# Simultaneous computational basis measurement

Many quantum algorithms requires measurements at the end of the processing and it seems like the quantum state measurement setup is required to be made for measuring the two computational basis.

I know that it is possible in photonics, especially using polarization degrees of freedom; for example, using a polarization beam splitter (PBS) with two detectors at each output of the PBS, we can check whether the quantum state was |H> or |V>.

I'm not saying whether simultaneous projection measurement in two computational bases, |0> and |1>, is possible, but saying whether two computational basis measurement setup can be made in a same computing system simultaneously.

As far as I know, it is very important for boosting the computational speed up for checking the computing result at once; for example, when we try to perform the Bernstein-Vazirani algorithm with a hidden secret string S=1, we need to check it in one query by putting all possible computational basis measurement set in the computer that we use.

But I'm wondering whether making two computational basis measurement setup simultaneously is experimentally available in ion trap and super conducting system.

Is it really available to measure those two setups at the same time?

If possible, I would like to know how it is implemented experimentally.

• I think you may have a misunderstanding of how measurement works. A qubit measured in the computational basis will be either $|0\rangle$ or $|1\rangle$. The measurement "collapses" the qubit into the respective basis state. There's nothing simultaneous about such measurement. Feb 18 at 22:44
• Feb 18 at 23:32
• So do you mean to ask "how is the physical measurement being setup in the lab for superconducting and trapped ion system?" Feb 19 at 0:25
• What do you mean by “simultaneously” in this context? A coin flip is measured to be heads or tails, but there’s no simultaneity in the measurement. A qubit is measured as $|0\rangle$ or $|1\rangle$ but it’s not simultaneous measurement. We might say that the qubit collapses instantaneously to one of the two basis states, but that’s not simultaneity. Perhaps you are wondering how measurement is made in superconducting qubit systems, and in ion trap systems. Those are reasonable questions, but they have nothing to do with simultaneity. Feb 19 at 3:34

Before we explain how ion traps and superconducting qubits are measured, let's clarify what's meant by a basis, as that might reduce some confusion about simultaneous measurements.

Following up on the example of BB84, suppose Alice sends precisely one qubit, which is measured by Bob.

Alice can prepare her photon to be in one of:

$$|0^\circ\rangle,|90^\circ\rangle,|45^\circ\rangle,|135^\circ\rangle.$$

Bob can measure the received photon in either the $$|0^\circ\rangle,|90^\circ\rangle$$ basis or the $$|45^\circ\rangle,|135^\circ\rangle$$ basis. If Alice prepares the photon as $$|0^\circ\rangle$$ and Bob measures the received photon in the $$|0^\circ\rangle,|90^\circ\rangle$$ basis, the photon will be measured as $$|0^\circ\rangle$$. If, however, Alice prepares the photon as $$|0^\circ\rangle$$ and Bob measures the received photon in the $$|45^\circ\rangle,|135^\circ\rangle$$ basis, the photon will "snap" to either be polarized as $$|45^\circ\rangle$$ or $$|135^\circ\rangle$$ (or, the wavefunction "collapses").

There's nothing simultaneous about the measurement - Bob measures either in the horizontal/vertical ($$|0^\circ\rangle,|90^\circ\rangle$$) basis or the diagonal ($$|45^\circ\rangle,|135^\circ\rangle$$) basis.

You are correct that Bob can use a PBS with two detectors at each output of the PBS, and Bob can check whether Alice prepared her photon in the $$|0^\circ\rangle,|90^\circ\rangle$$ basis, or he can rotate his PBS and measure in the $$|45^\circ\rangle,|135^\circ\rangle$$ basis, and a photon prepared as $$|0^\circ\rangle$$ and measured in the $$|45^\circ\rangle,|135^\circ\rangle$$ basis with photodetectors positioned in the appropriate space will measure a tick randomly.

Turning to how ion traps and superconducting qubits, I am not familiar enough with the physics. Nonetheless I think the respective Wikipedia articles are pretty good; instead of polarization of light going through beam-splitters and photodetectors, laser pulses for trapped ions and microwave pulses for superconducting qubits manipulate the qubits and they are measured with, I think, a CCD camera for trapped ion systems and with microwave resonators for superconducting qubits.

• You are at least confusing me about "making two computational basis measurements". Perhaps you are asking how you can convert a trapped ion in the state $|0\rangle$ to one in the state $|+\rangle=\frac{1}{\sqrt 2}|\big (|0\rangle+|1\rangle\big )$ - this is done with a Hadamard transform, again with well-timed laser pulses. Perhaps consider asking another question? Feb 21 at 18:32