29
votes
Accepted
Oracle Construction for Grover's Algorithm
The oracle is basically just an implementation of the predicate you want to search for a satisfying solution to.
For example, suppose you have a 3-sat problem:
...
- 1,498
19
votes
Accepted
Entropy and computational complexity
Yes, but most of the work so far (except very recently, see below) has focused on turning irreversible computations into reversible ones, thereby hoping to avoid any entropy generation. (Note: there ...
- 36.9k
15
votes
Generating "infinite" randomness from a constant number of sources
This is a great question, Suresh!
Our randomness expansion result does not imply any complexity theoretic result. Here's one way to understand the result: we believe that quantum mechanics governs ...
- 3,758
9
votes
Accepted
Where can I find examples of error correcting codes of the following types?
If you just need any code $E : \{0,1\}^n \to \{0,1\}^m$ where $m=O(n)$ and where the distance is linear in $m$, then what you are looking for is called an "asymptotically good code". There are many ...
- 5,350
8
votes
Accepted
theorems for universal set of quantum gates for SU(d)
I'm not aware of any proof that the Clifford group + any non-Clifford element gives a universal set of quantum gates. The closest related result that I know is that the Clifford group + any non-...
- 743
8
votes
Accepted
Quantum Money where not even the Bank can counterfeit
There are proposals for quantum money where it appears that not even the bank can produce two copies of a quantum money state with the same serial number. See
Farhi et al's paper Quantum Money from ...
- 24.3k
7
votes
Accepted
Is it possible to encrypt quantum states under reasonable assumptions?
One can encrypt an n-qubit state using a 2n-bit classical secret key. The idea is to use the key to select a random Pauli operator, and apply that operator to the secret as an encryption. (The inverse ...
- 878
7
votes
States and Probability distributions that the 5-qubits IBM computer can produce
I just ran the first state you suggest (i.e. the GHZ state with negative phase in the Hadamard basis). Basically what I did was to write a circuit which creates that state, apply one of 5 stabilizer ...
- 13.6k
7
votes
Accepted
Fast classical simulation of quantum algorithms
Your question was inspired by the recent quantum-inspired classical advance in recommendation algorithm. Note that it is not the firs time such a thing happens. In 2015, similar developments happened ...
6
votes
Accepted
Why is it impossible to work with polylog length encoding schemes for quantum circuits?
Clearly you can work with abstract compressed representations of circuits. You can reason about them and manipulate them and turn them into concrete lists of gates. We do it all the time.
But in ...
- 1,498
6
votes
When is a non-unitary quantum system only theoretical?
Look at John Preskill's Lecture Notes; particularly Section 3.2.
As you noted, you can do a NAND gate by using a Toffoli gate and throwing away some of the output qubits. This results in decoherence, ...
- 24.3k
6
votes
Accepted
Simulating quantum measurements by unitaries
The basic idea here is that any operation that uses measurement can be replaced by an operation that instead CNOTs qubits onto ancillae.
Any circuit with an intermediate measurement can be converted ...
- 1,498
6
votes
How can I get $\sum_n e^{i a_n} |n\rangle$ from $\sum_n a_n |n\rangle$?
It's not unitary, so it's impossible because all quantum transformations have to be unitary.
Consider the states
$$
\frac{3}{5} |0\rangle + \frac{4}{5} |1\rangle \quad \mathrm{and} \quad \frac{3}{5} |...
- 24.3k
5
votes
Accepted
When is a non-unitary quantum system only theoretical?
I complete here Peter’s answer with a characterization of physical maps as CPTP maps.
As you know, if the system is isolated, the only operations you can implement are the unitary operations. But, ...
5
votes
Accepted
Is it possible to MAC a quantum state with a classical key under reasonable assumption?
Howard Barnum, Claude Crepeau, Daniel Gottesman, Adam Smith, Alain Tapp. "Authentication of Quantum Messages", FOCS 2002.
http://www.cse.psu.edu/~ads22/pubs/PS-CSAIL/BCGST02-focs-final.pdf
As with ...
