Measuring a qubit in a different basis

[shahab_sh]

Hi,
I'm a student of computer science and am writing a simulator for Measurement-based quantum computing or one-way quantum computing (it's based on the paper "The Measurement Calculus" by Elham Kashefi et al.).

Anyway, I'm still a bit confused when it comes to the calculations. If we write the state in let's say the z-basis that is in the Bloch sphere along the z-axis (up) is |0> and the opposite direction (down) is |1> a typical state might be [1/2, 1/2, 1/2, 1/2]^T then if we measure the first qubit also in the z-basis, it has (1/2)²+(1/2)² = 0.5 probability of collapsing into 0 (or 1). And now because one of the qubits has collapsed into a classical state we can remove it from the state and the new state will be [1/√2, 1/√2] (which includes only the second qubit). But what if we want to measure it in an arbitrary basis (as required for one-way quantum computers)? For example in the paper mentioned above, it says:
Measurement M_i^α is defined by orthogonal projections on
|+_α> := 1/√2 (|0> + e^iα |1>)
|-_α> := 1/√2 (|0> - e^iα |1>)
followed by a trace-out operator. The parameter α ∈ [0, 2π] is called the angle of the measurement. For α = 0, α = π , one obtains the X and Y Pauli measurements.

My question is how do we do the calculations and what will the state after the measurement look like. I have found some materials on the Internet which basically says we should first change the basis, calculate the probability of the outcome of the measurement, do the measurement and remove the collapsed qubit from the state, and then change the basis back to whatever it was.

But first, it seems like an awfully expensive operation (for a simulator) and if that's the only way, how can we change the basis of a system with more than 1 qubit to an arbitrary basis?

Thanks for your help :)

 

[eljose79]

Hi there,

As a fellow computer scientist, I understand your confusion and concerns with the calculations involved in one-way quantum computing. Let me try to break it down for you.

Firstly, the state of a qubit in the Bloch sphere can be represented as a vector in a two-dimensional Hilbert space. The z-basis you mentioned, with |0> and |1>, is just one possible basis for this space. In general, a qubit can be in a superposition of multiple states, which is why we use vectors with coefficients to represent them.

Now, when we perform a measurement on a qubit, we are essentially collapsing its state into one of the basis states with a certain probability. This is where the calculation you mentioned comes in, with the probabilities of each outcome being determined by the coefficients of the vector.

When it comes to measuring in an arbitrary basis, the process is similar but with a slight difference. The basis states |+α> and |-α> are just two different basis states for the qubit, and the angle α helps us determine the coefficients of the vector in this new basis. So, the calculation involves first changing the basis of the qubit, calculating the probabilities of each outcome, and then collapsing the state and removing the collapsed qubit from the system.

As for changing the basis of a system with more than 1 qubit, the process is similar but involves more complex calculations. This is because the qubits in a system can be entangled, meaning their states are dependent on each other. To change the basis of such a system, we need to use a unitary transformation, which is essentially a matrix that operates on the entire system. This operation can be computationally expensive, but it is necessary for simulating one-way quantum computing.

I hope this helps clarify the calculations involved in one-way quantum computing. Keep up the good work with your simulator and don't hesitate to reach out if you have any further questions. Best of luck!