State
\|0>_L	\|+++> = ((\|0>+\|1>)/sqrt(2))^3
\|1>_L	\|---> = ((\|0>-\|1>)/sqrt(2))^3

Encoding Circuit

CODE
     ┌───┐          ┌───┐
q_0: ┤ X ├──■────■──┤ H ├
     └───┘┌─┴─┐  │  ├───┤
q_1: ─────┤ X ├──┼──┤ H ├
          └───┘┌─┴─┐├───┤
q_2: ──────────┤ X ├┤ H ├
               └───┘└───┘

Prepare logical state on qubit 0 (X gate for |1>)
Spread via CX gates (creates GHZ state in Z basis)
Apply H to all qubits (rotates to X basis)

The Hadamard Trick

The syndrome extraction works by temporarily converting back to the computational basis:

CODE
    ┌───┐              ┌───┐
d0: ┤ H ├──■────■──────┤ H ├   (data qubits)
    ├───┤  │    │      ├───┤
d1: ┤ H ├──┼──■─┼──■───┤ H ├
    ├───┤  │  │ │  │   ├───┤
d2: ┤ H ├──┼──┼─┼──┼───┤ H ├
    └───┘┌─┴──┴─┐│  │  └───┘
s0: ─────┤parity├┼──┼────────   (syndrome ancillae)
         └──────┘│  │
s1: ─────────────┴──┴────────

This is the same parity check as the bit-flip code, sandwiched between H gates.

Syndrome Table

Syndrome	Error Location	Correction
00	No error	None
01	Qubit 2	Z on q2
10	Qubit 0	Z on q0
11	Qubit 1	Z on q1

Running the Circuit

PYTHON
from circuit import run_circuit, verify_phase_flip_code

# No error
result = run_circuit(error_qubit=None)
print(f"Success: {result['success_rate']:.2%}")

# Inject Z error on qubit 1
result = run_circuit(error_qubit=1)
print(f"Corrected: {result['success_rate']:.2%}")

# Full verification
v = verify_phase_flip_code()
for check in v["checks"]:
    print(f"[{'PASS' if check['passed'] else 'FAIL'}] {check['name']}")

Expected Output

Scenario	Success Rate
No error	~100%
Single Z error	~100% (corrected)
Two Z errors	~0% (uncorrectable)
Single X error	~0% (wrong code!)

Limitations

Only corrects Z errors: X (bit-flip) errors are not corrected
Single error only: Two or more Z errors cause failure
Not universal: Need combined code (Shor code) for both X and Z

Building to Shor Code

The Shor code combines bit-flip and phase-flip codes:

Inner code: 3-qubit bit-flip code (protects each block against X)
Outer code: 3-qubit phase-flip code (protects across blocks against Z)
Result: 9-qubit code protecting against any single-qubit error

3-Qubit Phase Flip Code