Questions tagged [fault-tolerance]
For questions about achieving (almost) perfect logical operations with imperfect gates. The tag also applies to questions about the amount of tolerable noise in a quantum circuit, as well as designing circuits to prevent errors from spreading.
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Leakage Detection Circuit
Leakage happens when the state of the system is no longer in the computational subspace that defines a qubit. In Preskill's notes on Fault Tolerant Quantum Computation (Pg 40), a circuit for leakage ...
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Prospects of topological/anyon approach to QC
Given the recent articles/preprints by Google and Quantinuum claiming implementation of braiding of non-Abelian anyons on a quantum processor, what is the likely impact of these findings, and ...
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Effect of too many syndrome measurement cycles on surface code threshold
Assume I have a surface code with distance $d$ and an i.i.d error model with both single qubit depolarization and measurement errors, both with probability $p$.
In this case, one usually repeats the ...
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Shor FTEC scheme vs Steane FTEC scheme and other schemes
In the common stabilizer-based error correction procedure, one uses one syndrome qubit to measure the eigenvalue of a weight "w" syndrome by "w" times entangling the syndrome ...
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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)
On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
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Logical |+> State Preparation
From this paper (page 2), they said that
Surface code qubits can be easily initialized in the logical
+1-eigenstates Logical |0> and Logical |+> of Logical Z and Logical X by initializing all ...
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List of codes which can perform non-Clifford gates fault-tolerantly
I am looking to make a comprehensive list of all known quantum error correction codes which can perform one non-Clifford gate fault-tolerantly.
Currently, I have found two codes.
3D Color code
Tri-...
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Why is it necessary for the Fault-tolerant error correction condition to take into account two qubit errors?
The Fault-tolerant error correction (FTEC) condition states that for a code distance of 3, if there are no input errors, even if up to one gate fault (including a two qubit error occurring on a two ...
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Steane's method for fault-tolerant syndrome measurement
In Steane's method for fault-tolerant syndrome measurement, as explained in arXiv:quant-ph/0504218, ancilla is prepared in the logical 0 state for the circuit performing the X-type stabilizer ...
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Definition of fault-tolerant measurement in QCQI
In the book "Quantum Computation and Quantum Information," one of the definitions for fault-tolerant measurement is mentioned as "any single component in the procedure results in an ...
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Is there non-fault-tolerant quantum error correction?
See this sectino on Wikipedia for Quantum error correction
Peter Shor's 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a ...
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Can we have a constant-overhead threshold theorem?
The threshold theorem states that any abstract circuit in BQP can be computed by another polynomial-depth circuit that succeeds in the presence of noise. The original construction from 1996 requires ...
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Resource Estimation for Ion Trap Architecture
Most resource estimation papers (computing physical qubit count, algorithm run time, power consumption, etc.) target the superconducting qubit architecture. Some examples include
How to factor 2048 ...
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Can you give an intuitive idea behind how the Minimum Weight Perfect Matching (MWPM) decoder work?
The Minimum Weight Perfect Matching (MWPM) decoder seems to be the most popular choice for decoding error syndromes in Surface Code quantum error correction. Can anyone give an intuitive idea of how ...
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What are the transversal gates of the $ [[4,2,2]] $ code?
Consider the $ [[4,2,2]] $ stabilizer code with stabilizer generators $ XXXX,ZZZZ $.
Logical $ X $ on the first qubit is
$$
XIXI
$$
Logical $ Z $ on the first qubit
$$
ZZII
$$
Logical $ X $ on the ...
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Logical error rate of topological cluster state under erasure channel
Assume I use the topological cluster state (https://arxiv.org/abs/quant-ph/0703143) to protect against erasure. Assume also that the only noise or erasure that I have in my system is loss in the ...
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Weakly transversal gates for the $ [[5,1,3]] $ code
For the $ [[5,1,3]] $ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code $ X^{\otimes 5} $ implements logical $ X $ and $ Z^{\otimes 5} $ implements logical $ Z $. A less common gate ...
