New answers tagged mathematics
2
votes
Accepted
Does every code have transversal Pauli group?
I don't think so - consider e.g. the 'diagonal' representation $$\phi:SU(2) \rightarrow GL((\mathbb{C}^2)^{\otimes 4})$$
$$U \mapsto U^{\otimes 4}. $$
The Clebsch-Gordan series tells us that, for spin ...
- 502
2
votes
Does every code have transversal Pauli group?
Every code that can be implemented by a stabilizer circuit (this includes stabilizer codes, gauge codes, floquet codes, etc) has this type of subset-transversal Pauli gate.
In such a code, the X, Y, ...
- 28.7k
0
votes
Accepted
Find min of a quantum state L2 norm
For unit vectors $|\psi\rangle, |\phi\rangle \in \mathbb{C^n}$ we have:
$$
||e^{i\alpha}|\psi\rangle-|\phi\rangle ||^2 = \left(e^{-\alpha i}\langle\psi|-\langle \phi| \right) \left(e^{i \alpha} |\psi\...
- 2,033
0
votes
Weight enumerators for Hermitian operator
This is just fleshing out some of the themes from Adam Zalcman's answer, which I have already accepted.
The standard proof that $ B_j\geq A_j $ for a projection $ H $ uses the spectral theorem to ...
- 2,200
2
votes
Accepted
Weight enumerators for Hermitian operator
TL;DR: No. We can actually blow $A_j$ up to infinity while simultaneously sinking $B_j$ negative.
Sneaky plan
Coefficients $A_j$ cannot be negative, but if $H$ squares to identity and anticommutes ...
- 18.2k
2
votes
Accepted
Weight enumerators for Hermitian operator (wrong $ B_j $ definition)
TL;DR: No. Suppose that $H=\alpha H'$ for some scale factor $\alpha\in\mathbb{R}$. The key observation is that $A_j$ is independent of $\alpha$, but $B_j$ is linear in $\alpha$. We can use this to ...
- 18.2k
3
votes
Why can all quantum circuits be converted into circuits that use only real matrices?
To give a more general perspective on this: in linear algebra in general you can replace complex numbers with real numbers by enlarging the space and suitably redefining operations.
A simple way is to ...
glS♦
- 21.2k
13
votes
Why can all quantum circuits be converted into circuits that use only real matrices?
The fact that "we need work only with quantum Turing machines (QTMs) with real-valued transitions" is proved by Bernstein and Vazirani in their paper Quantum complexity theory (1993).
...
- 7,109
4
votes
Why can all quantum circuits be converted into circuits that use only real matrices?
As Mark notes, the result is quite old, possibly dating back to the 90s. However, a quick search yields this paper, which briefly discusses this result. Then it cites this thesis for proofs.
Technical ...
- 658
3
votes
Accepted
Where am I going wrong in my understanding of qubit associativity?
Looks like a simple error in the left-associative calculation:
$$
\left(
\begin{pmatrix}
0\\
1\\
\end{pmatrix}
\otimes
\begin{pmatrix}
1\\
0\\
\end{pmatrix}
\right)
\otimes
\begin{pmatrix}
0\\
1\\
\...
- 303
4
votes
Accepted
Building universal gate set for $SU(d^n)$ from universal gate set for $SU(d)$
Let $ G $ be a finite set of gates, of size $ |G| $. Let $ <G> $ be the words of $ G $. That is, $ <G> $ is the group generated by $ G $. Suppose that $ <G> $ is dense in $ SU(d) $. ...
- 2,200
3
votes
Accepted
What is the difference between Gate.power() and Gate.repeat()?
The method Gate.repeat() accepts positive integers only. While Gate.power() accepts real values. So, you can use it, for eaxmple,...
- 7,109
2
votes
Accepted
Bounding operator norm by total variation distance
No you cannot, here's a counterexample.
Let $U=I$ be the identity matrix and let $S = \sum_{i} (-1)^{\delta_{0,i}} |i\rangle \langle i|$ where $\delta_{i,j}$ is the Kronecker delta. That is, $S$ is ...
- 4,516
1
vote
Modular Addition general explanation
To add to hft's great answer, addition modulo 2 is often used as a synonym for the bit-wise XOR function, as you can see from a truth table:
...
- 823
1
vote
Accepted
Modular Addition general explanation
What exactly does a modulo 2 addition stand for?
It means that numbers are considered congruent (often stated as "equal" or "equal... mod 2") when they have the same remainder ...
- 606
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