# Measuring Qubits: Basics for Quantum Computing Beginners

• Ananthan9470
In summary, the conversation discusses the concept of quantum computing and measuring qubits in different bases. It is explained that in order to measure a qubit in a non-computational basis, one must apply a gate operation to transform the basis, then measure in the standard basis. This process is compared to measuring spin in a Stern and Gerlach apparatus. It is also mentioned that in some cases, a controlled-U operation can be used instead of a gate. The conversation ends with an example of rotating the SG apparatus to measure spin in a different basis for an electron in a hydrogen atom.
Ananthan9470
I am newly learning quantum computing and am confused about some concepts. Suppose your qbit is the electron of a hydrogen atom and its in the state α|0> + β|1> . As far as I can understand, this means that if you measure the qbit in |0>, |1> basis, you will get a ground state electron with the probability |α|2 and an exited one with the probability |β|2. So what does it mean when you say you are measuring the state in some other basis? say |+>, |-> basis. How do you physically do this? Thanks.

It is easier to see that things when you measure a spin along a direction with a Stern and Gerlach. You get 0> or 1> You will get another orthogonal basis by rotating the SG apparatus.

Ananthan9470
naima said:
It is easier to see that things when you measure a spin along a direction with a Stern and Gerlach. You get 0> or 1> You will get another orthogonal basis by rotating the SG apparatus.

I kind of understood the thing about spin and how it is measured but I am trying to do understand the same thing for the H atom system. There has to be some similar measuring in a different basis thing for this as well right?

Ananthan9470 said:
I am newly learning quantum computing and am confused about some concepts. Suppose your qbit is the electron of a hydrogen atom and its in the state α|0> + β|1> . As far as I can understand, this means that if you measure the qbit in |0>, |1> basis, you will get a ground state electron with the probability |α|2 and an exited one with the probability |β|2. So what does it mean when you say you are measuring the state in some other basis? say |+>, |-> basis. How do you physically do this? Thanks.

Generally what you do is apply some operations that do a basis transform so the basis you want gets mapped to the computational basis, then measure, then undo the basis transform.

For example, suppose you want to measure the X observable (instead of the Z observable that the computational basis corresponds to). The Hadamard operation happens to switch between those two basises, so you can use this circuit:

Code:
    |ψ> ──[H]──[Measure]──[H]──

or this:

Code:
    ψ ──H──•──H── ψ [collapsed]
│
0 ─────X───── result

If you don't want to figure out the basis transform, but have a gate whose operation is the observable's matrix U, and are able to do a controlled-U, then you can always use this circuit:

Code:
    ψ ─────U───── ψ [collapsed]
│
0 ──H──•──H── result

(The X observable is unique in that *both* solutions use the Hadamard gate so you can reverse which gates go on which wire and it would still work.)

Try measuring in the Y basis with this simulator.

Last edited:
Ananthan9470
Strilanc said:
Generally what you do is apply some operations that do a basis transform so the basis you want gets mapped to the computational basis, then measure, then undo the basis transform.

For example, suppose you want to measure the X observable (instead of the Z observable that the computational basis corresponds to). The Hadamard operation happens to switch between those two basises, so you can use this circuit:

Code:
    |ψ> ──[H]──[Measure]──[H]──

or this:

Code:
    ψ ──H──•──H── ψ [collapsed]
│
0 ─────X───── result

If you don't want to figure out the basis transform, but have a gate whose operation is the observable's matrix U, and are able to do a controlled-U, then you can always use this circuit:

Code:
    ψ ─────U───── ψ [collapsed]
│
0 ──H──•──H── result

(The X observable is unique in that *both* solutions use the Hadamard gate so you can reverse which gates go on which wire and it would still work.)

Try measuring in the Y basis with this simulator.

Thank you so much. It makes much more sense now. So basically, you apply a gate to the hydrogen atom and then you measure it in the standard basis. After the gate, the electron will be in a corresponding superposition and then when you measure, you will get outcomes depending on the new state and the new probability amplitudes(which depends on the gate you used). Did I get everything right? Thanks again.

here u> and d> are spin of a particle along z.
If you couple your atom with this particle, the state is g> d> + e> u>
= g> (|+> + |->) + e> (|+> - |->) = (|g> + |e>)|+> + (|g> - |e>) |->
So if you rotate your SG along x and measure its spin, your atom is projected on one of your Schrodinger cat's state.

Ananthan9470

## 1. What is a qubit?

A qubit is the basic unit of quantum information in quantum computing. It is similar to a classical bit in traditional computing, representing either a 0 or a 1. However, a qubit can also exist in a superposition state, representing both 0 and 1 at the same time. This property allows qubits to perform complex calculations and provide much more computing power compared to classical bits.

## 2. How are qubits measured?

Qubits are measured using quantum gates, which are operations that manipulate the state of the qubit. The most common gate used for measurement is the Hadamard gate, which collapses the qubit's superposition state into a classical state of either 0 or 1. Other gates, such as the Pauli gates, can also be used for measurement.

## 3. What is entanglement and how does it relate to qubits?

Entanglement is a phenomenon in quantum mechanics where two or more particles become connected in such a way that their states are dependent on each other. This means that measuring the state of one particle will affect the state of the other particle, even if they are physically separated. Qubits can be entangled with each other, allowing for faster and more efficient communication and computation.

## 4. How many qubits are needed for quantum computing?

The number of qubits needed for quantum computing varies depending on the task at hand. Generally, the more qubits that are available, the more complex calculations can be performed. For example, to factor large numbers, it is estimated that at least 10,000 qubits are needed. Currently, quantum computers have around 50-100 qubits, but this number is increasing as technology advances.

## 5. What are the challenges in measuring qubits?

One of the main challenges in measuring qubits is maintaining their delicate quantum state. Any disturbance from the environment, such as noise or vibrations, can cause the qubit to lose its superposition state and collapse into a classical state. This can result in errors in the measurement and affect the overall performance of the quantum computer. Researchers are working on developing better methods to control and measure qubits with minimal disturbance from the environment.

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