# Tag Info

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TL;DR: There is no universal gateset with transversal implementation in the repetition code. This is the case not due to Eastin-Knill theorem, whose assumptions the code fails to satisfy, but due to the fact that logical states have varying amount of entanglement between physical qubits and that transversal operators cannot change the amount of it. ...

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The accuracy of measuring the correct result is given by a sine $\sin^2((r + \frac{1}{2})\theta)$ where $r$ is the number of Grover iterations and $\theta$ is the angle between starting state (before Grover iteration) $|s\rangle$ and $|s'\rangle$. $|s'\rangle$ is a state perpendicular to our winner, desired output state $|\omega\rangle$. $\theta$ is given by ...

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Topological quantum memory by Eric Dennis, Alexei Kitaev, Andrew Landahl and John Preskill is a very nice introduction to the surface code. It develops intuitive description of the code in topological language from the low level combinatorial properties of the lattice focusing on the planar surface code with hole-free encoding. It is notable for its breadth ...

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How to understand the boundary condition they are using to create this logical qubit. What did they precisely mean in the text. They just mean that the boundaries are unspecified but far enough way to be irrelevant. It could be an infinite grid, it could be a really big donut, it could be a really big square patch with a variety of X or Z cuts running ...

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An X error on a data qubit places an edge between the two Z stabilizers adjacent to it. The edges must form a contiguous path from one side to the other to be an observable (or equivalently an undetectable logical error). The problem with your third example is you're missing one of the errors, which puts a gap in the path (resulting in the product ...

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No, there's no simple built-in way. You have to do it for yourself. This was an intentional design choice, which I will now attempt to justify because I do realize it's inconvenient. Stim has no concept of an error model separate from the concept of a circuit. Errors are nothing more than a certain type of instruction that can appear in a circuit file. You ...

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The simplest thing to do would be to make a pair of collections.Counters, sample the circuit millions of times, and add the result-flipped cases into one counter and result-not-flipped to the other. Then for decoding you would look up the current case to see which one had more counts: flip or don't flip. circuit = ... assert circuit.num_observables == 1 ...

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In order to apply Nyquist-Shannon sampling theory, we need to know the maximum frequency that will be present in the signal we intend to measure. We will do this by rewriting a time-dependent observable in terms of the frequencies present in the Hamiltonian $H$. Consider an arbitrary observable $O$ which will time evolve under application of $H$ in the ...

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The current gold standard method for this is to simulate various error rates and code distances, and do a linear fit of code distance vs log logical error rate. This allows you to project the code distances needed for other error rates. For example, we did this for the surface code in "A Fault-Tolerant Honeycomb Memory" for StandardDepolarizing ...

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In this example we then only use 2 dimensions out of the $2^3$ conceptually available (one logical qubit out of the three conceptually available). Is that correct? You need to specify the boundary conditions in order to do that kind of calculation. If you're using periodic boundaries (a donut) then yes there are two other logical qubits, but they have ...

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There are two reasons this works and you have identified the first one. Namely, the fact that a Pauli operator anywhere in a Clifford circuit is equivalent to a Pauli operator immediately preceding a terminal measurement. The second reason is the simple predictable effect that a Pauli operator immediately preceding a computational basis measurement has on ...

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The threshold is relative to the noise model and the decoder that you are using. In general you can't compute it analytically, you can only estimate it numerically. If you have the right tools available, the simplest way to estimate the threshold is to just simulate running the circuit under various levels of noise and see how often decoding succeeds at ...

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Each bulb becomes an outside corner and each side-square because an inside corner. You get a zig-zaggy pattern of cuts:

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The ancilla being noisy has two effects: making the measurement output noisier, and creating "hook errors" which look like multiple data qubit errors. An $n$-qubit stabilizer can have hook errors equivalent to up to $\lfloor n/2 \rfloor$ data errors (which occurs when the error on the ancilla occurs exactly halfway through the stabilizer ...

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Is my little summary of techniques more or less complete (and correct). Yes, basically. What is the most seriously considered technique to perform logical gates for the surface code? Lattice surgery is the current best technique. What is a good pedagogic reference to learn it? For the basics I'd recommend reading "Surface code quantum computing by ...

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A list of up-to-date techniques and references have been compiled at the Error-correction zoo for this and other codes: https://errorcorrectionzoo.org/c/surface.

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Syndromes are analyzed in space and time. An error on ancilla (also equivalent in effect to measurement error) seems as a sign flip of the syndrome in time, in only 1 ancilla. While error on data qubit will affect the 2 ancillas next to him: In the general case, data qubits will show even number of errors, while ancilla error, will create an odd number of ...

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The correction of errors during computations is required for large-scale quantum information processing. A qubit is encoded in a subspace of many physical qubits in quantum error correction so that faults can be actively rectified without altering the stored data. We must account for the fact that errors occur not only when something like a gate or ...

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If you don't do the data qubit measurements then chains of X errors can end at that location. Chains of X errors are what can result in your initialization going wrong. Ultimately this isn't a disastrous error, but it does decrease the code distance of the initialization. It takes fewer physical errors to break it. If you use a hole of width of w, and ...

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This answer provides a "combinatorial" approach to computing the dimension of the logical subspace and is meant to complement the "topological" approach described in Craig Gidney's answer. Importance of generator independence Our starting point is the fact that a stabilizer group $S$ on $n$ physical qubits generated by $g$ independent ...

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Use qtcodes Python library which is using qiskit. It is still a baby library, but this is the only thing I found. I am actually planning to override it and expand it.

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$P_L \approx 0.03(p/p_{th})^{d_e}$ $p$ - per-step error rate. Essentially you want to prepare some state $|a\rangle$ but due to imperfections you prepare $|b\rangle$ with probability $p$. $p_{th}$ - threshold error rate. It's the value of $p$ below which logical error falls exponentially with $d$ and above which it increases with $d$. $d$ - minimal number of ...

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Lowest order approximation of logical error We can generalize equation $(12)$ as follows $$P_L=\sum_{k=1}^\infty N(k)p_e^k\tag{a}$$ where $P_L$ is the probability of a logical error, $N(k)$ is the number of error chains of weight $k$ which are misidentified by the decoder and $p_e$ is the probability of an error on a physical data qubit$^1$. All error ...

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Repetition codes are great for helping to demonstrate some understanding of how things work. But they are not good quantum codes. In fact this is part of the reason why they are so good for demonstrating some of the ideas - they help bridge the gap between the classical intuition that we're used to and the quantum world which often feels a bit less ...

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Every gate has a cost because each gate of NISQ computers are noisy so each one introduces a little bit of error. This means that if you use fewer gates to do the same task, you'll end up with less error. Also, coherence time is limited so one can only perform so many gates...meaning that if you reduce your gate-count, you are able to do more meaningful ...

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