quantum computer

A quantum computer can solve computational problems considerably faster than classical computers. There are a variety of models of quantum computers, including the quantum circuit model and the quantum Turing machine. Central to the heart of quantum computers are the individual elements, called qubit’s, which equate to the classical computational bit. They are represented in many states, including a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0. This multi-permutation holds the advantage over binary systems, and thus opens up the possibilities of far reaching computation power based on quantum theory.

 

How does a quantum computer work?

Quantum computing began when Benioff proposed a quantum model of the Turing machine, in the 1980’s. Feynman and Manin later suggested that a quantum computer had the potential to simulate things that couldn’t be achieved classically. An algorithm was later developed for factoring integers that had the potential to decrypt RSA-encrypted messages. Investment into quantum research has increased and an example in 2019, Google AI and NASA performed a quantum computation that is infeasible on any classical computer.

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer. Quantum computers obey the Church–Turing method. They can quickly solve certain problems that no classical computer could solve in any feasible amount of time, often referred to as quantum supremacy.  Quantum complexity theory is the field of study.

 

Definition of a quantum computer

The model of quantum computation explains the computation in terms of a network of quantum logic gates. This is an abstract linear-algebraic sumamry of a classical circuit. Obeying quantum mechanics leads to a quantum computer capable of efficiently running these circuits and is theoretically believed to be physically realisable.

 

Scaling quantum computing

To maximise computational power we optimise along two dimensions, qubit count and low error rates. The greater the number of qubits, the greater number of states that can be manipulated. Low error rates are also needed to to be accurate and perform more precise sequential operations.

Quantum volume is a critical metric and forms the relationship between number and quality of qubits, circuit connectivity and error. Achieving higher quantum volume will lead to applications which can offer a real computational advantage for real world challenges.

 

Quantum algorthims

Quantum algorithms focus on the quantum circuit model. Quantum algorithms are classified by the uplift achieved. Algorithms that output a polynomial uplift over the best known classical algorithms e.g. Shor’s algorithm or Pell’s equation. These algorithms rely on the quantum Fourier transform. Oracle problems like Simon’s problem do give speedups, though this is a restricted model and doesn’t always translate for practical problems.

Further, simulations of quantum physical processes from chemistry, or the quantum algorithm for linear systems have quantum algorithms that give super-polynomial speedups. Some quantum algorithms, like Grover’s algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.

 

Quantum computer applications and advantages

Cryptography

A simple application would be the defence of cryptographic systems. Factorisation is agreed to be practically impossible for large integers if they are derived from several prime numbers. A quantum computer could solve this using Shor’s algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve algorithms could be solved. These are used to protect secure web pages and encrypted email. Lattice-based crypto systems are also a benchmark and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, is an interesting problem.

 

Machine learning

As quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.  For example, the quantum algorithm for linear systems of equations, or HHL Algorithm, is believed to provide speedup over classical counterpart.

 

Quantum supremacy

Quantum supremacy refers to the hypothetical speedup advantage that a quantum computer would have. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year, whilst IBM said in 2018 that the best classical computers will be beaten within five years. In 2019, a Sycamore processor (Google AI Quantum) was reported to have achieved quantum supremacy, with calculations more than 3,000,000 times as fast as those of Summit, the world’s fastest computer.

 

Disadvantages of quantum computing

There are a number of technical challenges in building a quantum computer.

  • Scalability of qubits
  • Qubits that can be initialised to arbitrary values
  • Quantum gates that are faster than decoherence time
  • Universal gate set

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables.

 

Quantum computer models

There are a number of models. The four main models of practical importance are:

  • Quantum gate array (computation decomposed into a sequence of few-qubit quantum gates)
  • One-way quantum computer (computation decomposed into a sequence of one-qubit measurements applied to a highly entangled cluster state)
  • Adiabatic quantum computer (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)
  • Topological quantum computer (computation decomposed into the braiding of anyons in a 2D lattice)

 

In 2020, Honeywell forged ahead with the method of trapped ions. They are taking on this method, against the grain as other global progress has not seen this as the preferred route. IonQ have since revealed they are looking at a trapped-ion machine, to compete with the technologies arising from IBM. Trapped-ion quantum computers were the actually studied before progress in superconducting loops.

 

Quantum computing in the finance world

In finance, the gains from huge computing performance could be game changing. For example, in valuation, the ability to identify an optimal risk-adjusted portfolio would offer a leading edge in the market. Across loans, fast estimates of more detailed credit exposures would mitigate risk. Additionally, multi-capital investment could be analysed and fine tuned in a far more eloquent manner. As highlighted previously, data encryption would move into a new field.

The advantages for equity would be enormous. The financial sector spends billions in investment into computational power.

 

 

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References

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