QUANTUM COMPUTING BASICS

QUANTUM 2

WHAT IS QUANTUM COMPUTING?

Quantum computing is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as quantum computers

Quantum computing harnesses the phenomena of quantum mechanics to deliver a huge leap forward in computation to solve certain problems.

For some problems, supercomputers aren’t that super

Until now, we’ve relied on supercomputers to solve most problems. These are very large classical computers, often with thousands of classical CPU and GPU cores. However, supercomputers aren’t very good at solving certain types of problems, which seem easy at first glance. This is why we need quantum computers.

Why quantum computers are faster

Quantum computers can create vast multidimensional spaces in which to represent these very large problems. Classical supercomputers cannot do this.

Algorithms that employ quantum wave interference are then used to find solutions in this space, and translate them back into forms we can use and understand.

Quantum computers perform calculations based on the probability of an object’s state before it is measured – instead of just 1s or 0s – which means they have the potential to process exponentially more data compared to classical computers. … A single state – such as on or off, up or down, 1 or 0 – is called a bit.

Types of Quantum Computers

There are three types of quantum computers that are considered to be possible by IBM. They range from a quantum annealer to a universal quantum.

The quantum annealer has been successfully developed by Canadian company D-Wave, but it is difficult to tell whether it actually has any real “quantumness” thus far. Google added credibility to this notion in December 2015, when it revealed tests showing that its D-Wave quantum computer was 3,600 times faster than a supercomputer at solving specific, complex problems.

Expert opinion, however, is still skeptical on these claims. Such criticisms also shed light on the major limitation of quantum annealers, which is that they may only be engineered to solve very specific optimization problems, and have limited general practicality.

The holy grail of quantum computing is the universal quantum, which could allow for exponentially faster calculations with more generality.

However, building such a device ends up posing a number of important technical challenges. Quantum particles turn out to be quite fickle, and the smallest interference from light or sound can create errors in the computing process.

Doing calculations at exponential speeds is not very useful when those calculations are incorrect.

Real-World Example of a Quantum Computer 

Google (GOOG) is spending billions of dollars on its plan to build its quantum computer by 2029. The company has opened a campus in California, called Google AI, to help it meet its goal. Google has been investing in this technology for years. As well, so have other companies, such as Honeywell International (HON) and International Business Machine (IBM). IBM expects to hit major quantum computing milestones in the coming years. 

While some companies have built personal (although expensive) quantum computers, there is still nothing available on the commercial side. And there’s interest in quantum computing and its technology, with JPMorgan Chase and Visa looking into the technology. Once developed, Google could launch a quantum computing service via the cloud.

Companies can also gain access to quantum technology without having to build a quantum computer. IBM plans to have a 1,000-quibit quantum computer in place by 2023. For now, IBM allows access to its machines if they’re part of its Quantum Network. Those that are part of the network include research organizations, universities, and laboratories. 

Microsoft also offers companies access to quantum technology via the Azure Quantum platform. This is unlike Google, which doesn’t sell access to its quantum computers.

Quantum Computer vs. Classical Computer

Quantum computers process information differently. Classical computers use transistors, which are either 1 or 0. Quantum computers use qubits, which can be 1 or 0 at the same time. The number of qubits linked together increases the quantum computing power exponentially. Meanwhile, linking together more transistors only increases power linearly.

Classical computers are best for everyday tasks that need to be completed by a computer. Meanwhile, quantum computers are great for running simulations and data analyses, such as for chemical or drug trials. These computers must be kept ultra-cold, however. They are also much more expensive and difficult to build.

Classical computing advances include adding memory to speed up computers. Meanwhile, quantum computers help solve more complicated problems. While quantum computers might not run Microsoft Word better or faster, they can run complex problems faster.

For example, Google’s quantum computer that’s in development could help with many processes, such as speed up machine-learning training or help create more energy-efficient batteries.

Quantum computing has a number of other applications, including securely sharing information. Other methods include fighting cancer and various health concerns, such as cancer and developing new drugs. As well, quantum computers can help improve radars and their ability to detect such things as missiles and aircraft. Other areas include the environment and using quantum computing to keep water clean with chemical sensors.

