Quantum Computing Stack Exchange
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Questions tagged [logical-gates]

39 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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Implementing error correction (from physical qubit point of view)

I have worked with physical qubits, and I am fairly familiar with gates and sequences from initialisation to readout. I don't know much about error correction, and I love to learn how error correction codes are implemented on real hardware. I have…
Rex
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Understanding the operation of commutation of stabilizer operators

I want to show that the stabilizer operators ($M_{0}, M_{1}, M_{2}, M_{3}$) for the 5-qubit quantum error correcting code: If $M_{1} = [XXZIZ]$ and $M_{2} = [XZIZX]$ They commute iff $[M_{1},M_{2}]=0$. I am given to believe that $[M_{1}, M_{2}] =…
am567
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