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dwierichs
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Amazing work @drdren! It was really easy to follow and the scope, structure and flow really work out very nicely! 🎉
I had a serious of comments and questions, but nothing dramatic/major :)
I think the "largest" is that the semi-Clifford section could use some additional connectors to the remaining demo; 1) anticipate its usefulness for the question "Why T gates?" and 2) highlight that the cheaper teleportation circuit can be used for
| The trouble with universality and quantum error correction | ||
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| It would be nice if we were certain that applying a finite sequence of gates could lead to any arbitrary quantum state -- a property called universality [#universality]_. However, Clifford gates such as the `Hadamard <https://docs.pennylane.ai/en/stable/code/api/pennylane.Hadamard.html>`__ $H$, `Phase <https://docs.pennylane.ai/en/stable/code/api/pennylane.S.html>`__ $S$, or `CNOT <https://docs.pennylane.ai/en/stable/code/api/pennylane.CNOT.html>`__ gates and all Pauli gates :math:`\{X,Y,Z\}` are not enough because they can only achieve :math:`90^{\circ}` or :math:`180^{\circ}` rotations. |
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I feel like in this statement:
It would be nice if we were certain that applying a finite sequence of gates could lead to any arbitrary quantum state
we are missing some relaxation; either, the sequence is not required to be finite, or we are happy with approximate states. Any discrete gate set can arguably only produce a discrete set of states for sequences of length
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| It turns out that all you need to achieve universal quantum computing are the Clifford gates and at least one non-Clifford gate [#qecbook]_! In principle, you could select any non-Clifford gate, but a common gate set is `{Clifford}+T <https://pennylane.ai/compilation/clifford-t-gate-set>`__. | ||
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| The `T gate <https://docs.pennylane.ai/en/stable/code/api/pennylane.T.html>`__ applies a :math:`45^{\circ}` rotation about the $Z$ axis. On the surface, this doesn’t seem too special. But with the additional of a non-Clifford gate, the `Solovay-Kitaev theorem <https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem>`__ guarantees that any state can be reached by a finite sequence of gates, and the gate sequence can be found with the Solovay-Kitaev algorithm [#SK_alg]_ or gridsynth algorithms [#gridsynth]_. So, you can finally obtain, say, a :math:`1^{\circ}` rotation about the $Z$ axis to a :math:`10^{-2}` error with a sequence of $T$, $H$, and $S$ gates. |
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| The `T gate <https://docs.pennylane.ai/en/stable/code/api/pennylane.T.html>`__ applies a :math:`45^{\circ}` rotation about the $Z$ axis. On the surface, this doesn’t seem too special. But with the additional of a non-Clifford gate, the `Solovay-Kitaev theorem <https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem>`__ guarantees that any state can be reached by a finite sequence of gates, and the gate sequence can be found with the Solovay-Kitaev algorithm [#SK_alg]_ or gridsynth algorithms [#gridsynth]_. So, you can finally obtain, say, a :math:`1^{\circ}` rotation about the $Z$ axis to a :math:`10^{-2}` error with a sequence of $T$, $H$, and $S$ gates. | |
| The `T gate <https://docs.pennylane.ai/en/stable/code/api/pennylane.T.html>`__ applies a :math:`45^{\circ}` rotation about the $Z$ axis. On the surface, this doesn’t seem too special. But with the addition of a non-Clifford gate, the `Solovay-Kitaev theorem <https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem>`__ guarantees that any state can be reached by a finite sequence of gates, and the gate sequence can be found with the Solovay-Kitaev algorithm [#SK_alg]_ or gridsynth [#gridsynth]_. So, you can finally obtain, say, a :math:`1^{\circ}` rotation about the $Z$ axis to a :math:`10^{-2}` error with a sequence of $T$, $H$, and $S$ gates. |
I think?
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And same comment about finite sequence + "any state can reached" (without "approximately" or so) as above. All statements I know are of the form "Given a target and some precision
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| The `T gate <https://docs.pennylane.ai/en/stable/code/api/pennylane.T.html>`__ applies a :math:`45^{\circ}` rotation about the $Z$ axis. On the surface, this doesn’t seem too special. But with the additional of a non-Clifford gate, the `Solovay-Kitaev theorem <https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem>`__ guarantees that any state can be reached by a finite sequence of gates, and the gate sequence can be found with the Solovay-Kitaev algorithm [#SK_alg]_ or gridsynth algorithms [#gridsynth]_. So, you can finally obtain, say, a :math:`1^{\circ}` rotation about the $Z$ axis to a :math:`10^{-2}` error with a sequence of $T$, $H$, and $S$ gates. | ||
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| But to achieve fault-tolerant universal quantum computing, quantum states must be encoded with quantum error correction (QEC) codes. Many QEC codes such as the `CSS <https://pennylane.ai/qml/demos/tutorial_stabilizer_codes>`__, colour, surface, and qLDPC [link to Utkarsh's upcoming demo] codes have transversal (therefore fault-tolerant) implementations of Clifford gates. However, the `Eastin-Knill theorem <https://arxiv.org/pdf/0811.4262>`__ dictates that there can be no quantum error correction code that can implement both Clifford and non-Clifford gates transversally. Therefore, it appears that universal quantum computing is impossible to do with error correction. |
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Having some background knowledge, I understand this paragraph. However, to a reader less familiar with the topic, it might not be clear at this point what transversality is, and whether/why it would be the only way to get fault tolerance.