Google, in 2019, proved that a quantum computer can solve a problem in minutes, while it would take a classical computer 10,000 years.

APPLICATION OF QUANTUM COMPUTING IN PROBLEM SOLVING

Let’s assume we have a function f. It takes a single bit as an input. Either 0 or 1. And it provides a single bit as its output, too. Again, either 0 or 1.

There are four different possible functions.

  • Function f_0 always returns 0.
  • Function f_1 returns 0 if the input is 0 and it returns 1 if the input is 1.
  • Function f_2 returns 1 if the input is 0 and it returns 0 if the input is 1.
  • Function f_3 always returns 1.

The functions f_0 and f_3 provide constant outputs. No matter what their input is, they always return the same result. f_0 returns 0.f_3 returns 1. Always.

The functions f_1 and f_2 provide balanced outputs because for half of the inputs they return 0 (f_1 if the input is 0 and f_2 if the input is 1) and for the other half they return 1 (f_1 if the input is 1 and f_2 if the input is 0).

If you’re given one of these functions at random, how can you determine whether the function is constant (f_0 or f_3) or balanced (f_1 or f_2)? We don’t care about the specific function we got. We only care about whether it is constant or balanced.

Classically, you have to run the function twice. You run it with the input 0. If you get 0 as the result, the function at hand could be the constant function f_0 that always returns 0. And it could be the balanced function f_1 that returns 0 if the input is 0. You have to run the function again with the input 1 to decide whether the function is constant or balanced. The constant function f_0 still returns 0. But the balanced function f_1 returns 1 for input 1.

The same situation applies if you get 1 as the result of running the function for the input 0. You would need to distinguish between the balanced function f_2 and the constant function f_3.

It does not help to run the function with the input 1 first, either. With either result, 0 and 1, the function at hand could be constant or balanced. You need to run the function again, with the input 0.

In quantum computing, we only need to run the function once.

We start with a new quantum gate. The gate O. This gate should represent our four functions f_i. Mathematically, this would be

It transforms an arbitrary quantum state |x⟩ into the quantum state |f_i(x)⟩ — the output of the function f_i given the input x.

For O is a quantum transformation gate, it must be reversible. Therefore, it must have the same size of input and output. And each output must be uniquely identity the input it originates from.

But that’s a problem. The constant functions always return the same value regardless of their inputs. Given their output, you can’t tell what the input was.

We can deal with this problem with a little trick. We add a second qubit |y⟩ and we use the exclusive or (XOR, ⊕) operator to keep track of the result of the function f_i.

Mathematically, the refines gate O is:

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We can safely state that not changing a state is reversible.

When i=1, we apply the function f_1 that returns 0 for x=0 and 1 for x=1. Thus, f_1(x)=x.

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When we apply f_2 that returns 1 for x=0 and 0 for x=1, we can say f_2(x)=x XOR 1.

The truth table discloses that the term |x⟩⊗|y⊕x⊕1⟩ is reversible, too.

Finally, f_3 always returns 1.

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The output is like the input but with a reversed y.

Our refined O-gate is a valid transformation gate for all our functions f_i.

But, we’re not too interested in how the functions f_i work. Rather, we regard O as a black-box. The two important things are

  • O is a valid two-qubit gate for all i
  • The output of O

Collectively, the two qubits are in the state

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Now, we apply the gate O. Thus, we replace the four basis states with the terms we calculated in the truth table above.

We can rearrange this term by putting the basis states of the first qubit outside the brackets.

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Let’s have a closer look at the term |f_i(0)⟩−|f_i(0)⊕1⟩.

  • For f_i(0)=0, the term is |0⟩−|1⟩.
  • For f_i(0)=1, the term is −|0⟩+|1⟩.

We can rewrite it as (−1)^f_i(0)*(|0⟩−|1⟩).

The term (−1)^f_i(0) takes care of the signs. For f_i(0)=0, it is 0 because anything with exponent 0 is 1. For f_i(0)=1, it is −1, yielding −|0⟩+|1⟩.

