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Questions tagged [error-correction]

Quantum error correction (QEC) is a collection of techniques to protect quantum information from decoherence and other quantum noise, to realise fault-tolerant quantum computation. Quantum error correction is expected to be essential for practical quantum computation in the face of noise on stored quantum information, faulty quantum gates, faulty state preparations, and faulty measurements. (Wikipedia)

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Intuitions for magic quantification in a particular state

Suppose we define a magic state on $2n + 1$ qubits as: \begin{equation} \prod_{i = 1}^{n} CCZ(0, i, i+n) |+\rangle^{\otimes 2n + 1}. \end{equation} Does anyone have an intuition for the scaling of the ...
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What is the syndrome in Hastings and Haah honeycomb code?

In the honeycomb code proposed by Hastings and Haah, by measuring 2-body checks and multiplying these values, they obtain the values of plaquette stabilizers. In the decoding, do they use only the ...
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`predictions` and `fault_ids` in Stim and PyMatching for surface code decoding

When I tried to use Stim and PyMatching to decode errors in a Rotated Planar code, I noticed that, Regardedless of the code distance, predictions.shape, as well as ...
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How exactly does measuring a syndrome operator work for 'non-discrete' errors?

I am reading a summary of the stabilizer formalism in this paper, which considers the following encoding of a single qubit in three qubits: $$ |\overline{0}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\...
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Shadow weight enumerators

Is there any physical intuition or motivation behind the "shadow weight enumerator" of a quantum error correcting code?
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Effect of rounds in surface-code simulation with Stim

I'm learning Stim with the official tutorial and have a question. I made a small program: ...
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Can IBM Quantum hardware handle any CSWAP at all?

I am designing quantum algorithms where the quantum circuit uses CSWAP gates a lot. The result is very noisy on the quantum hardwares. So I designed really simple circuits to test whether it is indeed ...
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AGP Fault-tolerance of the flag qubit QEC for 7-qubit Steane code

I was trying to apply the flag qubit QEC (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.050502) for 7-qubit Steane code. From the AGP method (https://arxiv.org/pdf/quant-ph/0504218), ...
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How to verify that a certain gate was applied to a quantum code

Suppose I have a quantum error correcting code $|\psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$, say the $[[7,1,3]]$ Steane code for concreteness. Suppose there is a black box that either ...
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Deriving |𝐶𝐶𝑍⟩ magic states from |𝐶𝐶𝐶𝑍⟩?

Analogous to the $|T\rangle$ and $|CCZ\rangle$ magic states, one can define a $|C^n Z\rangle$ magic state. Is there any known quantification of the amount of magic of this state, and is there a way to ...
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Obtaining the stabilizer outcome by small operators

Suppose we want to obtain the measurement outcome of the stabilizer $X_1X_2X_3X_4X_5X_6$. Of course, by measuring $X_1X_2$, $X_3X_4$, $X_5X_6$ in this order and combining these three outcomes, we can ...
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Does one have multiple degrees of freedom in defining logical states and logical operators of a QEC?

Consider a rotated surface code. Let the surface code have $Z$ stabilizers along the top and bottom boundary and $X$ stabilizers along the left and right boundary. If I initialize all the physical ...
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The understandings of logical operator

In quantum error correcting code, such as shor 3-qubit code, code space is spanned by the basis {|000>, |111>}. The logical X operator is XXX and the logical Z operator is IIZ. When the code ...
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Measurement outcomes in gauge fixing

I consider measuring Pauli operator m in either normal stabilizer codes and subsystem codes. In normal stabilizer codes, if m commutes with the stabilizer group, measurement outcome is deterministic (...
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What is the physical operation behind "moving edges" and "moving corners" in Litinski's game of surface codes paper?

I was reading Litinski's A game of surface codes (https://quantum-journal.org/papers/q-2019-03-05-128/pdf/). In the introduction (page 2), the paper talks about operations like "moving edges"...
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What is the logical gate speed of a superconducting quantum computer?

What is the logical gate speed of a photonic quantum computer? says In a simple world the speed of a photonic quantum computer would just be the speed at which it’s possible to make small (fixed ...
2 votes
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How do Union-Find Decoders deal with Measurement errors through multiple measurement rounds?

