Quantum Bit (Qubit) Coherence Time Calculator
Calculate and compare qubit coherence times across different quantum computing technologies. Select from superconducting (transmon), trapped ion, photonic, spin qubit, topological (Majorana), and NV center qubits. Enter custom T₁ and T₂ values, gate time, gate fidelity, qubit count, and circuit depth. Understand the maximum number of operations within the coherence window, algorithm success probability, and requirements for fault-tolerant quantum computing.
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Time to execute one quantum gate operation
Single-gate fidelity percentage
Total qubits in the processor
Number of sequential gate operations in the algorithm
Survival Probability = exp(-Algorithm_Time / T₂)
Algorithm Success ≈ (1 - Error_Rate)^(Gates × Qubits)
1/T₂ = 1/(2T₁) + 1/Tφ
where:
T₁ = Energy relaxation time
T₂ = Dephasing time
Tφ = Pure dephasing time
Fault-tolerant threshold: error rate < 10⁻³ per gate
Surface code overhead: ~1,000 physical qubits per logical qubit
Max ops: 50µs / 50ns = 1,000 gates per qubit
Total alg time: 100 × 50ns = 5,000 ns = 5 µs
Survival: exp(-5/50) = 90.5% coherence probability
Error rate: 0.1% per gate
Total gates: 100 × 100 = 10,000
Alg success: (0.999)^10000 = 0.0045% (need error correction)
Physical qubits for FT: ~1,000
Pure dephasing rate: 1/(50×10⁻⁶) - 1/(2×100×10⁻⁶) = 15,000 Hz
How is qubit coherence time calculated and why does it matter?
Qubit coherence time is characterized by two parameters: T₁ (energy relaxation time) and T₂ (dephasing time). T₁ measures how long a qubit stays in the |1⟩ state before decaying to |0⟩ due to energy loss to the environment. T₂ measures how long the quantum phase information is preserved before decoherence. The relationship is: 1/T₂ = 1/(2T₁) + 1/Tφ, where Tφ is the pure dephasing time. Coherence time determines the maximum number of quantum operations possible: N_ops = T₂ / t_gate. For a 100 µs T₂ with 100 ns gate time: N = 1,000 operations. For fault-tolerant quantum computing, error rates must be below the threshold of ~10⁻³ per gate.
What is the difference between T₁ and T₂ coherence times?
T₁ (longitudinal relaxation) is the energy decay time from the excited |1⟩ state to the ground |0⟩ state. It is limited by interactions with the thermal environment (phonons, thermal photons). T₂ (transverse relaxation or dephasing time) is the time over which the quantum phase relationship between |0⟩ and |1⟩ is maintained. T₂ is always ≤ 2T₁ (usually much shorter). Superconducting qubits: T₁~100 µs, T₂~50 µs. Trapped ions: T₁~500 s, T₂~200 ms. The ratio T₂/T₁ indicates the quality of phase protection. A high T₂/T₁ ratio means good isolation from magnetic noise.
What limits qubit coherence in current quantum processors?
The main factors limiting coherence are: (1) Material defects - two-level systems (TLS) in amorphous dielectrics couple to qubits, causing energy loss; current research focuses on cleaner materials and surfaces. (2) Magnetic flux noise - unpaired electron spins on surfaces create 1/f magnetic noise; mitigated by spin echo techniques. (3) Quasiparticle poisoning - broken Cooper pairs in superconductors cause energy fluctuations; improved shielding and lower temperatures help. (4) Control electronics - noise in microwave pulses and DC biases causes gate errors; filtered lines and cryogenic electronics reduce this. State-of-the-art: Google's Willow processor achieves ~100 µs T₁ with 99.99% gate fidelity.
How many quantum operations can be performed within the coherence time?
The number of useful quantum operations is: N_gates = T₂ / t_gate. A single-qubit gate takes 10-100 ns for superconducting qubits, 1-10 µs for trapped ions. With T₂ = 50 µs and t_gate = 50 ns: N_gates = 1,000. For error correction: surface code requires ~1,000 physical qubits per logical qubit with ~1,000 gates per error correction cycle. This means T₂ must support ~10⁶ gates for practical quantum computing. This requires T₂ > 10 ms with 10 ns gates, or error rates below 10⁻⁶. Current best: ~1,000 operations before error correction overhead.
🔗 Related Calculators
📐 Formula
Survival Probability = exp(-Algorithm_Time / T₂)
Algorithm Success ≈ (1 - Error_Rate)^(Gates × Qubits)
1/T₂ = 1/(2T₁) + 1/Tφ
where:
T₁ = Energy relaxation time
T₂ = Dephasing time
Tφ = Pure dephasing time
Fault-tolerant threshold: error rate < 10⁻³ per gate
Surface code overhead: ~1,000 physical qubits per logical qubit
📝 Example Calculation
Max ops: 50µs / 50ns = 1,000 gates per qubit
Total alg time: 100 × 50ns = 5,000 ns = 5 µs
Survival: exp(-5/50) = 90.5% coherence probability
Error rate: 0.1% per gate
Total gates: 100 × 100 = 10,000
Alg success: (0.999)^10000 = 0.0045% (need error correction)
Physical qubits for FT: ~1,000
Pure dephasing rate: 1/(50×10⁻⁶) - 1/(2×100×10⁻⁶) = 15,000 Hz