CHAPTER 9

Outlook and Open Problems

Machines have always made me uneasy. Thinking more, I realize my insecurity

comes from a fundamental distrust of machines. What will happen if powerful

machines take over our world?

I believe quantum information science will enable us to “see” the colorful quan-

tum world and will bring us exciting new technologies. Elaborating on an idea of

M. Freedman and X.-G. Wen, we can consider wave functions as new numbers.

While place values have a linear array of holders for a fixed number of digits, wave

functions have Hilbert space bases as holders and complex numbers as digits. In

principle a Hilbert space basis can form any shape of any dimension, though bases

for qubits are linear arrays. It is bound that we can count more eﬃciently with wave

functions. Science makes a leap when we can count more things eﬃciently. Before

we start to count things with wave functions, we have to be able to control them.

Take one qubit as an example: we need to reach every point on the Bloch sphere

with arbitrary precision. This might be diﬃcult, but seems not impossible. As

a reminder to ourselves, I consider this endeavor as analogous to mountaineering:

reach every point on our sphere, such as the daunting K2. K2 has been conquered;

qubit states

CP2n−1

are the new frontier.

There are many open problems and new directions. Some are mentioned in the

earlier chapters; here we list a few more.

9.1. Physics

The central open problem in TQC is to establish the existence of non-abelian

anyons. The current proposal is to use the candidate materials to build a small

topological quantum computer [DFN]. More theoretical questions include:

(1) Define topological phases of matter. They are likely in 1-1 correspondence

with pairs (C,c), where C is a UMTC and c is a positive rational number

such that ctop = c mod 8.

(2) Develop a theory of phase transitions between topological phases of mat-

ter, especially from abelian to non-abelian phases.

(3) Study stability of topological phases of matter under realistic conditions

such as thermal fluctuations or finite temperatures.

(4) Develop tools to decide whether a given Hamiltonian has a gap in the

thermodynamical limit. There are many interesting model Hamiltonians

for wave functions in FQH states.

(5) Formulate mathematically and prove Holo=Mono (Sec. 8.3).

(6) More speculatively, extend Landau’s theory from group symmetry to fu-

sion category symmetry.

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