Questions tagged [logical-gates]
40 questions
6
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Smallest stabilizer codes with transversal CCZ or CS gates?
It has been shown that $[[15,1,3]]$ quantum Reed-Muller code is the smallest quantum error correcting stabilizer code with a transversal T gate.
What are the smallest codes with transversal CCZ or CS gates (which are the other examples of level 3…
tomek
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5
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Transversal CNOTs on CSS codes with multiple logical qubits
I am interested in the theory of implementing logical gates on quantum error correcting codes. From a practical view, transversal gates are very attractive. I have a question about transversal gates.
Background
Here on stack exchange, I find many…
Vincent
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4
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1 answer
Why is the first logical unitary gate in this example fault tolerant?
From Arthur Pesah's blog on "Computing with Quantum Codes using Transversal Gates", found here: https://arthurpesah.me/blog/2023-12-25-transversal-gates/
He gives the following examples of logical gates that are fault tolerant and non-fault…
am567
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4
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Can quantum algorithms include conditional jumps/change an instruction pointer?
From what I've seen (in talks for a general physics audience, but I'm not in the field of quantum computing), all or most quantum algorithms are fixed sequences of instructions applied to registers made of qbits. These instructions are built from…
Jim Pivarski
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3
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How to initialize the Surface Code in the $\pm Y_L$ State, perform logical $Y$-basis measurement, and $S$ logical gate clarification?
I am trying to learn more about logical state initialization, logical operators, and measurement for the surface code. I am having some trouble understanding the nitty-gritty details of the logical $\pm Y_L$ initialization and logical $Y$…
buddhazapha
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3
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Are toric codes (surface codes) doubly even, therefore have transversal $S$?
From this site and this post, doubly even codes have transversal $S$.
Based on this post, surface codes don't have transversal $S$ gates. We can check the boundary stabilizers of surface codes are not doubly even, so surface codes are generally not…
Yunzhe
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3
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For stabilizer codes, is a certain logical operation unique？
Suppose we have a $[[n, 1]]$ stabilizer code $Q$ and a single-qubit unitary $U$. We define the logical counterpart of $U$ as $\bar{U}$. My question is: Is there just one $\bar{U}$ up to stabilizers of $Q$?
I am asking this because I have seen the…
Yunzhe
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3
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1 answer
Teleportation of Transversal Hadamard Gate from the $[[8,3,2]]$ to $[[4,2,2]]$ codes
I'm trying to understand the circuit from Appendix A of the paper Fault-Tolerant One-Bit Addition with the Smallest Interesting Colour Code.
Here the top 3 qubits represent the 3 logical qubits of the $[[8,3,2]]$ color code (i'll call…
tbg
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2
votes
2 answers
Do logical operations map stabilizer group to itself?
Say we have a $n$-qubit stabilizer code $Q$ with stabilizer group $S$. If unitary $\bar{K}$ is a logical operation that preserves the codespace, what conclusion can we draw for $\bar{K}$ and $S$?
If $\bar{K}$ is Clifford, I believe it's…
Yunzhe
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2
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How to use the embedding operator to find logical operators?
In the following paper: https://arxiv.org/pdf/2409.18175
They give a $[[4,2,2]]$ code with generator matrix $$G:=\begin{bmatrix} X & X & X & X \\ Z & Z & Z & Z \end{bmatrix}$$
We are told that the check matrix of the embedded code is:
$$G_{V} :=…
am567
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2
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How to transform a 2-qubit state in general?
I have the following initial state
$$
|\psi\rangle=\frac{1}{\sqrt{5}}(|00\rangle+\sqrt{2}|10\rangle+\sqrt{2}|11\rangle)
$$
and I am trying to find the right "circuit" as combination of quantum gates (within the pool of Hadamard, CNOT, Pauli-X/Y/Z…
Randomize
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2
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Minus sign in logical operators
Logical operators are defined as $C(S)\backslash S$, where $S$ is a stabilizer group, but I am confused about this definition. For instance, in the 3-qubit repetition code, the stabilizer group is $\langle ZZI, IZZ\rangle$, and logical $Z$ operators…
lassel
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2
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Why are the following gates implemented transversally/weakly transversally?
I have read that logical gates can be implemented transversally ($U_{L} = V^{\otimes n}$) or weakly transversally ($U_{L} = \otimes _{i=1} ^{n} V_{i}$).
I have verified that the $[[8,3,2]]$ colour code has the logical gates:…
am567
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2
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1 answer
Trying to prove $U$ implements a logical $CCZ$ gate on the $[[8,3,2]]$ code?
I know from several different papers that the gate $U=TT^{\dagger}TT^{\dagger}TT^{\dagger}TT^{\dagger}$ actually implements a logical $CCZ$ gate on the $[[8,3,2]]$ quantum code. However, I am having difficulty proving this to myself.
My…
am567
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2
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2 answers
Fault Tolerance of 2-transversal gates
Suppose I have a single block $n$-qubit stabilizer code that can correct a weight 1 error (so the distance is $d=3$). If I apply a $1$-transversal gate of the form $U = U_1 \otimes U_2 \otimes \cdots \otimes U_n$, then if there was 1 error before I…
Eric Kubischta
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