nLab topological quantum computation

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Contents

Context

Computability

constructive mathematics, realizability, computability

intuitionistic mathematics

propositions as types, proofs as programs, computational trinitarianism

Constructive mathematics

Realizability

Computability

Quantum systems

quantum logic


quantum physics


quantum probability theory – observables and states


quantum information


quantum computation

qbit

quantum algorithms:


quantum sensing


quantum communication

Topological physics

Contents

Idea

The idea of topological quantum computation is to implement quantum computation on quantum systems whose dynamics is described by topological quantum field theory (TQFT), so that the defining invariance of TQFTs under local perturbations implements protection of the quantum coherence by fundamental physical principles, instead of after the fact by quantum error correction.


General idea


The Problem of Contemporary Quantum Computing.

Common quantum computing architectures

(such as based on superconducting qbits, trapped ions, spin resonance, …)

suffer from an intrinsic tension:

  1. quantum gates are implemented via interaction between subsystems,

  2. but coherence requires avoiding interaction.

(cf. CCEHRSZ 07 p 272: “Quantum logic gates involving two atomic qubits must overcome the problem of the short range coherent interaction of neutral atoms, while maintaining atom confinement and suppressing decoherence. The main challenge is to perform the gate sufficiently fast compared to typical decoherence and relaxation times.”)

This problem haunts contemporary NISQ devices (cf. HHT 23),

whence the “Quantum Winter” (cf. McKenzie 24).


The bold idea of Topological Quantum Computing is to cut this Gordian knot:

Find quantum gates operating without interaction!

Can this work? In principle: Yes!

(By a phenomenon known as the “quantum adiabatic theorem”.)


For this we need a quantum system (say a crystalline quantum material)

with the following properties:

  1. ground states degenracy:

    even when completely frozen at absolute zero temperature

    the system has more than one state to be in (even up to phase)

    (Meaning: The Hilbert space ℋ\mathscr{H} of quantum states with energy eigenvalue E=0E = 0 is of dimension ≥2\geq 2.)

  2. an energy gap ϵ>0\epsilon \gt 0:

    every excitation of the ground state has energy larger than ϵ\epsilon

    so that (light) quanta impacting the material have no interaction as long as they carry energy <ϵ\lt \epsilon.

    (Meaning: The material’s Hamiltonian has a spectral gap above the ground state.)

  3. control parameters:

    the above properties hold for a range of continuously tunable parameters

    such as strain or voltage applied to the crystal

    (Meaning: We have vector bundle of ground state Hilbert spaces ℋ p\mathscr{H}_p over a topological space PP of parameter values p∈Pp \in P.)


Adiabatic Transformation of Ground States.

Under these conditions:

  • after cooling such a system to its ground state,

  • sufficiently slow (“adiabatic”) variation of the external parameters

  • does not excite the system: it remains in a ground state

  • but the different ground states may transform into each other

    (Meaning: Each parameter path γ:p→p′\gamma \,\colon\, p \to p' induces a Berry phase unitary operator U γ:ℋ p→ℋ p′U_\gamma \,\colon\, \mathscr{H}_p \xrightarrow{\;} \mathscr{H}_{p'}.)


This is part of a general phenomenon of quantum physics:

While quantum fluctuations are a little like thermal flcutuations

one key difference is that they remain present at absolute zero.

(Compare quantum phase transitions.)


Such operations on ground states are called holonomic quantum gates.

(From “holonomy” for the parallel transport of a Berry connection along loops.)

These are protected from external noise quanta of energy <ϵ\lt \epsilon

but may still be sensitive no noise in the parameter paths.


Topological invariance.

To overcome this last issue, look for such quantum systems which in addition have:

  1. parameter topology

    The parameter space has “holes”, in that

    some closed parameter paths cannot continuously be deformed to constant paths.

    (Meaning: The fundamental group of parameter space is non-trivial.)

  2. local parameter independence

    All parameter paths with the same endpoints

    that are continuous deformations of each other

    yield the same transformation on ground states.

    (Meaning: The Berry connection is flat.)

Quantum materials with all these properties are topologically ordered;

the resulting adiabatic quantum processes are topological quantum gates.


In principle, such topological quantum gates are:

  1. protected against external noise quanta of energy <ϵ\lt \epsilon, and

  2. protected against any noise in the parameter path.