- 878
5
votes
Accepted
Dimension of the Fourier transform for $S_5$
A quantum Fourier transform is a unitary operation, so the number of basis states of the input and output must be the same.
The number of basis states before the Fourier transform is 120, the number ...
- 24.3k
5
votes
Is it proved that error rate of quantum computation is bounded by constant rather than a function dependent on time and environment by quantum theory
As far as I know, nobody has come up with a convincing physics reason that fault-tolerant quantum computing is fundamentally impossible. However, it is a formidable engineering task, which is why we ...
- 24.3k
4
votes
Is there any connection between the diamond norm and the distance of the associated states?
Following up on the line of thinking presented by Alex Monras, there is actually a quite generic argument for this kind of bound that goes beyond diamond norm and applies to many other channel ...
- 141
4
votes
Accepted
From CHSH inequality to CHSH game
I think your history is entirely right. Computer scientists are used to thinking about protocols as games, while physicists are not, and this is probably why these results weren't formulated as games ...
- 24.3k
4
votes
Accepted
Question about discarding the second register in the standard approach of hidden subgroup algorithm
Discarding in this context can be carried out by performing a Partial Trace. The system you're interested in is that of the first register. If you want to know only its state, as opposed to the ...
- 977
3
votes
Accepted
Is there any hidden subgroup of a symmetric group which can be efficiently determined?
From what I understand, it is partially because we don't have any techniques currently that take advantage of structure of the hidden subgroup itself. Weak Fourier sampling solves the problem whenever ...
- 36.9k
3
votes
Accepted
Noisy channel coding theorem in quantum information
You have two questions here.
Why can't Shannon's Noisy Coding Theorem be used for a quantum channel?
What gaps are there to proving a quantum noisy channel coding theorem?
I will concentrate on the ...
- 24.3k
3
votes
Accepted
Quantum circuit simulation divergence in results
Peter Shor nailed the reason. This is because OpenQASM and Quirk have different conventions for how the order of wires maps into the order of bits shown in the kets. The simplest way to check this is ...
- 1,498
2
votes
Is there any connection between the diamond norm and the distance of the associated states?
If it should prove useful, this is a derivation of the inequality
$$
\frac{1}{n} \| \Phi_0 - \Phi_1 \|_{\Diamond} \leq
\| J(\Phi_0) - J(\Phi_1) \|_1
$$
in John Watrous' answer. We will make use of ...
- 21
2
votes
Accepted
Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?
I think your focus on the rigid case of GI limits you too much. Instead phrase (non-rigid) GI as an HSP in the same way, but now the goal is to determine the size of the hidden subgroup, or a ...
- 36.9k
2
votes
Accepted
How are new probabilities computed when simulating measurement on a set of qubits?
Let
$$P(0) = |a|^2 + |c|^2 + |e|^3 + |g|^2.$$
This is the probability of observing $0$.
Then
\begin{eqnarray*}
w &=& a \big/ \sqrt{P(0)}, \\
x &=& c \big/ \sqrt{P(0)}, \\
y &=&...
- 24.3k
2
votes
What are the general classes of measured systems
I don't think there will be any result of the form you want. There's no single "most general system" that has this property.
(Instead, it'd be better to choose some specific pair of classes of ...
- 11.7k
2
votes
How can I get $\sum_n e^{i a_n} |n\rangle$ from $\sum_n a_n |n\rangle$?
I wouldn't think so. I'm assuming your $a_n$'s are normalized, i.e. that $||\sum_n a_n |n\rangle|| =1$. But then your wish output is not normalized since $$||\sum_n e^{ia_n} |n\rangle|| = \sqrt{\sum_n ...
- 960
2
votes
Accepted
Computing 'Robustness of Magic' of $n$-bit W states
Here is a proof that $R(W_n) \in \Omega(\sqrt[4]{n})$.
Given a $W_n$ state, it is possible to create $\lg n$ T states with Clifford operations and a single additional T gate. Basically:
Xor together ...
- 1,498
2
votes
QPIP minimal client quantum capabilities
According to this paper:
https://arxiv.org/pdf/1509.09180.pdf
The client only needs the ability to prepare random single-qubit pure states.
You may also look at this paper for more information:
...
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