Hypothetical fault-tolerant quantum computer based on topological condensed matter

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) bumping into a wall. Alexei Kitaev proposed topological quantum computation in 1997. While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.

Anyons are quasiparticles in a two-dimensional space. Anyons are neither fermions nor bosons, but like fermions, they cannot occupy the same state. Thus, the world lines of two anyons cannot intersect or merge, which allows their paths to form stable braids in space-time. Anyons can form from excitations in a cold, two-dimensional electron gas in a very strong magnetic field, and carry fractional units of magnetic flux. This phenomenon is called the fractional quantum Hall effect. In typical laboratory systems, the electron gas occupies a thin semiconducting layer sandwiched between layers of aluminium gallium arsenide.

When anyons are braided, the transformation of the quantum state of the system depends only on the topological class of the anyons' trajectories (which are classified according to the braid group). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories. In 2005, Sankar Das Sarma, Michael Freedman, and Chetan Nayak proposed a quantum Hall device that would realize a topological qubit. In a key development for topological quantum computers, in 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou claimed to have created and observed the first experimental evidence for using a fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. It should also be noted that non-abelian anyons, a species required for topological quantum computers, have yet to be experimentally confirmed. Possible experimental evidence has been found,[1] but the conclusions remain contested.[2]

Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum Turing machine model[citation needed]. That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.

To live up to its name, a topological quantum computer must provide the unique computation properties promised by a conventional quantum computer design, which uses trapped quantum particles. Fortunately in 2002, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do.[3]

They found that a conventional quantum computer device, given an error-free operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.

Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. Simulating the dynamics of a topological quantum computer may be a promising method of implementing fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf, Harrington, and Goyal have studied one model, with promising simulation results.[4]

One of the prominent examples in topological quantum computing is with a system of fibonacci anyons.[5] These anyons can be used to create generic gates for topological quantum computing. There are three main steps for creating a model:

Fibonacci Anyons are defined by three qualities:

The last fusion rule can be extended this to a system of three anyons:

Thus, fusing three anyons will yield a final state of total charge {displaystyle tau } in 2 ways, or a charge of 1 {displaystyle 1} in exactly one way. We use three states to define our basis.[6] However, because we wish to encode these three anyon states as superpositions of 0 and 1, we need to limit the basis to a two-dimensional Hilbert Space. Thus, we consider only two states with a total charge of {displaystyle tau } . This choice is purely phenomenological. In these states, we group the two leftmost anyons into a 'control group', and leave the rightmost as a 'non-computational anyon'. We classify a | 0 {displaystyle |0rangle } state as one where the control group has total 'fused' charge of 1 {displaystyle 1} , and a state of | 1 {displaystyle |1rangle } has a control group with a total 'fused' charge of {displaystyle tau } . For a more complete description, see Nayak.[6]

Following the ideas above, adiabatically braiding these anyons around each-other with result in a unitary transformation. These braid operators are a result of two subclasses of operators:

The R matrix can be conceptually thought of as the topological phase that is imparted onto the anyons during the braid. As the anyons wind around each-other, they pick up some phase due to the Aharonov-Bohm effect.

The F matrix is a result of the physical rotations of the anyons. As they braid between each-other, it is important to realize that the bottom two anyonsthe control groupwill still distinguish the state of the qubit. Thus, braiding the anyons will change which anyons are in the control group, and therefore change the basis. We evaluate the anyons by always fusing the control group (the bottom anyons) together first, so exchanging which anyons these are will rotate the system. Because these anyons are non-abelian, the order of the anyons (which ones are within the control group) will matter, and as such they will transform the system.

The complete braid operator can be derived as:

B = F 1 R F {displaystyle B=F^{-1}RF}

In order to mathematically construct the F and R operators, we can consider permutations of these F and R operators. We know that if we sequentially change the basis that we are operating on, this will eventually lead us back to the same basis. Similarly, we know that if we braid anyons around each-other a certain number of times, this will lead back to the same state. These axioms are called the pentagonal and hexagonal axioms respectively as performing the operation can be visualized with a pentagon/hexagon of state transformations. Although mathematically difficult,[7] these can be approached much more successfully visually.

With these braid operators, we can finally formalize the notion of braids in terms of how they act on our Hilbert space and construct arbitrary universal quantum gates.

