Decoherence

The loss of quantum coherence due to interaction with the environment, and the primary obstacle to quantum computing.

Decoherence is the process by which a quantum system loses its quantum properties (superposition, entanglement) through interaction with its environment. It’s the primary obstacle to building useful quantum computers.

The Problem

An isolated quantum system maintains coherent superpositions: $$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

But real systems interact with their environment. This interaction:

1. Leaks quantum information to the environment
2. Destroys superposition
3. Turns pure states into mixed states

Two Types of Decoherence

Relaxation (T1 Decay)

Energy loss to the environment: $$|1⟩\to |0⟩$$

Think: Excited atom spontaneously emitting a photon.

Characterized by T1 time.

Dephasing (T2 Decay)

Loss of phase information without energy exchange: $$\alpha |0⟩+\beta |1⟩\to \text{mixed state with no off-diagonal terms}$$

Think: Random fluctuations in the qubit frequency.

Characterized by T2 time.

Relationship: $T_2\le 2T_1$ (dephasing can’t be slower than relaxation).

Mathematical Picture

Initial state (pure): $$\rho =|\psi ⟩⟨\psi |=\left(\begin{array}{cc}|\alpha {|}^2& \alpha {\beta }^{\ast }\\ {\alpha }^{\ast }\beta & |\beta {|}^2\end{array}\right)$$

After decoherence (mixed): $$\rho \to \left(\begin{array}{cc}{p}_0& 0\\ 0& p_1\end{array}\right)$$

Off-diagonal terms (coherences) vanish.

Sources of Decoherence

Why It’s Devastating

Circuit Depth Limit

If gate time $\approx$ 50 ns and $T_2\approx$ 100 μs: $$\text{Max gates}\approx \frac{T_2}{\text{gate time}}=\frac{100,000}{50}=2000$$

After ~2000 gates, coherence is lost.

Error Accumulation

Each gate has some error. Without error correction: $$\text{Final fidelity}\approx (1-ϵ{)}^{N_{gates}}$$

Decoherence makes $ϵ$ larger over time.

Fighting Decoherence

The Race

Quantum computing is a race between:

• Gate operations (doing useful work)
• Decoherence (destroying quantum information)

Success requires: $\text{Gate time}\ll T_1,T_2$

See also: T1 Time, T2 Time, Quantum Error Correction, Mixed State