Fault-Tolerant Quantum Computing

Quantum computing with error correction, enabling arbitrarily long and accurate computations.

Fault-Tolerant Quantum Computing (FTQC) is the ultimate goal: quantum computers that can run arbitrarily long computations with arbitrarily low error rates.

NISQ vs FTQC

Requirements for FTQC

1. Below-Threshold Error Rates

Physical error rate $p<p_{th}$ (threshold). For surface code: $p_{th}\approx 1\%$

2. Enough Physical Qubits

Ratio of physical to logical qubits:

• Near threshold: ~1000:1
• Below threshold: Can be ~100:1

3. Fast Classical Processing

• Decode error syndromes in real-time
• Faster than errors accumulate
• Challenging at scale

4. Mid-Circuit Measurement

• Measure syndromes during computation
• Feed-forward based on results

5. Scalable Architecture

• Millions of physical qubits eventually
• Manageable control systems

What FTQC Enables

The FTQC Stack

┌────────────────────────────┐
│    Quantum Applications    │
├────────────────────────────┤
│    Logical Operations      │
├────────────────────────────┤
│    Error Correction        │
│    (Surface code, etc.)    │
├────────────────────────────┤
│    Physical Qubits         │
│    (Hardware platform)     │
└────────────────────────────┘

Resource Estimates

For useful computations:

Timeline

Timelines are highly uncertain.

Current Progress

• Below-threshold operations: Demonstrated
• Logical qubits: Created, but not yet better than physical
• Logical operations: Demonstrated at small scale
• Full FTQC: Not yet achieved

The Challenge Ahead

Physical error rate:     ~0.1%    ✓ Achieved
                              ↓
Logical error rate:      ~10⁻⁶   ? In progress
                              ↓
Required for utility:    ~10⁻¹⁰  ✗ Far to go

Scaling from ~100 qubits to millions while maintaining quality.

See also: NISQ, Quantum Error Correction, Surface Code, Logical Qubit, Fault Tolerance