Therefore:

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The same logic applies to the term |f_i(1)⟩−|f_i(1)⊕1⟩.

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We insert these terms into our qubit state equation

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And we rearrange it, by putting the terms (−1)^f_i(0) and (−1)^f_i(1) outside the brackets.

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Then, we move anything except the term |0⟩−|1⟩ outside the brackets, too.

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Finally, we apply the common factor 1/2 to each of the resulting terms, note 1/2=1/sqrt(2) * 1/sqrt(2).

The result is a two-qubit state in the form of |x⟩⊗|y⟩. |x⟩ and |y⟩ are the two output qubits with the qubit |x⟩ is the top one.

Let’s have a closer look at the state of qubit |x⟩.

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Inside the brackets, we see the usual sum of our two basis states, |0⟩+|1⟩. Yet, the output of the function f_i determines the signs of the basis states. f_i(0) controls the sign of |0⟩. It is + for f_i(0)=0f because (−1)^0=1 and it is − for f_i(0)=1 because (−1)^1=−1. Accordingly, f_i(1) controls the sign of |1⟩.

The following table depicts the possible values of the qubit |x⟩ in dependence of f_i.

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The same accounts for the state |−⟩, respectively

The upfront constant amplitude of 1/sqrt(2) applies to both basis states. Its square (1/sqrt(2))²=1/2 is the probability of measuring either one state. So, half of the time we measure the qubit as 0. And half of the time we measure the qubit as 1. This applies to both states |+⟩ and |−⟩. The measurement probabilities are the same for all four functions f_i.

Unless we apply another transformation gate to this qubit. If we apply the Hadamard gate to the first qubit, it transforms the state |+⟩ into |0⟩. And it transforms |−⟩ into |1⟩.

Thus, we measure the qubit as 0 with certainty, if the output of our O-gate is |+⟩. This is the case for f_0 because f_0(0)=0 and f_0(1)=0. And it is the case for f_3 because f_3(0)=1 and f_3(1)=1. These are the two constant functions.

Accordingly, we always measure 1 if the O-gate outputs |−⟩. This is the case for f_1 and f_2. These are the balanced functions.

So, no matter what function f_i we plug into our circuit, we can tell whether it is constant or balanced by running it only once. Something we can’t achieve classically.

ADVANTAGES AND DISADVANTAGES OF QUANTUM COMPUTING

The main advantages and strengths of quantum computers

Used correctly, quantum computers are incredibly fast and effective. They can perform calculations in a few seconds for which today’s supercomputers would need decades or even millennia. This fact is also referred to by experts as quantum superiority . For a long time, this was just a theory. In 2019, however, Google’s quantum computer protoype was able to perform such a calculation and verify quantum superiority in practice.   Calculations with quantum computers are particularly promising wherever incredibly complex processes with huge amounts of data are to be analyzed or simulated. In addition to digital marketing, the natural science disciplines in particular see great potential here. Quantum computers could contribute to a better and more detailed understanding of the interaction of individual particles, elements and the processes in living cells. But there are also potential applications in medicine.   Most of all, researchers hope that quantum computers will take artificial intelligences (AI) a big step forward. These could then safely and reliably take over tasks such as data evaluation or forecasting in the future. 

Caution: The new technology has these disadvantages and weaknesses

Current quantum computers are largely prototypes that are still bulkycomplicated and expensive . At the same time, they still have many teething problems that their developers have not yet been able to fully address. The entanglement of many qubits at once is currently just as difficult as maintaining the necessary state for quantum operations. At the same time, the developers of the systems are struggling with an error ratethat is still far too high (AmPnBsP);  But even if quantum computers were perfect at some point, they would not only have advantages. Due to their incomparable computing power, all currently used encryption mechanisms would be useless from one day to the next. Secure communication or any kind of transaction over the internet could be cracked and the data misused or resold . Even many cryptocurrencies would no longer be secure and anonymous. To prevent this from happening, researchers are already working on so-called post-quantum cryptography. With new methods, secure communication should also be possible in the future.

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