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Understanding The Theory And Math Behind Qubits

Superposition, Entanglement and Quantum Information

A bit basically represents information at a fundamental level. It can take either of the two states, a 1 or a 0, an ON or an OFF, TRUE or FALSE, two stable levels of current or voltage, and so on. Any physical entity that has two stable states can be used to store a bit of information. In traditional computers, we generally use a transistor for this, which basically acts like a switch than can be turned ON or OFF.

The fundamental unit of information of a Quantum computer is termed as the “Qubit”.

The logic is simple, it’s like a bit that can have infinite states instead of two.

But how? Enter superposition. At any instant of time, a Qubit represents the superposition of the two states, (say 0 and 1) in any proportion. And when you measure it, it collapses to one of its states.

Representing the state of a qubit

Let’s say you have two bits. You can represent four states using them- 00, 01, 10, 11. At any instant of time, your two-bit system can have one of these four states.

Now, let’s say you have two qubits. As mentioned, a qubit can have infinite states, but when you measure it, it’s still either a 1 or a 0. Fundamentally, two qubits also represent four states, but the catch here is in superposition.

Until you measure it, the state = 𝛂1.[00] + 𝛂2.[01] + 𝛂3. [10] + 𝛂4.[11]

It’s some combination of probabilities of each state, where the alphas are some coefficients.

Imagine a sequence of three bits. A bit can be either a 1 or a 0. Let’s say, the sequence is 101. It represents a number ‘5’.

Now consider a sequence of 3 Qubits. [0+1][0+1][0+1]. Depending on our measurement, this sequence can represent any number from 000 (i.e zero) to 111 (i.e seven)

Similarly, 4 bits can store a single number in the range from 0 to 15, while 4 qubits can store all the numbers from 0 to 15.

To generalize it, n bits can store 1 number between 0 and 2^n while n qubits can store all the numbers from 0 to 2^n.

Everything sounds good on the paper but how can we physically realize this? Didn’t we mention that when we measure the state of the qubits, it still collapses to a single value? In fact, we can only get one state of information from a Qubit, the state we finally measure, as if it’s a classical bit.

So, how to actually use these multiple states simultaneously?

Unraveling Qubits With Math

Imagine a function f(x). Let’s say you want to find the values f(x1) and f(x2), where x1 and x2 are some constants. A classical computer would compute this in a sequence, right? So, it would take two cycles of execution.

Based on your current understanding of a quantum computer, you would have guessed, it would simultaneously process both f(x1) and f(x2) in a single cycle of execution. But how exactly? What makes it possible are the inherent properties of quantum particles.

To understand the situation better, think of f(x) as a function that transmits a bit of information between Alice and Bob. let x1 and x2 be two bits.

Can you transmit two bits of information at once? No classical algorithm can. Let’s see how things play out when you go quantum.

A two-qubit system can be represented mathematically as-

State = 𝛂1.[00] + 𝛂2.[01] + 𝛂3.[10] + 𝛂4.[11]

A superposition of multiple states it can exist in.

Be we go further, let us state two facts as it is, without further explanation to keep things simple.

The probability of measuring any particular state = (coefficient of that state)²

Hence, ⅀(coefficient)² = 1 (Probability of measuring all the states)

Let’s take an example now.

Say, we have two qubits.

Q1 = 3/5.[0] + 4/5.[1]

Q2 = 1/√2.[0] + 1/√2.[1]

The state of this two-qubit system is,

Q1.Q2 = 3/(5.√2)[00] + 3/(5.√2)[01] + 4/(5.√2)[10] + 4/(5.√2)[11]

Here we simply multiply the coefficients of the states, to obtain the combined state. This is also known as a Tensor Product. Now, can you say that any two-qubit system like this can be represented as a product of states of the individual qubits? Let’s find out with another example.

Q’ = (1/√2).[00] + (1/√2).[11]

In this case, let’s say,

Q1 = 𝛂1.[0] + 𝛃1.[1]

Q2 = 𝛂2.[0] + 𝛃2.[1]

For Q = Q1.Q2 to be true, we have (just multiplying the coefficients and equating them),

𝛂1.𝛂2.[00] + 𝛂1.𝛃2.[01] + 𝛃1.𝛂2.[10] + 𝛃1.𝛃2.[11] = (1/√2).[00] + (1/√2)[11]

Comparing the coefficients,

  • 𝛂1.𝛃2 = 𝛃1.𝛂2 = 0
  • 𝛂1.𝛂2 = 𝛃1.𝛃2 = 1/√2

By simple observation, you can see that either of the alpha or the beta coefficients must be zero to satisfy the first condition. But if they are zero, the second condition isn’t satisfied. Hence, this particular two-qubit system cannot be separately described by the states of the individual qubits.

Hmm, looks like we are heading somewhere.

Observe something here. The two states of the qubit we have are [00] and [11]. If the first qubit is known to be a 0, what will be the value of the second qubit? Since the state 01 isn’t possible in this system, you can certainly tell that the second qubit will also be a 0. Similarly, if the first qubit is a 1, the second qubit will also be 1, no matter what.

Sounds like they are mysteriously connected? Such a pair of Qubits are described to be entangled.

With this understanding, let us go back to our Alice and Bob problem statement. (Please note that this is a simplified explanation and the real picture is much bigger)

Alice wants to send two bits of information in one transmission to Bob.

Let Alice have a qubit Q1 and Bob, qubit Q2. Let these two be a pair of entangled qubits represented by the above-discussed state.

Q = (1/√2).[00] + (1/√2)[11]

Assume Alice wants to send the bits ‘00’ to Bob. To do that, all Alice needs to do is send her qubit Q1 which is measured to be 0 to Bob.

Do you see where this is going? Bob receives the qubit and measures it to be 0. At that moment, what else does Bob know? The qubit Q2, which he has is definitely 0 too because they are entangled. He need not even measure it to confirm.

Together, Bob now knows that the information required is ‘00’. As you see, Bob received two bits of information required in a single transmission. ‘11’ can also be transmitted in a similar manner.

While this is a very simple case, how about transmitting the sequence ‘10’ or ‘01’? It can be done as well. What I described here right now is known as ‘Super Dense Coding’.

There are more questions to ponder at this moment, “How the hell do we realize such a system physically in reality? And how does Bob actually measure the qubit to be zero?” Take a moment to reflect on what you learned till now before we explore further.

Quantum Entanglement is the Real Deal

It is, in fact, phenomena like Quantum Entanglement, that make it possible to leverage the efficiency of multiple states in a Qubit. As seen in our previous post, you can use a Quantum computer to compute a function multiple times simultaneously.

If you measure the state of one qubit of an entangled pair, you can confidently tell the state of the other qubit, irrespective of where it is, without even needing to measure it. Hence, you indirectly obtain two values from a single measurement, effectively overcoming the previously discussed idea of how when you measure a qubit, its state collapses into a single value.

Stay tuned for upcoming articles where I will discuss Quantum gates and the physical realization of Qubits.

Do check out the resources below to learn more.

References and Additional Learning Resources

  • This is a great video to start your journey of learning quantum computing.

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