It looks like you're asking about this part of the paper:
Therefore, a quantum computation is hidden as long as these measurements are successfully hidden. In order to achieve this, the BQC protocol exploits special resources called blind cluster states that must be chosen carefully to be a generic structure that reveals nothing about the underlying computation (see Figure 1).
-"Experimental Demonstration of Blind Quantum Computing" (2011)
That last part, about how they want a "generic structure that reveals nothing about the underlying computation" might make a reader wonder about how a computer's structure could leak information about its computations.
As a simple example of structure leaking information about a cypto scheme, suppose that Bob asks Sally question to which we assume that Sally'll respond
no. Sally directly encrypts her response using their shared one-time pad (OTP), resulting in the ciphertext
rk4. Despite the OTP scheme having perfect secrecy in general, it's clear that Sally responded
In this case, the computer was structured to leak information about the length of a message given that message, which was especially disastrous in this contrived example. In general, structure can leak information about the computation. Avoiding such leaks would be necessary for a blind-computation server like the one the paper intends to discuss.
Generally speaking, attacks that operate like this are called side-channel attacks.
In the case of this paper (disclaiming that I just skimmed it quickly), it looks like they're basically talking about creating a generic computational structure that doesn't leak information through its structural traits. For example, if the structure behaved differently in any way based on a secret aspect of the message, then it may leak secret information to the server when the server observes its own computational behavior.
The paper appears to be trying to point out that the computational unit needs to be designed to avoid such information leaks.
Later in the paper, they discuss stuff about blinding:
In cryptography, blinding is a technique by which an agent can provide a service to (i.e., compute a function for) a client in an encoded form without knowing either the real input or the real output. Blinding techniques also have applications to preventing side-channel attacks on encryption devices.
-"Blinding (cryptography)", Wikipedia
And, really, blinding's what this paper's all about: figuring out a way to have a server do work for clients without the clients revealing their secrets to the server.
One way to enable blind computation is for the client to use homomorphic encryption on its job request before sending it to the server:
Homomorphic encryption is a form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on the plaintext. The purpose of homomorphic encryption is to allow computation on encrypted data.
-"Homomorphic encryption", Wikipedia