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"Rigorous" understanding that one should correct Pauli drift before non-Clifford gates
It is a well-known fact that in fault-tolerant quantum computing, it is sufficient to actually correct the Pauli errors only right before the next non-Clifford gate.
My purpose is to check all the ...
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Threshold estimate for 7-qubit code with flag-qubit syndrome extraction
I am looking for a paper where the threshold for 7-qubit code having a syndrome extraction based on flag qubits has been done.
In the original paper about flag qubits, there is an estimation of pseudo-...
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How to correct error during the syndrome measurement or recovery process?
Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
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Difference between Repeat-until-success circuit and Distillation procedure
On the fault-tolerance quantum computer (FTQC), the errors will be corrected by the error correction code (like surface code). In this case, the small angle rotation gate is difficult to apply on the ...
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Transversal Hadamard gates and multiple logical qubits
This question is linked to this post.
I wanted to work through the details in the main answer to check I understood the conditions for the existence of logical Hadamard gates. But I am slightly ...
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Conditions for Fault-tolerant QC to work better than not doing any EC (except the ideal decoder)
I've been exploring fault-tolerant QC with the simplest 7-qubit Steane code, yes I'm able to obtain all the predicted thresholds for various transversal gates and the EC gadgets. I then tried to ...
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Fault tolerance of MBQC
Is there a proof that measurement-based quantum computation can be fault tolerant under any noise model? Or at least, under any noise model with small enough correlation length?
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Computation for threshold of CNOT gate
I'm trying to reproduce some threshold results using 7-qubit Steane code in those legacy papers. I managed to obtain the graph for circuit noise and single qubit gate
e.g. the pseudocode for computing ...
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Logical error rate for QECC using qiskit
I've been coding QECC with qiskit recently. As I'm not very skilled in programming I encountered a lot of questions.
Now I'm trying to generate the logical error rate against physical error rate graph ...
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Qiskit, possible to manipulate mid-circuit measurement outcomes?
I'm currently coding up an error correction gadget using qiskit.
The one I'm doing now is 7-qubit Steane code with Steane's error correction gadget, in which the ancilla qubits are measured and then ...
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Estimate of threshold values for 7-qubit Steane code with qiskit (or stim)
I have trouble recreating the graphs shown in this paper
https://arxiv.org/pdf/quant-ph/0508176.pdf
i.e. the quadratic trend of the logical error against physical single qubit error for [7,1,3] code
I ...
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Why the perfect 5-qubit code was used for magic state distillation?
I am currently trying to understand magic state distillation. So far, my understanding is that the general idea is to find a code where a non-Clifford gate is transversal (very well explained in https:...
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How is the size of the circuit derived, in proving the threshold theorem?
In chapter 10.6.1 in Nielsen and Chuang, the section on concatenated codes and the threshold theorem (pages 480-481) states:
The size of the simulating circuit goes as $d^k$ times the size of the
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Why can't we implement fault-tolerant single qubit rotations continuously?
My understanding is that in order to implement single-qubit rotations on logical fault-tolerant qubits one has to do smth like discretizing rotations and using $T$ gates. So, is it the case that given ...
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Necessity of decoding in fault-tolerant quantum computation
On page 476 of Nielsen/Chuang's book it is stated:
The basic idea of fault-tolerant quantum computation is to compute directly on encoded quantum states in such a manner that decoding is never ...
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How to intuitively understand why $T$ gate can't be implemented transversally?
Suppose we have all to all connectivity and can implement fault-tolerant logical gates in the Clifford group. Why can't we just apply a physical $T$ gate to every physical qubit in the code block? (...
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Distance of one dimensional quantum error correcting code
Consider one dimensional quantum codes $[[ n,k=0]] $.
One way to describe them is using stabilizer framework with $n$ independent Pauli matrices.
Usually, one considers them in the graph state model ...
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What informs which choice of stabilizer generators to measure?
