14 votes
Accepted

Does the Quantum Fourier Transform require universality?

Yes, the QFT requires universality. No, there isn't a non-universal gate set that implements the QFT. Just having the QFT as an operation is already computationally universal, because it can generate ...
Craig Gidney's user avatar
7 votes

How can quantum error correction correct small rotations/continuous errors?

The trick is to rewrite your continuous rotations as a perturbative sum. For example, consider applying $R_Z(\theta)$ to all data qubits of an $n$-qubit code. You can rewrite: $$R_Z(\theta) = I \cos \...
Craig Gidney's user avatar
6 votes

Is the $[\![7,1,3]\!]$ Steane code a surface code?

There is not "the" $[\![7, 1, 3]\!]$ code in the sense that there is "the" $[\![5, 1, 3]\!]$ code. While the Steane code is not a surface code (it is a color code, locally ...
Andrew J. Landahl's user avatar
5 votes
Accepted

Compute the exact minimum distance of a QECC with integer linear programming method

Here is python code that generates the stabilizers and observables of the IBM code, then produces a stim circuit measuring the Z stabilizers and observables after applying noise, then turns the ...
Craig Gidney's user avatar
4 votes

Can you produce Non-Clifford gates via Lattice Surgery?

Merge and split can be performed with stabilizer operations (up to feedback due to ZX defining merge to use postselection whereas real computers have measurement instead): Therefore if merge or split ...
Craig Gidney's user avatar
3 votes
Accepted

In context of stabilizer codes, are logical gates and Pauli operators the same?

Let $\mathcal{S}$ be the set of stabilizers. It is not true that $N(\mathcal{S}) = C(\mathcal{S})$. There can exist an operator $U$ such that for all $S\in \mathcal{S}$, $USU^\dagger = S'$ for some $S'...
rnva's user avatar
  • 772
3 votes
Accepted

how to construct a ((7,2,3)) code and verify its properties

This is Eric (other author). If $\lambda = \chi_3$ then $\lambda^* = \chi_3$ (not $\chi_2$). You check this explicitly in GAP using $$\texttt{ComplexConjugate(\lambda)=\lambda;}$$ which evaluates to $\...
Eric Kubischta's user avatar
3 votes
Accepted

How to convert stim circuit to a detector check matrix and a logical observable matrix?

You might want to consider encapsulating your decoder as a CustomDecoder to benefit from stim/sinter framework for your simulation. Other people have done this (e.g....
AG47's user avatar
  • 476
3 votes
Accepted

How can quantum error correction correct small rotations/continuous errors?

When you encode in an error correcting code, you select a subspace that your encoded qubit sits in. You can identify this with some projector $P$. For instance, if you know your logical 0 and logical ...
DaftWullie's user avatar
3 votes

What is the definition of physical gate error rate?

Here is a partial answer: Error rate is not a well defined quantity. In experimental papers, the numbers reported are most often measured using a technique called randomized benchmarking (RB) which ...
Frederik Ravn Klausen's user avatar
3 votes
Accepted

How to get smaller stabilizer codes with Gottesman's construction?

I think that Gottesman's proof implicitly assumes that the stabilisers of the code all have weight which is at least that of the distance of the code: his construction of the distance says that assume ...
DaftWullie's user avatar
2 votes

Is there a "simple" way to tell if a stabilizer code is degenerate?

A distance $ d $ code is nondegenerate (with respect to its set of correctable errors) if and only if the weight enumerator values $ A_i=0 $ for all $ 0<i<d $. This criterion has the advantage ...
Ian Gershon Teixeira's user avatar
2 votes

Why I am continuously getting "failed to decompose hyperedges into edges" in Stim?

It sounds like you don't want decomposition and so should set decompose_errors=False (the default in stim). What version of sinter are you using? When you don't ...
Craig Gidney's user avatar
2 votes

Can Interaction with the environment cause $|0\rangle$ to evolve into a superposition like $|+\rangle$?

It entirely depends on what "environment" and "interaction" you're talking about. As an extreme example, your "environment" could be a qubit in the state $|+\rangle$, and ...
glS's user avatar
  • 24.9k
2 votes
Accepted

Can Interaction with the environment cause $|0\rangle$ to evolve into a superposition like $|+\rangle$?

