In the early 20th century, new findings about quantum world revolutionized mathematics and physics and created a completely new research field of quantum mechanics. In the late 20th century, it was discovered that quantum theory applies not only to incredibly small physical elements, but to bits and logic operations in computers. Especially the phenomenon of entanglement provides a completely new approach to storing and processing information. This finding has birthed a new field of quantum information processing.
On this page we will discuss what is the great benefit of quantum computing and what it can mean for the future of information processing.
Digital computers
Digital or binary computers are the most common computers. Basically every computer you interact with on daily basis is a binary computer. Their key feature is in storing information in discrete binary digits, known as bits.
Each bit holds either value 0 or value 1.
On its own, that is not a lot. However they are fast and reliable for computation. And since the expansion of the circuty industry, it is inexpensive to produce mass amount of binary storage - several of these bits together can hold larger amounts of information.
Analog computers
Through the years, analog computers have largely fallen out of favor, since they are difficult to programm and much less reliable. They store information in an analog value with electrical, mechanical or hydraulic quantities. For example one piece of information can be encoded in amount of charge in a transistor.
Each single information element holds a single value in continous specter between 0 or 1.
Quantum computers
Quantum computer also has discrete elements of information - quantum bits or qubits.
Each qubit can hold a value 0, 1 or a linear combination of both, which is similar to an analog computer.
However, a quantum computer also takes advantage of a special kind of superposition (most common superposition in quantum computing is entanglement) that allows for exponentially many logical states at once, raging from |0⟩ to |1⟩. Entangled qubits together represent more information than simply the sum of information of each qubit.
The counterintuitive principles of quantum physics are:
- A physical system in a definite state can still behave randomly.
- Two systems that are too far apart to influence each other can nevertheless behave in ways that, though individually random, are somehow strongly correlated.
A huge problem in quantum physics is that these principles are inconsistent with classical principles and laws.
We have to avoid the temptation to try and explain these principles with classical knowledge - "in terms that we understand". This is simply not possible - at least we haven't yet been sucessful in this regard. So, the best way to approach quantum physics is as a blank slate.
Every quantum computational operation has to be logically reversible.
Meaning: Result of every operation (or series of operations) must be able to produce the original input.
For example: A negation operation is logically reversible in itself. It is possible to get the orifinal input form the result, simply by using another negation operation. This is not true for an XOR operation (as well as AND, OR,...). In the latter case we cannot map resulting value of 1 to the original input, since we do not know which of the two input variables was positive.
Standard XOR operation
| A B | | C |
| 0 0 | | 0 |
| 0 1 | => | 1 |
| 1 0 | | 1 |
| 1 1 | | 0 |
It is not possible to clearly and uniquely revert XOR operation
| C |
| 0 | => A = B | A=B=1 OR A=B=0
| 1 | => A != B | A=0, B=1 OR A=1, B=0
In oder to bypass this issue, in quantum computing, a CNOT operation is used instead.
This operation can be understood simpy as an XOR operation with added control bit.
(source: https://en.wikipedia.org/wiki/Controlled_NOT_gate)
CNOT operation
| A B | | C | A' |
| 0 0 | | 0 | 0 |
| 0 1 | => | 1 | 0 |
| 1 0 | | 1 | 1 |
| 1 1 | | 0 | 1 |
CNOT is a reversible operation, since we have enough information, to clearly and uniquely define the input values from results
| C | A' | | A B |
| 0 | 0 | | 0 0 |
| 1 | 0 | => | 0 1 |
| 1 | 1 | | 1 0 |
| 0 | 1 | | 1 1 |
Unlike classical computation, where we know exactly which bit is affected by each operation, in quantum computation, all qubits in an operation get equally effected.
Although we often use names like conrol and target qubit, we have to keep in mind that these qubits can be used interchangibly.
CNOT operation is often used to create a Bell state, in which qubits are maximally entangled.
A (control) qubit: 1/√2 (|0⟩ + |1⟩)
B (target) qubit: |0⟩
After we apply CNOT operation, we get the result: 1/√2 (|00⟩ + |11⟩),
where each qubit has 50/50 chance in resolving into each of the states. In effect, these qubits are undefined.
But these qubits are also entangles, such as if we chose the same basis to measure both qubits, the measurements will perfectly correlate.
This state is called a Bell state |Φ+⟩.
(source: https://en.wikipedia.org/wiki/Controlled_NOT_gate)