Quantum Lab Session 5 Simulation Q14
๐Ÿ“‰ Before we begin โ€” why qubits die
A qubit is an extraordinarily fragile object. The superposition and entanglement that make it powerful also make it exquisitely sensitive to the environment. Every stray photon, every vibration, every fluctuating electromagnetic field can destroy the quantum state โ€” a process called decoherence.

Decoherence happens in two ways. T1 relaxation: a qubit in |1โŸฉ spontaneously drops to |0โŸฉ by releasing a tiny amount of energy to the environment (like a drop of dye spreading in water). T2 dephasing: a qubit in superposition loses track of its phase โ€” the arrow on the Bloch sphere drifts to a random position on the equator, then the equator itself shrinks to the centre.

These timescales โ€” T1 and T2 โ€” are the fundamental limits on every quantum computation. Every gate must complete within a tiny fraction of T2. Every circuit must finish before the qubits decohere. This is why error correction exists and why quantum hardware is so difficult.
๐ŸŒ€ The numbers that matter
Best superconducting T2 in 2024: ~500 ยตs. Single-qubit gate time: ~20 ns. This means you can fit about 500,000/20 = 25,000 gates before decoherence ruins the computation. Sounds like a lot โ€” but a useful fault-tolerant quantum circuit might need millions of logical gates, each requiring hundreds of physical gates for error correction. The math is brutal.
๐Ÿ“‰ Decoherence ยท Session 5 ยท Q14

Decoherence Simulator

Watch qubits lose their quantum state in real time. Animate T1 relaxation and T2 dephasing on the Bloch sphere. See why temperature kills coherence โ€” and why 15mK is the magic number.

๐Ÿ“‰ T1 Relaxation
๐ŸŒ€ T2 Dephasing
๐ŸŒก๏ธ Temperature
๐Ÿ”Š Noise Channels
๐Ÿ† Badge
๐Ÿ“‰

T1 โ€” Relaxation

Qubit in |1โŸฉ spontaneously emits energy and drops to |0โŸฉ. Exponential decay with time constant T1. Typically 100โ€“500ยตs.

๐ŸŒ€

T2 โ€” Dephasing

Phase of superposition randomises. Bloch sphere arrow spirals inward on the equator. T2 โ‰ค 2ร—T1 always. The binding constraint for circuit depth.

๐ŸŒก๏ธ

Temperature kills qubits

Thermal photons at room temperature (300K) have enough energy to flip qubits constantly. 15mK suppresses thermal noise by a factor of 20,000.

๐Ÿ”Š

Noise channels

Bit-flip, phase-flip, depolarising, amplitude damping. Each models a specific physical mechanism. Error correction must handle all of them.

๐Ÿ“‰
Wizzy ยท Quantum Guide
T1 relaxation: a qubit in |1โŸฉ spontaneously flips to |0โŸฉ as it loses energy to the environment. Watch the Bloch sphere arrow decay from the south pole (|1โŸฉ) toward the north pole (|0โŸฉ). The probability of finding |1โŸฉ decays as e^(-t/T1). Press Start T1 to watch it happen.
๐ŸŒ€ Spontaneous emission โ€” the same process as light
T1 relaxation is the quantum equivalent of a radioactive atom decaying or a photon being spontaneously emitted. The qubit in |1โŸฉ is in an excited state. It interacts with electromagnetic vacuum fluctuations and drops to the ground state |0โŸฉ. No external trigger โ€” it just happens randomly, with average lifetime T1.

T1 Relaxation โ€” Energy Decay

Bloch Sphere (decaying)
P(|1โŸฉ) over time
Press Start to watch the qubit in |1โŸฉ spontaneously decay to |0โŸฉ. The curve follows P(|1โŸฉ) = e^(-t/T1).
๐ŸŒ€
Wizzy ยท Quantum Guide
T2 dephasing destroys superposition without flipping the qubit. The phase of (|0โŸฉ+|1โŸฉ)/โˆš2 randomly drifts โ€” the Bloch sphere arrow spirals inward on the equator. After time T2, all phase information is lost. The qubit becomes a classical mixture โ€” 50% |0โŸฉ or |1โŸฉ with no quantum coherence. Press Start T2.
๐ŸŒ€ T2 is always โ‰ค 2ร—T1 โ€” a fundamental limit
T2 can be at most 2ร—T1. This is because phase information can only survive as long as the qubit stays in |1โŸฉ โ€” T1 limits how long that can be. If T1 = T2/2, the qubit is in an "energy-limited" dephasing regime. In practice, environmental magnetic noise often makes T2 much shorter than this limit.

T2 Dephasing โ€” Phase Decay

Bloch Sphere (dephasing)
Coherence |โŸจฯˆ|+โŸฉ| over time
Press Start to watch superposition phase decay. The Bloch sphere arrow spirals inward โ€” quantum coherence lost.
๐ŸŒก๏ธ
Wizzy ยท Quantum Guide
Temperature is the enemy of coherence. At room temperature (300K), thermal photons have 20,000ร— more energy than at 15mK โ€” enough to constantly flip qubits. Drag the temperature slider from 15mK upward and watch the coherence time plummet from hundreds of microseconds to nanoseconds.
๐ŸŒ€ The Boltzmann factor โ€” why cold matters so much
The probability of a thermal photon flipping a qubit scales as e^(-โ„ฯ‰/kT). At 300K, kT โ‰ˆ 26 meV โ€” much larger than typical qubit energy gaps (~6 GHz ร— โ„ โ‰ˆ 0.025 meV). So thermal excitations dominate. At 15mK, kT โ‰ˆ 1.3 ยตeV โ€” far smaller than the qubit gap. Thermal flips become exponentially rare.

