Use Of Complex Numbers in Quantum Computers
Complex Numbers might come in handy in mathematical calculations in physics but we rarely find a need to use them in our day-to-day life. Though they play a very important role in the understanding of quantum mechanics and quantum computers. Quantum computing is a rapidly-emerging technology that harnesses the laws of quantum mechanics to solve problems too complex for classical computers. Here we will see how are complex numbers used in quantum computers, more specifically in the representation of qubits.
<b><h2><center>Properties Of Complex Numbers</b></h2></center>
We have written one article explaining the <a href = "https://www.s-tronomic.in/post/61">dynamics of complex numbers</a>. If you want to understand complex numbers in more detail, you can read more about it there and come back here. Here I will talk about some properties of complex numbers in short.
<b>A complex number is the combination of a real number and an imaginary number</b>. The complex number is in the form of <b><i>a+ib</b></i>. While a real number can be represented on one axis, a complex number requires 2 (one for real part, another for complex part). Though we can also represent complex numbers in terms of amplitude and the angle it makes with the real axis (phase angle).
Addition and subtraction are quite intuitive in complex numbers. Like in real numbers, when both the numbers are on the same side of the axis, the sum of the number gives a result larger number. Similarly, if the complex numbers are facing a "similar" direction, their "value" will increase otherwise the value will decrease. We can represent complex numbers in different forms. Euler's form of complex numbers makes a lot of calculations easier with complex numbers.
<center><h4>Z = re<sup>iθ</sup></center></h4></b>
<b><h2><center>What's A Qubit?</b></h2></center>
Like a bit in classical computers, qubits can have values in form of 1 or 0. But unlike classical bits, it exists in a <b>superposition state</b> which means <i>at that time, it has both the values (0 and 1) at the same time</i>. But when we try to observe the superposition state, it collapses to one of the values and it will remain in that state for the rest of the time.
But the collapse of the superposition state is indeterministic or completely random. We cannot tell beforehand in which state the superposition will collapse to, but we can find the probability of the superposition state to collapse in one of the two states. We can write or represent these probabilities in terms of complex numbers.
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<b><h2><center>Why Do We Use Complex Numbers To Represent Qubits?</b></h2></center>
A question that might arise in your mind is that why do we use complex numbers only. We can't represent qubits like classical bits as qubits are represented in form of probabilities of two possible states they can be in. We are using the sum of the sine wave to represent the probability wave in quantum physics. It has an amplitude and an angle. This is what a complex number representation looks like.
Though we can use matrices to represent the probabilities of the two possible states of the qubits. That is completely fine. <b>But Euler's form of complex numbers make calculations like multiplication easier</b> in comparison to matrices. This makes complex numbers very helpful for representing qubits.
- Ojas Srivastava, 10:25 PM, 05 Jan, 2022