What is the difference between T1 and T2?

T1 and T2 describe different aspects of decoherence. T1 (energy relaxation) measures how quickly the qubit loses energy from the excited state |1> to the ground state |0>. It represents an irreversible process: the qubit's energy is absorbed by the environment (substrate, control lines, radiation, quasiparticles). After time T1, the probability of finding the qubit in |1> decays to 1/e of its initial value. T2 (dephasing) measures how quickly the qubit loses phase information in a superposition state (alpha|0> + beta|1>). The relative phase between |0> and |1> components accumulates random fluctuations from low-frequency noise (charge noise, flux noise, photon number fluctuations), causing the phase to become unpredictable. T2 has a fundamental upper bound of 2*T1 because energy relaxation also destroys phase information (you cannot maintain phase coherence if the qubit has decayed to |0>). In practice, T2 is usually less than 2*T1 due to pure dephasing processes. T2* (measured in Ramsey experiments) includes all dephasing sources including slow quasi-static noise; T2 echo (measured via Hahn echo) refocuses the slow noise component.

What limits coherence time in superconducting qubits?

Multiple decoherence mechanisms contribute, with different mechanisms dominating in different qubit designs and quality levels. Dielectric loss in the substrate and interfaces (silicon-oxide and metal-oxide layers) is currently the dominant T1 limitation in state-of-the-art transmon qubits, contributing a loss tangent of 10^-6 to 10^-5 that limits T1 to 100 to 1000 microseconds. Material improvements (tantalum instead of aluminum, high-purity sapphire or silicon substrates, surface cleaning to remove oxides) have pushed T1 from 1 microsecond (2007) to over 500 microseconds (2024). Quasiparticle tunneling across Josephson junctions creates a T1 floor that scales with junction size and quasiparticle density. The equilibrium quasiparticle density at 20 mK should be negligible, but non-equilibrium quasiparticles generated by stray radiation (cosmic rays, infrared photons leaking into the cryostat) create a measured density of 10^-8 to 10^-6 per Cooper pair. Photon shot noise in the readout resonator causes dephasing through the dispersive interaction (chi*n_photon frequency shift), limiting T2 when residual photons from readout or thermal noise are present. Charge noise (1/f spectrum) and flux noise (1/f spectrum, approximately 1 micro-Phi_0/sqrt(Hz)) cause dephasing in charge-sensitive and flux-sensitive qubits, respectively.

How does coherence time determine quantum gate count?

The maximum number of useful quantum gates before errors accumulate is approximately N_gates = T2 / t_gate, where t_gate is the duration of a single gate operation. For a transmon qubit with T2 = 100 microseconds and single-qubit gate time of 20 nanoseconds: N_gates = 100e-6 / 20e-9 = 5,000 gates. Two-qubit gates (CNOT, CZ) are slower (50 to 300 ns), giving 300 to 2,000 two-qubit gates per coherence time. In practice, the gate fidelity F is limited by T1 and T2: F approximately equals 1 - t_gate/(3*T1) - t_gate/(2*T2_phi) for a single-qubit gate, where T2_phi is the pure dephasing time. With T1 = 200 microseconds, T2 = 100 microseconds (T2_phi = 200 microseconds), and t_gate = 20 ns: F = 1 - 20/(3*200000) - 20/(2*200000) = 1 - 3.3e-5 - 5e-5 = 0.99992. This exceeds the surface code threshold of approximately 0.999 (0.1% error rate), meaning error correction can work. Achieving fault-tolerant quantum computation requires T2/t_gate ratios above approximately 1,000, which current transmon qubits achieve for single-qubit gates but not yet reliably for two-qubit gates.

Quantum Computing RF

Coherence Time

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Coherence time measures how long a qubit maintains its quantum superposition. T₁ (energy relaxation): excited state decay to ground, 100 to 500 μs for state-of-the-art transmons. T₂ (dephasing): phase coherence decay, bounded by T₂ ≤ 2T₁, typically 50 to 300 μs. Gate count ≈ T₂/t_gate: with 20 ns gates and T₂ = 100 μs, ~5,000 single-qubit gates before decoherence limits fidelity.