If the qubit is in connection with an environment, most likely it will develop quantum correlations (entanglement) and will become a mixed state. If the state is pure, the most general state is a ...
Zarathustra's user avatar
2 votes

Compute the exact minimum distance of a QECC with integer linear programming method

The author of the paper provided the implementation in the repository.
Inm's user avatar
  • 505
2 votes
Accepted

What are the main motivations for pursuing CSS codes over general non-CSS codes?

I think the main reason is just that they're simpler to analyze and to prove things about. Why do people look at stabilizer codes instead of considering non-stabilizer codes? Why do people treat "...
Craig Gidney's user avatar
1 vote

Are there standard forms for non-additive quantum codes?

If you impose some structure, you can get some type of standard form for things like codeword stabilized codes, which I think of as non-additive codes "lite". But at their heart, "...
squiggles's user avatar
  • 890
1 vote

Fault tolerance of turning off a stabilizer measurement for later rounds of error correction?

If you run this circuit, you will find that the error rate of the logical measurement is not improving with code distance. It only takes one stabilizer being consistently wrong to get the logical ...
Craig Gidney's user avatar
1 vote
Accepted

Surface code with measurement errors

For each stabilizer (i.e. each ancilla qubit), there will indeed be three nodes in the syndrome graph: between layer 1 and 2, between layer 2 and 3 and between layer 3 and stabilizer reconstruction. ...
AG47's user avatar
  • 476
1 vote

Implement parity check gadget using stim

Executing your second code, you will see that final_measurements is of length 12, which suggests that the simulator starts over ...
AG47's user avatar
  • 476
1 vote
Accepted

How is a surface code state measured?

It depends on your error model. In the absence of any error, you could only physically measure all the data qubits from the support of one representative of the logical operator (observable) $Z$. Your ...
AG47's user avatar
  • 476
1 vote

What is the definition of physical gate error rate?

Here is another partial answer: Infidelity does not tell us the error rate of a physical gate. It is a poor definition for speaking about the physical error rate of the gates. Fidelity is something ...
quest's user avatar
  • 636
1 vote
Accepted

Rotated surface code vs unrotated surface code - Boundary stabilizers

For the surface code in Figure 1, why do we drop the stabilizers in yellow circles? How did we know to drop those but keep the other stabilizers? After deleting data qubits outside the black-outlined ...
Craig Gidney's user avatar
1 vote

Rotated surface code vs unrotated surface code - Boundary stabilizers

First, as a remark, removing four qubits and four linearly independant stabilizers means that the unrotated and the rotated surface codes encode the same number of logical qubits: 1. The removed ...
AG47's user avatar
  • 476
1 vote

Rotated surface code vs unrotated surface code - Boundary stabilizers

Your terminology for "surface code" and "rotated surface code" seems to match mine. I have seen other references to "surface codes" with different definitions; for ...
unknown's user avatar
  • 2,146
1 vote
Accepted

Initialization of surface code into a logical state

just assume after the first round of stabilizer measurements that my surface code encodes $\vert \Psi\rangle = \vert 1\rangle_L$ instead. Or more generally, I could assume it is in the state $\vert \...
Craig Gidney's user avatar
1 vote
Accepted

Fault-tolerant correlated measurement on logical qubits

This approach will get you the correct measurement result, but will also collapse the encoded state. Consider first the case with no underlying stabilizer code. You start with your Bell pair $\left|00\...
AG47's user avatar
  • 476
1 vote

Why I am continuously getting "failed to decompose hyperedges into edges" in Stim?

It seems that your circuit experiment includes some error mechanisms that flip more than two detectors. This means your syndrome decoding graph is actually an hypergraph, with some hyperedges ...
AG47's user avatar
  • 476
1 vote
Accepted

Relating different surface code diagrams

Like this The stars are the XXXX stabilizers measured using your circuit (c), the plaquettes the ZZZZ stabilizers measured using circuit (b).
ChrisD's user avatar
  • 316

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