I've read a few papers regarding to Surface Code and its decoding algorithms. I've learned that a Union-Find decoder need up to $d$ measurement rounds to deal with measurement errors. These ...
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How to find the undetected errors for general stabilizer codes in Stim?

In Stim, we use the detectors to track syndrome flips and infer the error pattern. However, the syndrome stays the same if the actual error pattern is a logical operator of the code by coincidence. It'...
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A better name for "weakly self-dual CSS codes"

Does anyone know a better name than "weakly self-dual"/"self-orthogonal" for CSS codes where $H_X=H_Z$, for example the Steane code, and the color codes? Details In the discussion ...
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Equivalent $[\![n,k,d]\!]$ codes and transversal gates

An operator on the Hilbert space of $ n $ qubits is called a local unitary if it is of the form $$ U=\bigotimes_{i=1}^n g_i $$ where each $ g_i $ is a $ 2 \times 2 $ unitary matrix. In other words if ...
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Are transversal entangling gates possible for stabilizer codes other than CSS?

It is well known that CSS codes can have lots of transversal entangling gates. For example, $ CNOT $ is exactly transversal on 2 blocks of any $ [[n,1,d]] $ CSS code. And https://arxiv.org/abs/1304....
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Is working with the |+> , |-> basis any harder than the |0>, |1> basis?

Say I have a code, for example the $ [[5,1,3]] $ code, and I want to (fault tolerantly) prepare the logical $ |+ \rangle $ state. Is that any harder than preparing the logical $ | 0 \rangle $ state? ...
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Qutrit Steane Code

There is a well known 5 qubit code $ [[5,1,3]] $ with stabilizer generators $$ XZZXI \\ IXZZX \\ XIXZZ \\ ZXIXZ $$ There is a corresponding $ [[5,1,3]]_3 $ code for qutrits given by \begin{align*} &...
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Commutation relationship and measurement results

There are things I do not understand about the following circuit, and I would appreciate it if you could explain. ...
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3 answers
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Highest theoretical threshold to fight single-qubit depolarizing noise for noiseless error-correction

Let's consider that each qubit in the lab faces a single-qubit depolarizing channel $\mathcal{N}(\rho)=(1-p) \rho + p \mathbb{I}/2$. Is there a theoretical result indicating the largest value of $p$ ...
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Is there any machine learning method for finding quantum error correction codes?

To define a quantum error correction code, first one needs to model noise, such as Pauli noise, dephasing noise, etc. Then according to the noise, look for the code space, stabilizer, and logical ...
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Is it sufficient to assume a constant coherent error?

I've recently started working with quantum errors and noise and came across an intriguing but simple question. When we consider coherent errors in quantum gate operations, it's common to model them as ...
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Boundary conditions for surface code

I have a question about boundary conditions for surface codes. Do any surface codes have torus-like boundary conditions? Are there any surface codes that don't actually have boundary conditions, i.e. ...
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1 answer
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How is $(\langle \psi| E_{a}^\dagger E_{b} | \psi \rangle)^\dagger = C_{ba}^*\langle \psi| \psi \rangle $

I am reading through Daniel Gottesmans surviving as a quantum computer in a classical world. On page 36, he presents the following theorem: Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E ...
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Is every code with a universal set of transversal gates trivial?

The quantum repetition code is an $ [[n,1,1]] $ stabilizer code with stabilizer generators $ Z_iZ_{i+1} $ for $ i=1, \dots, n-1 $. The Eastin-Knill theorem states that a $ d >1 $ code cannot have a ...
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What does DETECTORs mean in the example circuit for rotated surface code in Stim?

In Stim, an example circuit for rotated surface code is provided: ...
4 votes
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Fault-tolerant syndrome extraction

My question concerns fault-tolerant measurement of the syndrome of an quantum error-correcting code. Somewhat recently, Rui Chao and Ben Reichardt, in https://arxiv.org/abs/1705.02329, proposed a ...
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Advantages and disadvantages of rotated surface code

I think one of the advantages of rotated surface code is that it can express surface code with fewer physical bits. Are there any other advantages? Also, are there any disadvantages compared to ...
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1 answer
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What is the easiest way to get path graph from Stim?