In fact, the quantum operation induced by a parameter variation will

depend only on the discrete data of

how much the path winds around the holes of parameter space


This way,

topological quantum architecture may in principle

(be necessary to) solve the problem of quantum computing.

(cf. Sau 2017, Das Sarma 2022)


So far this is the theory behind topological quantum gates.

On the one hand it is extremely general:

  • any kind of topologically parameterized quantum system could do.

On the other hand it is very ambitious:

  • suitable such system have yet to be devised in the labs.


But the range of possibilities has hardly been explored,

most attention has been given to a single approach:

  • braiding of anyonic defects in position space.


That’s what we discuss next.


Via anyon braiding

In specialization of the above general idea, the

original proposal due to Kitaev 2003 and FKLW 2003 (which has become canonized in the literature) is to envision

The motion in such a parameter space is a braid (of “worldlines” of defect points) and

if this acts non-trivially on the material’s ground states by adiabatic transformations,

then one refers to these defects as defect anyons:

(graphics from SS24)


Status of anyonic quantum computation.

Experimentally,

at least abelian anyons

(whose braid representation factors through a representation of the symmetric group)

are seen in fractional quantum Hall systems,

though their controlled movement along the above lines seems out of reach.


Theoretically,

such defect anyons are expected to be effectively described by

some kind of Chern-Simons theory/Reshetikhin-Turaev theory

(and hence ultimately classified by modular tensor categories),

though attempts to derive some of these expectations from microscopic physics are rare.

(A more microscopic argument for anyonic defects not in position-space but in “reciprocal” momentum-space – the Brillouin torus –, appearing there as the familiar band nodes of topological semimetals, is made in SS23.)


The open problem of strongly-coupled quantum systems.

A general problem on both of these fronts is that

anyonic topological order is supposed to arise in strongly correlated systems which, like

all non-perturbative physics, remains ill-understood in general.


The directly analogous problem in particle physics,

where the the energy gap is known as the mass gap,

has been termed a Millennium Problem by the Clay Math Institute

(cf. the Mass Gap Problemap#ReferencesMassGapProblem))


One plausible approach on this front is to find

geometric engineering of anyonic topological order on M-branes

or their holographic dual bulk gravity.

(cf. CGK20, SS24)


Loosely related approaches.

Much attention in the 2010s had been given to claims of experimental detection of anyons in the form of “Majorana zero modes” (MZMs) – but these claims seem not to hold water. In any case, these MZMs are by design stuck at the end of nanowires and hence are not movable and in particular not braidable in the above sense.

More recently the idea of quantum simulation of anyons on ordinary quantum hardware has found more attention, though the relevance of this, if any, to the original idea of quantum-error protection by fundamental physical principles may remain to be discussed.


Topological quantum protocols.

Topological quantum computation protocols with defect anyons are often assumed to start by creating anyon/anti-anyon pairs out of the “anyon vacuum”, then braid their worldlines and finally annihilate them again — so that the total process is described by a link which, when regarded as a Wilson loop, may be understood as parameterizing a quantum observable of the Chern-Simons theory mentioned above:

Adapted from Rowell Wang 18, Rouabah 20


In any case, in this scheme the topological quantum gates are encoded by braid group-elements and are executed by actions through braid representations on the space of quantum states:

(from Sati-Schreiber 21)
(from Lahtinen-Pachos 17)


Topological Quantum Compilation.

This means that quantum gates based on anyon braiding are (going to be) quite different from the standard quantum gates traditionally considered in qbit-based quantum circuits.

For example, the following lengthy braid has been proposed [Hormozi, Zikos, Bonesteel & Simon (2007)] as a possible topological implementation of a single CNOT gate:

Hence if and when such topological quantum hardware becomes available, a major issue on the quantum software side will be the compilation of quantum algorithms to the exotic-seeming machine-level gates (cf. topological quantum compilation).

It has been argued that the complexity of this process will make its formal verification a practical necessity.


Extended TQC?

It may be interesting to note that:

from Sati-Schreiber 2021

Here

This means that while every individual loop in Conf N(ℝ 3)Conf_N(\mathbb{R}^3) is homotopically trivial (all “braid-gates” are equivalent) there is now non-trivial structure in higher-dimensional deformation families of braids (which is absent in Conf N(ℝ 2)Conf_N(\mathbb{R}^2)). Such structure would be reflected by extended TQFT.