Category: Quantum Computing RF

State-of-art T₁: 100 to 500 μs

Gate fidelity: >99.99%

Understanding Qubit Coherence Time

Quantum computation depends on maintaining fragile quantum superposition states long enough to perform useful calculations. A qubit in state α|0⟩ + β|1⟩ carries information in both the amplitudes (α, β) and the relative phase between the two components. The environment (thermal photons, material defects, electromagnetic noise, cosmic rays) continuously interacts with the qubit, causing it to lose energy (T₁ decay) and accumulate random phase errors (T₂ dephasing). After a time comparable to the coherence time, the quantum information is destroyed and the qubit is in a classical mixed state.

For superconducting qubits, the coherence time has improved by five orders of magnitude over two decades: from ~1 ns (first Cooper pair box, 1999) to ~500 μs (state-of-the-art transmon, 2024). This improvement comes from better materials (tantalum replacing aluminum, sapphire/silicon replacing silicon oxide), better designs (transmon's exponentially suppressed charge sensitivity vs Cooper pair box), and better shielding (infrared and cosmic ray protection, improved filtering on control lines). Each order-of-magnitude improvement enables qualitatively new capabilities: microsecond coherence enabled two-qubit gates, tens of microseconds enabled quantum error detection, and hundreds of microseconds are enabling early quantum error correction demonstrations.

Coherence Time Equations

T₁ Decay:
P_|1⟩(t) = P_|1⟩(0) × e^-t/T₁

T₂ Bound:
1/T₂ = 1/(2T₁) + 1/T_φ

Gate Fidelity (single-qubit):
F ≈ 1 - t_gate/(3T₁) - t_gate/(2T_φ)

Where T_φ = pure dephasing time. T₁ = 200 μs, T₂ = 100 μs (T_φ = 200 μs), t_gate = 20 ns: F = 0.99992. Surface code threshold: F > 0.999 (0.1% error). Maximum gates: T₂/t_gate = 5,000.

Qubit Coherence Time Comparison

Qubit Type	T₁	T₂	Gate Time	Gates/T₂
Transmon (2024)	100 to 500 μs	50 to 300 μs	20 to 40 ns	2,500 to 15,000
Fluxonium	200 to 1,000 μs	100 to 500 μs	20 to 60 ns	3,000 to 25,000
Trapped ion	~1 s	0.1 to 10 s	1 to 100 μs	1,000 to 10,000
NV center	~1 ms	1 to 100 μs	10 to 100 ns	100 to 10,000
Cooper pair box (2002)	~1 μs	~0.5 μs	1 to 10 ns	50 to 500

Common Questions

Frequently Asked Questions

T₁ vs T₂?

T₁: energy loss (|1⟩ → |0⟩), irreversible, caused by substrate dielectric loss, quasiparticles, radiation. T₂: phase coherence loss, caused by charge/flux noise, photon shot noise. T₂ ≤ 2T₁ always. T₂* (Ramsey) includes slow noise; T₂ (echo) refocuses it. Typically T₂* < T₂ < 2T₁.

What limits coherence in transmons?

Dominant T₁: dielectric loss in substrate/interfaces (tanδ 10^-6 to 10^-5). Quasiparticle tunneling: non-equilibrium QP from cosmic rays/IR photons. T₂: photon shot noise (dispersive shift χ×n), charge noise (1/f), flux noise (1/f). Materials progress: Al→Ta, Si/sapphire substrates, surface cleaning.

How does coherence determine gate count?

N_gates ≈ T₂/t_gate. T₂ = 100 μs, 20 ns gate: 5,000 single-qubit gates. Two-qubit (50 to 300 ns): 300 to 2,000 gates. Surface code needs <0.1% error (>99.9% fidelity). Current transmons: single-qubit >99.99%, two-qubit 99.5 to 99.9%. Fault tolerance needs T₂/t_gate > 1,000.

Related Terms

← Coherence Length Coherence Time Channel →