In Stim, we can get a detector graph with the probability of each error mechanism occurring. Now I want to construct a path graph from the detector graph, which is usually done by Dijkstra's algorithm....
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How does measurement based quantum computing (MBQC) behave under error propagation?

In the quantum circuit model, we know how to handle error propagation if we implement a unitary $U'$, which is $\varepsilon$-close to the ideal unitary $U$ and a state $|\psi'\rangle$, which is also $\...
2 votes
2 answers
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Are close states still close after measurement (regarding trace distance)?

We are given two states $|\psi_1\rangle, |\psi_2\rangle \in \mathbb{C}^2 \otimes \mathbb{C}^2$ with trace distance $\leq \varepsilon$, so they are very close to each other. Now, assume we measure the ...
2 votes
2 answers
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Kraus decomposition of merging in lattice surgery

I am reading about lattice surgery from this paper. I am interested in the merge operation which takes 2 qubits to 1 qubit. I want to understand the logical-level Kraus operation that the merge does. ...
3 votes
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Necessary condition for transversal Hadamard by family of stabilizer codes

A necessary and sufficient condition for a stabilizer code having transversal $CNOT$ is that the code is a CSS code (see Theorem 11.5 here or the question here). I know that a sufficient condition for ...
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1 answer
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Define the $k$-local transversal logical operation

For a $[[n, 1]]$ QEC code $\mathcal{Q}$, we say single logical gate $R$ is transversal if the logical $\bar{R}$ can be implemented with $R^{\otimes n}$. I am wondering if we could expand the ...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?

I have trouble understanding a proof in Nielsen & Chuang, specifically the identity in (10.20), which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger F_l P \sqrt{\rho}.$$ By ...
1 vote
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Why can Pauli errors $E$ be decomposed as $E=T(S)LG$ with $T(S)$ "pure errors"?

I have a question about the decomposition of Pauli errors. Pauli error $E \in \{I,X,Y,Z\}^{{\bigotimes}n}$ that satisfies the syndrome $S$ can be decomposed into a product of pure error $T(S)$, ...
1 vote
1 answer
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State injection in a surface code

I am reading this paper on lattice surgery by Dominic Horsman et al, and am struggling to understand a simple method they put forward in their Figure 8(a)(b)(c) to inject an arbitrary state into a ...
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1 answer
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Parallelizing decode_to_edges_array (PyMatching)

I am trying to parallelise the function: matching.decode_to_edges_array() to somewhat batch decode, but I am consistently running into the same problem: ...
2 votes
1 answer
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Is there a simple condition under which X-and-Z-error correctability leads automatically to Y-error correctability?

I had the impression and guess that in a quantum error correction code, once it can correct any single-qubit X and Z errors, it automatically can also correct all single-qubit Y errors. Now after ...
6 votes
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139 views

Universal gate set for the $ [[15,1,3]] $ code

The $ [[15,1,3]] $ triorthogonal code implements transversal $ T $. Since it is a CSS code, two blocks will also have a transversal $ CNOT $ gate. To get a universal gate set all that is required is ...
2 votes
1 answer
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Generating Equal Amplitude Superposition States from Another Equal Amplitude Superposition State

Can we prepare a state regarding a transformation in quantum computing that seems to generate another equal amplitude superposition state when applying a Hadamard gate? Specifically, I observed that ...
6 votes
3 answers
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Why focus on T gates and not some other single qubit rotation R making Clifford + R universal?

Background: In many error correction codes in particular the surface code, the Clifford operations generated by the S,H and CNOT are transversal for quantum computation (meaning that these logical ...
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Does two quantum error correcting codes having the same CSS Tanner Graph imply that they are locally equivalent?

I am studying CSS quantum LDPC codes and I am curious as to whether the Tanner Graph structure necessarily must have long-range connections in order to be a non-local qLDPC code. This is because the ...
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See error samples in Stim+Pymatching [duplicate]

I have a surface code circuit written in Stim. Following Stim's intro I can use sinter to get logical error rates. I'd like to see what error patterns could lead to ...
3 votes
1 answer
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Will logical clock cycle time be a limiting factor for quantum computations?

Fault-tolerant quantum computation promises to strongly suppress the errors by scaling up the size of the systems. Right now, different physical implementations of proto quantum computers have very ...

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