References

Need for topological protection

Highlighting the need for topological stabilization mechanisms:

  • Jay Sau, A Roadmap for a Scalable Topological Quantum Computer, Physics 10 68 (2017)

    “small machines are unlikely to uncover truly macroscopic quantum phenomena, which have no classical analogs. This will likely require a scalable approach to quantum computation. […] based on […] topological quantum computation (TQC) as envisioned by Alexei Kitaev and Michael Freedman […] The central idea of TQC is to encode qubits into states of topological phases of matter. Qubits encoded in such states are expected to be topologically protected, or robust, against the ‘prying eyes’ of the environment, which are believed to be the bane of conventional quantum computation.”

  • Sankar Das Sarma, Quantum computing has a hype problem, MIT Technology Review (March 2022)

    “The qubit systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. […] What is missing is the breakthrough […] bypassing quantum error correction by using far-more-stable qubits, in an approach called topological quantum computing.”

Topological quantum computation with anyons

The idea of topological quantum computation via a Chern-Simons theory with anyon braiding defects is due to:

and via a Dijkgraaf-Witten theory (like Chern-Simons theory but with discrete gauge group):

Monographs:

Review:

Focus on abelian anyons:

  • Jiannis K. Pachos, Quantum computation with abelian anyons on the honeycomb lattice, International Journal of Quantum Information 4 6 (2006) 947-954 [doi:10.1142/S0219749906002328, arXiv:quant-ph/0511273]

  • James Robin Wootton, Dissecting Topological Quantum Computation, PhD thesis, Leeds (2010) [ethesis:1163, pdf, pdf]

    “non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. […] Though universal non-Abelian models are admittedly the holy grail of topological quantum computation, and rightly so, this thesis has shown that Abelian models are just as useful as non-universal non-Abelian models. […] Abelian models are a computationally powerful, fault-tolerant and experimentally realistic prospect for quantum computation.”

  • Seth Lloyd, Quantum computation with abelian anyons, Quantum Information Processing 1 1/2 (2002) [doi:10.1023/A:1019649101654, arXiv:quant-ph/0004010]

  • James R. Wootton, Jiannis K. Pachos: Universal Quantum Computation with Abelian Anyon Models, Electronic Notes in Theoretical Computer Science 270 2 (2011) 209-218 [doi:10.1016/j.entcs.2011.01.032, arXiv:0904.4373]

see also:

Realization in experiment (so far via quantum simulation of anyons on non-topological quantum hardware, cf. FF24, Fig 5, as in “topological codes” for quantum error correction):

on superconducting qbits:

on trapped-ion quantum hardware:

Discussion of anyon braid gates via homotopy type theory:

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

Review:

in relation to modular tensor categories:

  • Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:

and understood in terms of anyon statistics:

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

See also:

  • R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary qq and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

Introduction and review:

Realization of Fibonacci anyons on quasicrystal-states:

Realization on supersymmetric spin chains:

  • Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains [[arXiv:2209.03822]]

See also:


Compilation to braid gate circuits

On approximating (cf. the Solovay-Kitaev theorem) given quantum gates by (i.e. compiling them to) cicuits of anyon braid gates (generally considered for su(2)-anyons and here mostly for universal Fibonacci anyons, to some extent also for non-universal Majorana anyons):

Approximating all topological quantum gates by just the weaves among all braids:

Anyons in quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):

Review:

Claims of experimental observation:

Anyons in topological superconductors

On anyon-excitations in topological superconductors.

via Majorana zero modes:

Original proposal:

  • Nicholas Read, Dmitry Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Phys. Rev. B61:10267, 2000 (arXiv:cond-mat/9906453)

Review:

  • Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana Zero Modes and Topological Quantum Computation, npj Quantum Information 1, 15001 (2015) (nature:npjqi20151)

  • Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview (arXiv:2009.07764)

  • Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors &lbrack;arXiv:2301.11614&rbrack;

Further developments:

via Majorana zero modes restricted to edges of topological insulators:

  • Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, Topological quantum computation based on chiral Majorana fermions, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 (doi:10.1073/pnas.1810003115)

See also:

  • Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors [[arXiv:2301.11614]]

Last revised on December 18, 2024 at 08:24:24. See the history of this page for a list of all contributions to it.

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