Projective measurements of quantum processor

In summary: These probabilities can be calculated using qubit gates such as Z, H, and S, which transform the state |1> into the states |±x>, |±y>, and |±z>. However, there is no single gate or series of gates that can transform |1> into |+y>.
  • #1
EightBells
11
1
Homework Statement
The quantum processor will provide information about the final state of
the system by performing projective measurement |1><1|. What are possible outcomes of
this measurement and what are the probabilities for this outcomes for each state listed
in Eq. (2). Explain, how you can perform projective measurements for states |±x> and
|±y> using the processor equipped with the standard single qubit gates and the projective
measurements |1><1|.
Relevant Equations
Eq. 2 gives the six states: |0>, |1>, |±x>=(1/sqrt(2))*(|0>±|1>), |±y>=(1/sqrt(2))*(|0>±i*|1>)
Am I correct in thinking that the system measures the probability |<f|1>|^2 for some state <f|? Then the probabilities for each of the six states would be:

|<0|1>|^2= 0
|<1|1>|^2= 1
|<+x|1>|^2= |(1/√2)|^2 = 1/2
|<-x|1>|^2= |(-1/√2)|^2 = 1/2
|<+y|1>|^2= |(-i/√2)|^2 = 1/2
|<-y|1>|^2= |(i/√2)|^2 = 1/2

However, what, specifically, is the question asking for when it asks for the possible outcomes of the measurements?

Performing projective measurements in|±x> and|±y>, am I looking for the probabilities |<f|±x>|^2 and |<f|±y>|^2, but in terms of state |1> which is done using the qubit gates? I know ZH|1> = |+x>, H|1> = |-x>, and SH|1> = |-y>, but I can't find a gate or series of gates that transforms |1> to |+y>, and I'm not sure how to use this to calculate the probabilities |<f|±x>|^2 and |<f|±y>|^2.
 

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  • #2
The possible outcomes of the measurements are the four states |±x> and |±y>. The probabilities for each of these states can be calculated using the equation |<f|1>|^2, where |f> is one of the four states. For example, the probability of measuring the state |+x> is |<+x|1>|^2 = |(1/√2)|^2 = 1/2. Similarly, the probability of measuring the state |-x> is |<-x|1>|^2 = |(-1/√2)|^2 = 1/2, the probability of measuring the state |+y> is |<+y|1>|^2 = |(-i/√2)|^2 = 1/2, and the probability of measuring the state |-y> is |<-y|1>|^2 = |(i/√2)|^2 = 1/2.
 

Related to Projective measurements of quantum processor

1. What is a projective measurement in a quantum processor?

A projective measurement in a quantum processor is a type of measurement that collapses the quantum state of a qubit into one of its basis states. This is done by applying a specific set of operations to the qubit, which results in a binary outcome.

2. How does a projective measurement differ from other types of measurements in quantum computing?

Unlike other types of measurements in quantum computing, such as weak measurements or von Neumann measurements, projective measurements always result in a definite outcome. This is because they are designed to collapse the quantum state into a specific basis state, rather than measuring the state in a continuous or probabilistic way.

3. What is the significance of projective measurements in quantum computing?

Projective measurements are an essential tool in quantum computing as they allow us to extract information from a quantum system without disturbing its state. This is crucial for performing operations and calculations on quantum processors, as any disturbance to the system can result in errors and loss of information.

4. How are projective measurements performed on quantum processors?

Projective measurements are typically performed by applying a specific set of operations, known as a measurement operator, to the qubit being measured. This operator is designed to collapse the qubit's state into a specific basis state, and the outcome is then read out using classical measurement techniques.

5. What are the challenges in performing projective measurements on quantum processors?

One of the main challenges in performing projective measurements on quantum processors is the potential for errors and decoherence. As quantum systems are highly sensitive to external interference, any noise or disturbances can affect the accuracy of the measurement. Therefore, careful calibration and error correction techniques are necessary to ensure the reliability of projective measurements in quantum computing.

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