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When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.

In some sense, these are rather mysterious objects, in that they perform "magic" discrete operations on the states. They are essentially black boxes, whose inner workings are not often dealt with while studying quantum algorithms. However, that is not how quantum mechanics works: states evolve in a continuous fashion following Schrödinger's equation.

In other words, when talking about quantum gates and operations, one neglects the dynamic (that is, the Hamiltonian) realising said evolution, which is how the gates are actually implemented in experimental architectures.

One method is to decompose the gate in terms of elementary (in a given experimental architecture) ones. Is this the only way? What about such "elementary" gates? How are the dynamics implementing those typically found?

glS
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    When expressing classical computations in terms of logical operations, one makes use of gates. In some sense, these are essentially black boxes, whose inner workings are not often dealt with while studying classical algorithms. However, that is not how nature works: states evolve in a continuous fashion describable by differential equations. When talking about classical algorithms, one neglects the dynamic realising said evolution, which is how the gates are actually realised in physical systems. But the dynamic generating a gate is unimportant, so long as the gate can in fact be realised. – Niel de Beaudrap Mar 22 '18 at 12:12
  • @NieldeBeaudrap I'm not sure whether you are rephrasing the question or asking something. Anyway, the dynamic generating a gate is important as soon as you need to actually implement the gate, it is just not directly dealt with when writing algorithms (though it is still considered, which is why most algorithms are expressed in terms of gates whose decomposition in terms of elementary gates are manageable). You don't need to worry about this in the classical case because we know very well how to make complex operations out of easy ones, but this is not the case with current quantum devices – glS Mar 22 '18 at 12:15
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    I'm making a rhetorical point: that the same argument could be directed at classical computation, but we allow ourselves the luxury of abstraction there because we know that the operations are realisable in principle, by a suitable application of manufacture and control. The only question is what level of 'principle' would satisfy you. Think about the analogy to the classical case: if you didn't know about consumer electronics, what level of detail would you hope for to be satisfied that NAND is physically realisable, rather than just as an intellectual abstraction for reasoning? – Niel de Beaudrap Mar 22 '18 at 12:32
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    @NieldeBeaudrap the kind of answer I expect is something highlighting that the way more complex gates (say, Toffoli gates) are implemented is through 1) gate decomposition using sets of gates which are "simple" in a given architecture (which brings the highly nontrivial problem of quantum compilation), 2) quantum control techniques, 3) using ancillary degrees of freedom, 4) implementing the gate as an effective dynamics in a larger Hilbert space, 5) possibly other methods – glS Mar 22 '18 at 12:35
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    So, you want to know how to decompose e.g. a Toffoli gate, but in some architecture, with an account of how the whole architecture works? Analogous to a description of how one realises NAND in terms of semiconductor physics, except for the more immature technological setting of quantum computation, where we are pretty sure that we have not yet completely solved the problem of scaling in any of our candidate technologies? I'm not sure your question will have an answer yet; and once it does have an answer, it will represent at least half of a semester course in engineering. – Niel de Beaudrap Mar 22 '18 at 12:50
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    No, I'm asking about the methodologies used today to implement gates, which are more or less the ones I mentioned above. This is different than asking about how gates are decomposed in terms of easier (in a given architecture) gates, because that is just one way to do this. I edited the question trying to make this point clearer. Here is an example of a paper using one such technique to implement a Toffoli: https://arxiv.org/abs/1501.04676, which might enlighten as to the kind of answer this question may have – glS Mar 22 '18 at 13:00
  • It's going to vary based on the 'implementation' of the system (i.e transmons will be different to linear optics, different to trapped ions etc.), so would you be able to narrow this down a bit further? – Mithrandir24601 Mar 23 '18 at 18:01
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    Chapter 1 and especially appendix D of my PhD thesis explain how abstract logic comes from the dynamics of superconducting qubits. – DanielSank Mar 23 '18 at 22:32

2 Answers2

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Generally speaking, a realization of a quantum gate involves coherent manipulation of a two-level system (but this is nothing new to you, maybe). For example, you can use two long-lived electronic states in a trapped atom (neutral or ionized in vacuo) and use an applied electric field to implement single-qubit operations (see trapped ions or optical lattices, for example).

Alternatively, there are solid-state solutions like superconducting qubits or silicon-defect qubits which are addressed by radio-frequency electronics. You can use microwave-addressed nuclear spin sublevels, or nitrogen vacancy cells in diamond. The commonality is that the manipulation and coupling of the qubits is via applied light fields, and there are a range of methods you can use to tune the level spacing in these systems to enable single-spin addressing or manipulate lifetimes.

The translation from the implementation to Hamiltonian is obviously dependent on your choice of system, but eventually it all boils back down to Pauli matrices in the end. The light field provides off-diagonal elements in your single-qubit operations, whereas two-qubit operations are trickier and techniques are very implementation-dependent.

GroundhogState
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I'll posit that, much as classical NAND and NOR gates can be generated with NMOS and PMOS transistors arranged in series and/or in parallel, quantum gates such as CCNOT (Toffoli) and CSWAP (Fredkin) and Hadamard gates can be generated with sums and/or differences of creation and annihilation operators.

Certainly this analogy that transistors are to classical gates as ladder operators are to quantum gates is not new; indeed, this appears to be at much of the heart of the intuition in Feynman's Quantum Mechanical Computers. Clearly CMOS technology is dominant nowadays while Feynman's paper illustrated a NAND gate with NMOS technology only.

By the time Lloyd envisioned his quantum mechanical simulator in 1993 he was focusing on electromagnetic pulses inducing CSWAP gates in molecules or quantum dot arrays or nuclear spins, and wanted to move away from designer Hamiltonians of the previous literature.

(This answer may be precisely against the motivation of the question and still too abstract, but @NeildeBeaudrap's comments made me think of transistors and of the connection to Feynman's paper.)

Mark Spinelli
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