Maybe you could spend one more sentence on introducing transversality, not on a technical level but in terms of its properties: they're simple realization of logic gates that do not propagate errors too quickly, so they are an easy way to get a fault-tolerant gate.
Connected to this:
Currently, with the little knowledge a reader might have, they might be confused by the strong premise
Therefore, it appears that universal quantum computing is impossible to do with error correction.
Following up on the above change, maybe the twist can be more like "what other way could there be to include non-Clifford gates if Eastin-Knill tells us it can't be transversal? -> If only there was some relationship..."
| You've probably seen a similar idea before | ||
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| The core idea of the Clifford hierarchy lurks beneath many of the concepts you may know: relationships between different gates can be exploited to simplify computation. For example, Clifford-only quantum circuits are known to be efficiently simulateable classically, as proven by the `Gottesman-Knill theorem <https://en.wikipedia.org/wiki/Gottesman%E2%80%93Knill_theorem>`__. |
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simulateable
Either "simulable" or "simulatable", I think?
| Pauli group ($C_1$) | ||
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| At the bottom of this hierarchy is the Pauli group, which contains the familiar Pauli gates :math:`C_1 = \{X, Y, Z\}`. |
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| At the bottom of this hierarchy is the Pauli group, which contains the familiar Pauli gates :math:`C_1 = \{X, Y, Z\}`. | |
| At the bottom of this hierarchy is the Pauli group, which contains the familiar Pauli gates and their tensor products :math:`C_1 = \{X, Y, Z\}^{\otimes n}` |
Something like this? :)
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| S.X. Cui, D. Gottesman, and A. Krishna | ||
| "Diagonal gates in the Clifford hierarchy" | ||
| `1608.06596v1 <https://arxiv.org/abs/1608.06596v1>`__, 2016. |
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| `1608.06596v1 <https://arxiv.org/abs/1608.06596v1>`__, 2016. | |
| `1608.06596 <https://arxiv.org/abs/1608.06596>`__, 2016. |
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| C. Chamberland, P. Iyer, and D. Poulin | ||
| "Fault-tolerant quantum computing in the Pauli or Clifford frame with slow error diagnostics" | ||
| `1704.06662v2 <https://arxiv.org/pdf/1704.06662>`__, 2017. |
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| `1704.06662v2 <https://arxiv.org/pdf/1704.06662>`__, 2017. | |
| `1704.06662 <https://arxiv.org/abs/1704.06662>`__, 2017. |
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| J. Hu, Q. Liang, and R. Calderbank | ||
| "Climbing the Diagonal Clifford Hierarchy" | ||
| `2110.11923v2 <https://arxiv.org/pdf/2110.11923>`__, 2021. |
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| `2110.11923v2 <https://arxiv.org/pdf/2110.11923>`__, 2021. | |
| `2110.11923 <https://arxiv.org/abs/2110.11923>`__, 2021. |
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| J.T. Anderson and M. Weippert | ||
| "Controlled Gates in the Clifford Hierarchy" | ||
| `2410.04711v3 <https://arxiv.org/pdf/2410.04711>`__, 2025. |
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| `2410.04711v3 <https://arxiv.org/pdf/2410.04711>`__, 2025. | |
| `2410.04711 <https://arxiv.org/abs/2410.04711>`__, 2025. |
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| L. Bastioni, S. Glandon, T. Pllaha, M. Stewart, and P. Waitkevich | ||
| "Climbing the Clifford Hierarchy" | ||
| `2603.12088v1 <https://arxiv.org/pdf/2603.12088>`__, 2026. |
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| `2603.12088v1 <https://arxiv.org/pdf/2603.12088>`__, 2026. | |
| `2603.12088 <https://arxiv.org/abs/2603.12088>`__, 2026. |
dwierichs
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dwierichs
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Amazing work @drdren! It was really easy to follow and the scope, structure and flow really work out very nicely! 🎉
I had a serious of comments and questions, but nothing dramatic/major :)
I think the "largest" is that the semi-Clifford section could use some additional connectors to the remaining demo; 1) anticipate its usefulness for the question "Why T gates?" and 2) highlight that the cheaper teleportation circuit can be used for
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