The Bloch Sphere

3.2.5 The Bloch Sphere

As we observed in the previous section, probabilities in the standard basis are not enough to specify a quantum state because it cannot capture the phase of the superposition. A convenient representation for a qubit is the Bloch sphere.

If we define a qubit state by |ψ⟩=cos(θ/2)|0⟩+eiϕ sin(θ/2)|1⟩ where θ and ϕ are defined in the picture, we see that there is a one-to-one correspondence between pure qubit states (C2，二度空間複數) and the points on the surface of a unit sphere (R3，三度空間實數). This is significant because now we can simply visualize qubit states and gates. We can reconstruct an arbitrary unknown qubit state |ψ⟩ by measuring the Bloch vector, whose vector components are the expectation values of the three （三個實數）Pauli operators, given by ⟨X⟩=tr(|ψ⟩⟨ψ|X), ⟨Y⟩=tr(|ψ⟩⟨ψ|Y), and ⟨Z⟩=tr(|ψ⟩⟨ψ|Z). The state is given by |ψ⟩⟨ψ|=(I+⟨X⟩X+⟨Y⟩Y+⟨Z⟩Z)/2.

Each expectation value ⟨Q⟩ can be obtained experimentally by first (a) preparing the state, (b) rotating the standard basis frame to lie along the corresponding axis, and (c) making a measurement in the standard basis. (d) The probabilities of obtaining the two possible outcomes 0 and 1 are used to evaluate the desired expectation value via ⟨Q⟩=P(0)−P(1). As an example, let's look at measuring the expectation value of X, ⟨X⟩=tr(|ψ⟩⟨ψ|X), depicted in the circuit below. Once |ψ⟩ is prepared we implement the gate H that exchanges Z to X, then we make a measurement in the standard basis. The desired expectation value is given by ⟨X⟩=P(0)−P(1). Similarly, we can use the S†−H gate to measure ⟨Y⟩.

To simplify performing these kinds of experiments, we provide a Bloch measurement circuit element that implements your quantum circuit three times with each of the above measurements, then plots the results on the Bloch sphere. The Bloch measurement element also takes some error into account, and corrects for it by scaling the Bloch vector using calibration experiments. A demonstration of the Bloch measurement element is given below.

Experiment log:

1. Superposition (+), already experimented in section 2.4, is resulted in 0 of 50% and 1 of 50%. However, the meaning here is different:

2. Superposition (+) X measurement, also already experimented in section 2.4, is resulted in 0 of 100% and 1 of 100%. However, the meaning here is different: here we’re measuring the expected value of the x component of |ψ⟩’s Bloch sphere. Because the meaning is different, here we RUN realistic condition to wait for the planned experiment result and get the result a bit different in the email as:

Executed on: May 17, 2016 2:13:12 AM

Results date: May 17, 2016 2:13:39 AM

Number of shots: 1024

Distribution

Device Calibration

Date Calibration: 2016-05-16 12:05

Fridge Temperature: 0.014356 Kelvin

So the desired expectation value is given by ⟨X⟩=P(0)−P(1)=0.967-0.033=0.934

3. Superposition (+) Y measurement here is different from the (+i) Y measurement in section 2.4 because it is |0⟩HS+H then measuring, not |0⟩HSS+H then measuring as in section 3.2.4. Here we’re measuring the expected value of the y component of |ψ⟩’s Bloch sphere. The simulated result is