Explicit braids that perform particular quantum computations with Fibonacci anyons have been given by [8]

See the original post:

Topological quantum computer - Wikipedia

- Solving problems by working together: Could quantum computing hold the key to Covid-19? - ITProPortal - July 2nd, 2020
- Spain Introduces the World's First Quantum Phase Battery - News - All About Circuits - July 2nd, 2020
- Professor tackles one more mystery about quantum mechanics and times flow - GeekWire - July 2nd, 2020
- This Week's Awesome Tech Stories From Around the Web (Through June 27) - Singularity Hub - June 29th, 2020
- Kudos: Read about faculty, staff and student awards, appointments and achievements - Vanderbilt University News - June 29th, 2020
- This Is the First Universal Language for Quantum Computers - Popular Mechanics - June 21st, 2020
- Universal Quantum raises $4.5 million to build a large-scale quantum computer - VentureBeat - June 17th, 2020
- Ethereum (ETH) Might Not have Quantum Resistance on its Roadmap, the QRL Team Reveals - Crowdfund Insider - June 17th, 2020
- Craig Knoblock Named Michael Keston Executive Director of the USC Information Sciences Institute - USC Viterbi School of Engineering - June 17th, 2020
- European quantum computing startup takes its funding to 32M with fresh raise - TechCrunch - June 11th, 2020
- SKT to expand use of new quantum-powered security solutions - The Korea Herald - June 11th, 2020
- Archer looks to commercialisation future with graphene-based biosensor tech - ZDNet - June 11th, 2020
- Dear NASA, please put a particle collider on the Moon - The Next Web - June 11th, 2020
- Top 10 emerging technologies of 2020: Winners and losers - TechRepublic - June 11th, 2020
- When Will Quantum Computing Come to Mainstream? - Analytics Insight - June 8th, 2020
- University announces 2020 winners of Quantrell and Graduate Teaching Awards - UChicago News - June 8th, 2020
- Physicists Found a Way to Save Schrdingers Cat - Dual Dove - June 8th, 2020
- Physicists hunt for room-temperature superconductors that could revolutionize the world's energy system - The Conversation US - June 3rd, 2020
- Covid 19 Pandemic: Quantum Computing Technologies Market 2020, Share, Growth, Trends And Forecast To 2025 - 3rd Watch News - May 24th, 2020
- Molecular dynamics used to simulate 100 million atoms | Opinion - Chemistry World - May 23rd, 2020
- Highest-performing quantum simulator IN THE WORLD delivered to Japan - TechGeek - May 18th, 2020
- Light, fantastic: the path ahead for faster, smaller computer processors - News - The University of Sydney - May 18th, 2020
- Wiring the quantum computer of the future - Space Daily - April 29th, 2020
- Technologies That You Can Explore Other Than Data Science During Lockdown - Analytics India Magazine - April 29th, 2020
- Will Quantum Computing Really Change The World? Facts And Myths - Analytics India Magazine - April 23rd, 2020
- Google's top quantum computing brain may or may not have quit - Fudzilla - April 23rd, 2020
- On the Heels of a Light Beam - Scientific American - April 23rd, 2020
- Advanced Encryption Standard (AES): What It Is and How It Works - Hashed Out by The SSL Store - Hashed Out by The SSL Store - April 23rd, 2020
- Google's Head of Quantum Computing Hardware Resigns - WIRED - April 21st, 2020
- COVID-19: Quantum computing could someday find cures for coronaviruses and other diseases - TechRepublic - April 21st, 2020
- The future of quantum computing in the cloud - TechTarget - April 21st, 2020
- Quantum computer chips demonstrated at the highest temperatures ever - New Scientist News - April 17th, 2020
- Alex Garland on 'Devs,' free will and quantum computing - Engadget - April 14th, 2020
- RAND report finds that, like fusion power and Half Life 3, quantum computing is still 15 years away - The Register - April 12th, 2020
- Quantum computing: When to expect the next major leap - TechRepublic - April 12th, 2020
- Cambridge Quantum Computing Performs the World's First Quantum Natural Language Processing Experiment - Quantaneo, the Quantum Computing Source - April 12th, 2020
- The Well-matched Combo of Quantum Computing and Machine Learning - Analytics Insight - March 23rd, 2020
- Picking up the quantum technology baton - The Hindu - March 23rd, 2020
- Research by University of Chicago PhD Student and EPiQC Wins IBM Q Best Paper - HPCwire - March 23rd, 2020
- Honeywell Achieves Breakthrough That Will Enable The Worlds Most Powerful Quantum Computer #47655 - New Kerala - March 23rd, 2020
- Is time broken? Physicists filmed a quantum measurement but the 'moment' was blurry - The Next Web - March 5th, 2020
