拓撲量子計算的前景？

01-01

PhD想做一些相關的方向，本人主要做理論和計算的（實驗殘廢），有人了解這個坑裡還有哪些問題有待解決嗎？

Introduction. --- Braiding Majorana Bound States(MBS) can be used as operation to switch within the ground state manifold and thus realizes topological quantum computation. In 1D, the kitaev spinless p-wave superconductor as a toy model can hold MBS. However, in reality, we don"t have spinless electrons, and p-wave superconducting pairing is also not so common ! Then, how do we realize them in experiment ?

Realization in 1D. --- The idea is simple:

spinless &<== spin polarized
p-wave &<== s-wave + spin-orbit coupling (OC)

There are mainly 3 theoretical proposals:

quantum spin hall edge + s-wave superconductor in L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
semiconductor (SOC) quantum wires + s-wave superconductor: Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010); R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010). J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011).
magnetic adatms on s-wave superconductor (with SOC): S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Phys. Rev. B 88, 020407(R) (2013); F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 88, 155420 (2013).

All of these proposals are being actively pursued in the laboratory !

Signatures for Detection. -- How do you know you realize MBS ?

One idea is to do braid experiment, in terms of interferometer.This is hard.
Transport experiment, using STM. Since MBS is at zero energy, a zero bias peak (in fact, it is quantized to $2e^2/h$ ) in a normal STM measurement can be a signature. But there are problems: in experiment the zero bias peak gets broadened by temperature, and never reaches the quantized value. Moreover, how do you know there are no other zero energy state which is not the MBS ? Here I recently have a paper in adressing this issue: http://arxiv.org/abs/1506.06763, we propose using superconducting STM, you can get robust signatures distinct from trivial zero energy state.
Still use STM, but one can in real space map out the Majorana wave function, to see the localization nature in real space ! Recently, the Princeton group had a great experiment: S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014). They found a very localized state at the end, and claimed to MBS. At first, people don"t understand why MBS can be very localized (at length much shorter then the superconductor coherence length !) It was solved this problem in my paper: Yang Peng, Falko Pientka, Leonid I. Glazman, and Felix von Oppen, PRL 114 106801. We proposed a Fermi velocity renormalization mechanism by proximity effect. This mechanism is very important for supproting localized MBS. Only with short localization length, the quantum information can be protected !
Topological Josephson Junction. the 4pi-josephson effect can be used, but it is also a bit delicate. But there are more signatures: I"m currently finish writing a paper on this topic.

Braiding Majoranas

In principle, you need 2D system to do braiding. However, you can also do braiding with a lot of 1D wires ! For example, using the T-shape junction in my advisor"s paper: Nature Physics 7, 412–417 (2011), Phys. Rev. B 91, 201102(R), 2015

The main issue that: when you move the Majoranas at a finite speed, you may generate quasiparticle excitations, which may lead to computational error. The above recent PRB(R) provided a way of error correction.
One can also use 2D p+ip superconductor, by moving vortices which bind MBS.

Beyond Majoranas

Although MBS is now a very promising candidate for TQC, there is a drawback. Braiding Majoranas cannot generate all kinds of unitary operation (known as the condition for universal quantum computation), hence one needs to use it with some non topologically protected operations.
The simplest Anyon that fit for universal TQC is the so-called Fibonacci anyon (the quantum dimension of N such anyons is Nth Fibonacci number). It was predicted that they appear in $u=13/5$ fractional quantum hall system (FQHS) (Read-Rezayi state), but it was not experimentally verified yet.
Another problem using FQHS is that the anyonic excitations are deconfined, rather than in p(+ip) superconductor the Majoranas are confined.
However, recently, there is a good proposal: using simple(abelian, e.g. $u=2/3$ ) fractional quantum hall liquid proximity coupled to superconductor:Phys. Rev. X 4, 011036 2015. They found out that in this system, besides the deconfined Fibonacci anyons, there are also FIbonnaci anyons bound to vortex flux, which are easier to operate.

Take-home Message

Close to the realization of MBS, far away from universal topological quantum computation.

感覺上最大的坑是現在只能找到 Ising Anyon 對應的物理系統(FQHE 和 Quantum wire)吧, 然而 Ising Anyon 只能實現 Clifford 門. 而要實現通用量子計算, 我們需要實現 Clifford+T, 也就是說至少找到 Fibonacci Anyon 對應的物理系統.

另外, 有個相關的工作. 大概十年前(?) Stephen Jordan 證明了估計某種 Jones Polynomial 是 DQC1-Complete 的. 2012年的時候還有人做過物理實現( DQC1 現在幾乎所有量子計算的物理系統都能實現). 聯繫的話, 因為 BQP-Complete 通常都是用各種量子計算模型對應的問題直接定義的, 其他的反而很難找(比如 Scott Aaronson 09年提出的 k-forrelation problem).

除此之外, 線路綜合或者 Error Correction 感覺並沒有什麼坑了. 前者這兩年似乎已經到了理論下界(Solovay-Kitaev 定理), 看起來更像是個工程問題. 何況改進到這個下界的 PhD 正在 MSR 的 QuArC group 做 postdoc 呢...

先佔個坑，但這個問題很難說。

這個問題暫時還在理論層面，現時正在找一個物理系統可以實現的。這是一個挑戰，因為實現這個要用anyons，其統計特性是介乎bosons和fermions之間的，有topological的特性。這些粒子不會是基本粒子，應該是准粒子（quasi-particles），那就看系統的excitations。早前有人說Majorana fermions是一個可能性，不過我不知道現在發展成怎樣。

這方面做得較活躍的可算是Microsoft Q-Station和馬里蘭大學的JQI和CMTC。

這個東西還有很多坑要填吧，這些坑不是量子計算方面的，而是凝聚態方面的，門檻較高。我懂的大概道途說的，還請高人補充。