A stabilizer code is a code with codespace defined as the joint $+1$ eigenspace of some set of stabilizers $\mathcal S = \{s_1, s_2,\ldots,s_j\}$ with eigenvalues $\pm 1$. Error correction involved ...
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What are the transversal gates of Shor's $[\![9,1,3]\!]$ code?
What are the transversal gates of Shor's $ [\![9,1,3]\!] $ code?
Since this is a stabilizer code we have transversal Pauli gates. Indeed, transversal $X^{\otimes 9}$ implements logical $\overline Z$ ...
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What are the transversal gates of the $[[5,1,3]]$ code?
What are the transversal gates of the perfect 5-qubit code ( [[5,1,3]] code) ?
Since this is a stabilizer code we have transversal Pauli gates.
Are there any others?
I am primarily interested in ...
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How many times does syndrome extraction need to be repeated?
As part of implementing a QEC code, we need to do some syndrome measurement. In order for this syndrome measurement, we need to make sure that any $k \leq t$ errors (with a distance $2t + 1$ code) ...
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What is the time overhead for fault-tolerant gate teleportation?
The gate teleportation protocol (https://arxiv.org/abs/quant-ph/9908010) has a step where it utilizes cat state measurement of transversal operators. Since cat states can not detect phase errors it is ...
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What is the smallest code for a single logical qubit
I am looking for a small stabilizer code that protect one logical qubit from $X$ and $Z$ noise.
Maybe not necessarily all of the physical qubits.
Something that has a good tradeoff between number of ...
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Why the boundary data qubits have to be put in the ground state for lattice surgery rough merging
I am trying to understand why we need to put the boundary data qubits in the ground state $|0\rangle$ for a rough merging procedure.
From my current understanding, if no errors occurred (perfect ...
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Questions on Definitions and Concepts Regarding Toric Codes
Currently, I reading the toric code section in the beginning of the paper Topological Quantum Memory. Here are a couple of sections that are somewhat confusing to me.
(1)
This quote below is under a ...
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A Particular Way of Erasing Ancilla Qubits
I am currently reading the paper Topological Quantum Memory by Dennis, Kitaev et. al. My main interest is to conceptually understand the idealized assumptions / requirements of a quantum computer ...
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Why does fault-tolerant transversal phase gate $P$ only work with doubly-even codes?
I read in this topic that to get a fault-tolerant transversal relative phase gate of $\pi/2$ -- $P$ gate in the reference --, one has to select a CSS code which satisfies doubly-even distances.
I don'...
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Diagrammatic language for fault-tolerance: I don't understand the difference between preparation correctness and propagation properties
I am learning the diagrammatic language for fault-tolerance which allows to show in an "easy" manner if circuits are fault-tolerant. I first put a reminder of the definitions and then I ask ...
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Magic state distillation with the $15$ qubit code. How are the transversal $T$ performed? Also via a (lower level) state injection?
I have a code $C1$ that admits native fault-tolerant gates for any element in the Clifford group (for instance via transversal implementation), but that then doesn't have an easy way to perform the $T$...
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Magic state distillation with the 15 qubit code. Why is the failure $O(p^2)$ if we include $|+_L\rangle$ preparation and decoding?
My goal is to understand why the magic state distillation (for instance based upon the $15$ qubit code) improves the fidelity of the magic state. I followed this lecture to try to understand it. I ...
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How to encode one stabilizer code for $H, CX, \sqrt{Z}$ operators
I understood that to get fault-tolerant computing, one can start from any stabilizer group and then unitarily modifying it to get a new stabilizer group which relates to some logical operators.
For ...
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Eastin Knill Theorem and groups of transversal gates
The Eastin-Knill Theorem shows that the transversal gates always form a group and that moreover this group is a finite subgroup of the group of all unitaries.
For many codes, for example all self dual ...
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What's the map from any linear classical codes to entanglement-assisted stabilizers codes
When a linear code is self-dual, its parity check matrix can be used to easily define a stabilizer code.
From my understanding, thanks to entanglement-assisted stabilizer codes, it is possible to ...
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