Decoherence

3.2.6 Decoherence

Real quantum computers must deal with decoherence, or the loss of information due to environmental disturbances (noise). The Bloch vector formalism we introduced in the previous section is sufficient to describe the state of the system under decoherence processes. The pure states we have studied so far have ρ=|ψ⟩⟨ψ| and a Bloch vector of length 1, touching the surface of the Bloch sphere. Decoherence causes our quantum states to become mixed states, which have a density matrix ρ that can be written as a sum over pure states

ρ=∑kpk|ψk⟩⟨ψk|

and a Bloch vector that sits inside the Bloch sphere

|⟨X⟩|2+|⟨Y⟩|2+|⟨Z⟩|2<1.

Energy relaxation and Time constant T1

One important decoherence process is called energy relaxation, where the excited |1⟩ state decays toward the ground state |0⟩. The time constant of this process, T1, is an extremely important figure-of-merit for any implementation of quantum computing, and one in which IBM has made great progress in recent years, ultimately leading to the prototype quantum computer you are now using. Experiment with the circuits below to see how adding many repetitions of additional do-nothing Idle gates (or Identity gates; these are gates that do nothing but wait) Id before measurement causes the state to gradually decay towards |0⟩.

Dephasing and Time constant T2

Dephasing is another decoherence process, and unlike energy relaxation, it affects only superposition states. It can be understood solely in a quantum setting as it has no classical analog. The time constant T2 includes the effect of dephasing as well as energy relaxation, and is another crucial figure-of-merit. Again, IBM has some of the world's best qubits by this metric. Experiment with the circuits below to see that when a state starts as a superposition (Bloch vector on the "equator" of the Bloch sphere) the qubit is subjected to more decay channels than when it starts in the computational state |1⟩.

So here are two facts: (1) decoherence happens for single-qubit system which may decay towards ground state (energy relaxation).  This means it has a tendency to be away from the Bloch surface towards center.  However, it can also be for multiple-qubit system.  It may be in a multiple-qubit system, not only energy relaxation and dephasing for each individual qubit, but entanglement also may decrease coherence??? (2) decoherence has to do with superposition: the Bloch equator state can diphase more than the state |1⟩. This means it has a tendency to be away from the Bloch equator towards other Bloch surface points.

Progress in decoherence with superconducting qubits

Because T2 is such an important quantity, it is interesting to chart how far the community of superconducting qubits have come over the years. Here is a graph depicting T2 versus time. 

Therefore, as time goes on, the experiment facility is improving, and the coherence time is getting greater exponentially.  It needs more time to decohere the same amount of phasing figure and relaxed energy, which mean getting better).

Experiment Log:

1.      Excited Bloch Tomography (X+Measuring) has the following results:

2.      Excited (4 idle) Bloch Tomography (X+4Id+Measuring) has the following results:

Note that it is moving away from the Bloch surface towards sphere center on the negative z-axis!

3.      Excited (16 idle) Bloch Tomography (X +Measuring) has the following results:

Note that it is moving away, further, from the Bloch surface towards sphere center on the negative z-axis!

4.      Superposition (+i) Bloch Tomography (H+S +Measuring) has the following results:

Note (1) it is ON the Bloch equator.  (2) it is OK to have negative expected value (on –y axis)!

5.      Superposition (+i) (4 idle) Bloch Tomography (H+S +4Id+Measuring) has the following results:

Note that it is moving away from the Bloch equator towards other surface point and sphere center on the negative y-axis!

6.      Superposition (+i) (16 idle) Bloch Tomography (H+S +16Id+Measuring) has the following results:

Note that it is moving away, further, from the Bloch equator towards other surface point and sphere center on the negative y-axis!