- What Is Quantum Computing? The Next Era of Computational ... - March 3rd, 2020
- Honeywell says it will soon launch the worlds most powerful quantum computer - TechCrunch - March 3rd, 2020
- Majority of Promising AI Startups Are Still Based in the US - Transport Topics Online - March 3rd, 2020
- 10 Revolutionary Technologies To Lookout For In 2020 - Fossbytes - March 3rd, 2020
- Quantum researchers able to split one photon into three - Space Daily - March 3rd, 2020
- Physicists Captured The Moment That An Atom Enters Quantum Measurement - Somag News - February 29th, 2020
- This Week's Awesome Tech Stories From Around the Web (Through February 29) - Singularity Hub - February 29th, 2020
- IC Breakthroughs: Energy Harvesting, Quantum Computing, and a 96-Core Processor in Six Chiplets - News - All About Circuits - February 29th, 2020
- Top 10 Strategic Technology Breakthroughs That Will Transform Our Lives - Analytics Insight - February 29th, 2020
- New Intel chip could accelerate the advent of quantum computing - RedShark News - February 28th, 2020
- Particle accelerator technology could solve one of the most vexing problems in building quantum computers - Fermi National Accelerator Laboratory - February 28th, 2020
- Top 10 breakthrough technologies of 2020 - TechRepublic - February 28th, 2020
- 21st ISQED Conference to Commence With Focus on Quantum Computing, Security, and AI/ML & Electronic Design - PRNewswire - February 25th, 2020
- NTT Research to Collaborate with UCLA and Georgetown on Cryptography and Blockchain - Yahoo Finance - February 25th, 2020
- Should decision makers be concerned by the threat of quantum? - Information Age - February 25th, 2020
- Keeping classified information secret in a world of quantum computing - Bulletin of the Atomic Scientists - February 11th, 2020
- A neural network that learned to predict the behavior of a quantum system - Tech Explorist - February 9th, 2020
- Deltec Bank, Bahamas A combination of Quantum Computing and Blockchain Technology Will Have a huge Impact on Banking - Press Release - Digital... - February 5th, 2020
- Could Photonic Chips Outpace the Fastest Supercomputers? - Singularity Hub - February 5th, 2020
- Google claims to have invented a quantum computer, but IBM begs to differ - The Conversation CA - January 22nd, 2020
- Xanadu Receives $4.4M Investment from SDTC to Advance its Photonic Quantum Computing Technology - Quantaneo, the Quantum Computing Source - January 22nd, 2020
- U of T's Peter Wittek, who will be remembered at Feb. 3 event, on why the future is quantum - News@UofT - January 17th, 2020
- Quantum Computing Technologies Market 2019, Size, Share, Global Industry Growth, Business Statistics, Top Leaders, Competitive Landscape, Forecast To... - January 17th, 2020
- This Week In Security: Windows 10 Apocalypse, Paypal Problems, And Cablehaunt - Hackaday - January 17th, 2020
- Kitchener's Angstrom Engineering is making a quantum leap with its next-generation technology - TheRecord.com - January 17th, 2020
- Xanadu Receives $4.4M Investment from SDTC to Advance its Photonic Quantum Computing Technology - Yahoo Finance - January 16th, 2020
- The dark side of IoT, AI and quantum computing: Hacking, data breaches and existential threat - ZDNet - January 16th, 2020
- 'How can we compete with Google?': the battle to train quantum coders - The Guardian - January 16th, 2020
- IBM heads US patent list for 27th consecutive year - Technology Decisions - January 16th, 2020
- New Technique May Be Capable of Creating Qubits From Silicon Carbide Wafer - Tom's Hardware - January 14th, 2020
- The hunt for the 'angel particle' continues - Big Think - January 13th, 2020
- How to verify that quantum chips are computing correctly - MIT News - January 13th, 2020
- Googles Quantum Supremacy will mark the End of the Bitcoin in 2020 - The Coin Republic - January 13th, 2020
- Bleeding edge information technology developments - IT World Canada - January 13th, 2020
- Jeffrey Epstein scandal: MIT professor put on leave, he 'failed to inform' college that sex offender made donations - CNBC - January 10th, 2020
- The teenager that's at CES to network - Yahoo Singapore News - January 10th, 2020
- AI, ML and quantum computing to cement position in 2020: Alibabas Jeff Zhang - Tech Observer - January 8th, 2020
- Perspective: End Of An Era | WNIJ and WNIU - WNIJ and WNIU - January 8th, 2020
- Volkswagen carried out the world's first pilot project for traffic optimization with a quantum computer - Quantaneo, the Quantum Computing Source - January 6th, 2020

## Recent Comments