Last year I attended Scott Aaronson's talk Hawking Quantum Wares at the Classical Complexity Bazaar. Being intrigued by his argument that "[e]ven if quantum mechanics hadn't existed, theoretical computer scientists would eventually have had to invent it", I have been thinking about how we can derive the first and second quantization in quantum mechanics from theoretical computer science.

First quantization is basically the quantization of properties of a unit quantum system and second quantization is the quantization of the electromagnetic field. Is there some way that theoretical computer science could entail these two ideas? I can recognize some "quantumnesses" in theoretical computer science, including

  1. enumeration of set elements,
  2. enumeration of interactions with oracles, and
  3. discrete states of Turing machine,

but I am having a hard time relating them to first and second quantization.

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    $\begingroup$ What are the "first and second quantizations" ? $\endgroup$ Mar 5 '13 at 23:29
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    $\begingroup$ I think at its core, Scott's argument is that even without quantum mechanics, TCS people would have eventually stumbled on quantum computation as a natural and powerful analog of probabilistic computation, specifically, by replacing probabilities with amplitudes. There would have been some vicious philosophical debates over what amplitudes "mean", which most TCS people would shrug off with "Whatever; the math works." $\endgroup$
    – Jeffε
    Mar 5 '13 at 23:46
  • $\begingroup$ @SureshVenkat, first quantization is, as I hinted in the question, that the properties (energy, number, spin etc.) of a unit quantum system will have discrete values. Second quantization is that the electromagnetic field is a superposition of discrete simple harmonic oscillators. The QM courses, I took, presented these concepts as fundamental. $\endgroup$ Mar 5 '13 at 23:57
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    $\begingroup$ what is "more fundamental" ? the "probabilities" in QM are different from classical probabilities, because of interference effects. $\endgroup$ Mar 6 '13 at 0:31
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    $\begingroup$ And one example of what @JɛffE says is in the realm of holographic computations. While one can't argue that this is "independent" of QM, the idea is that you can use cancellations in clever ways to count structures, and this is very reminiscient of what QCs give us. $\endgroup$ Mar 6 '13 at 0:32

I would amend your definition of second quantization. It is not a procedure that is applied only to the electromagnetic field. It is more properly a process in which something like the Schrodinger equation is re-interpreted as a classical field, which is then quantized. This is the standard old-fashioned way to move from single-particle quantum systems to multi-particle quantum systems in which particle creation and annihilation can be described via unitary operators.

I'm not sure that the paradigm of second-quantization makes sense in the context of quantum computation. To do so, one would need to interpret the "first-quantized" theory as a system which conserves something like particle number. The "second-quantized" theory allows for a description of systems with different particle numbers. Qubits obey neither Bose-Einstein nor Fermi-Dirac statistics and thus there is nothing analogous to "particle-number" in a qubit system. From this point of view it is hard to see that "second-quantizing" a qubit system would lead to something sensible.


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