Temperature vs Coherence Time

15 mK
Coherence time (T2)
~500 ยตs
Superconducting qubit โ€” excellent quantum regime
At 15mK โ€” the base temperature of superconducting quantum processors โ€” coherence times reach hundreds of microseconds. Drag the slider upward and watch coherence time collapse dramatically.
๐Ÿ”Š
Wizzy ยท Quantum Guide
Quantum noise is modelled as specific noise channels โ€” quantum operations applied randomly with some probability. Adjust the sliders to see how each type of noise affects the qubit's Bloch sphere. Understanding noise channels is essential for designing error correction codes.
๐ŸŒ€ Why noise channels matter for error correction
Different error correction codes protect against different noise channels. The 3-qubit bit-flip code (Q15) corrects bit-flip errors (X channel). A different 3-qubit code corrects phase-flip errors (Z channel). The Shor 9-qubit code corrects both. Understanding which noise dominates your hardware tells you which error correction strategy to use.

Noise Channels โ€” Adjustable Probability

Bit-flip (X) channel
Randomly flips |0โŸฉโ†”|1โŸฉ with probability p
p = 0%
Phase-flip (Z) channel
Randomly applies Z gate โ€” flips phase of superposition
p = 0%
Depolarising channel
Applies X, Y, or Z with equal prob p/3 โ€” worst-case noise
p = 0%
Remaining coherence (Bloch vector length)
100%
Adjust the noise sliders to see how each channel shrinks the Bloch vector โ€” reducing quantum coherence. Error correction (Q15) learns to counteract these effects.
๐Ÿ“‰
Wizzy ยท Quantum Guide
๐ŸŽŠ You now understand the physical challenge at the heart of quantum computing! Decoherence is the reason quantum computers need to be colder than outer space, why circuits must be short, and why error correction is not optional โ€” it's the only path to large-scale quantum computation.
๐Ÿง  What you actually learned today
  • T1 relaxation: a qubit in |1โŸฉ spontaneously decays to |0โŸฉ with time constant T1 (~100โ€“500ยตs for superconducting qubits). Limits how long you can store a qubit.
  • T2 dephasing: superposition phase randomises with time constant T2 โ‰ค 2T1. This is the binding constraint on circuit depth โ€” all gates must complete within T2.
  • Temperature: at 300K, thermal energy destroys superposition in nanoseconds. At 15mK, thermal excitations become exponentially suppressed โ€” coherence can last hundreds of microseconds.
  • Noise channels: bit-flip (X), phase-flip (Z), and depolarising noise each model different physical error mechanisms. Error correction codes target specific channels.
  • The key ratio: (T2) / (gate time) = maximum circuit depth before decoherence. Best superconducting: ~500ยตs / 20ns = 25,000 gates. Error correction extends this effectively.
๐Ÿ“‰

Decoherence Expert Badge!

You understand why quantum computers need to be colder than outer space!

๐Ÿ“‰ WhizzStep Quantum Lab
This certifies that
Student Name
has mastered Quantum Decoherence โ€” T1, T2, Temperature & Noise Channels
Decoherence Expert
T1 & T2
Noise Channels
๐Ÿ“– Quantum Vocabulary
Decoherence KEY

The process by which a quantum system loses its quantum properties (superposition, entanglement) due to interaction with its environment. The central challenge in building quantum computers.

T1 time KEY

Energy relaxation time โ€” how long a qubit in |1โŸฉ stays in |1โŸฉ before decaying to |0โŸฉ. Follows P(|1โŸฉ) = e^(-t/T1). Superconducting: 100โ€“500ยตs.

T2 time KEY

Phase coherence time โ€” how long superposition phase survives before randomising. Always T2 โ‰ค 2T1. The binding circuit depth limit.

Noise channel NEW

A mathematical model of an error โ€” a quantum operation applied randomly with some probability. Types: bit-flip (X), phase-flip (Z), depolarising, amplitude damping.

NISQ

Noisy Intermediate-Scale Quantum โ€” today's quantum computers. 50โ€“4000 physical qubits, significant error rates, no full error correction. The era we're currently in.

Fault-tolerant QC

Quantum computation with full error correction โ€” logical qubits encoded in many physical qubits, capable of arbitrary-length computation. Still years away from realisation.

Key Concepts from Q14

T1 vs T2

๐Ÿ“‰ Two ways to die

T1 kills the qubit's energy state. T2 kills the qubit's phase. T2 is always the tighter constraint โ€” it's the wall every quantum circuit runs into first.

Temperature

๐ŸŒก๏ธ 15mK matters

The Boltzmann factor e^(-โ„ฯ‰/kT) controls thermal excitation rate. Dropping from 300K to 15mK suppresses thermal noise by a factor of ~10 trillion โ€” enabling coherence times a million times longer.

Circuit depth

๐Ÿ“ T2 / gate time

The ratio (T2) / (gate time) is the maximum number of gates before decoherence. Best 2024: ~500ยตs / 20ns โ‰ˆ 25,000. Error correction effectively multiplies this by 100โ€“1000ร—.

Error correction

๐Ÿ›ก๏ธ The solution (Q15)

Error correction encodes logical qubits in many physical qubits, detecting and fixing errors faster than they accumulate. It's the only known path to fault-tolerant quantum computing.