In a comment to this question Peter Shor asked me for a reference about the third described in the question point of view, namely, that quantum computers can be described as computers that can manipulate units of information less than one bit independently. Although I cannot provide a reference, here is a brief clarification.

Suppose that each bit is divided into N sub-bits. Each sub-bit can be in two states: 0 or 1, similarly to the bit as a whole.

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Since the entropy of 1 bit is 1 (in bits), each sub-bit should have a factor, equal to 1/N so to sum up to 1 for the whole bit.

$$H=\sum_{i=0}^N \frac{1}{N} \ln 2= \ln 2 = 1$$

This factor, $1/N$ represents the probability that a certain sub-bit is measured when reading the bit with classical means. As such, the whole bit is equal to 1 when all its sub-bits are equal to 1, and to 0 when all the sub-bits are 0. In other cases there is a probability that the whole bit is read either as 0 or as 1. The probability is determined by the ratio of 0's and 1's in the sub-bits.

As such, the bits of sub-bit composition (10), (1100), (111000), (11110000), ... considered equal as each sub-bit can be infinitely devided.

The more the number N of sub-bits to which the bit is devided, the more points of the Bloch sphere is representable. At the analog limit, the whole Bloch sphere is covered.

enter image description here

As such, the digital quantum computer can be represented as a set of classical logical gates applied to sub-bits, very much similar to a classical computer with only one diffrernce: the final result cannot be read by each sub-bit separately, but only whole bits at minimum.

Thus my question is, whether the outlined model can be considered an exhaustive description of a digital quantum computer that uses a finite set of pure states to perform its operation?

Whether the presented model can represent any quantum gate quantum computer in a limit of infinite division?

Whether this somehow indicates that a qubit actually is the same unit of information as a classical bit, albeit treated somewhat differently?


This update clarifies some of the issues pointed by Niel de Beaudrap.

Quantum gates in sub-bit model can be represented by truth table.

Entanglement is realized as a fact that some states with the same Hamming-weight (and as such, the same expected outcome) may be opposite to each other. For example, qubits 1100 and 0011 are opposite to each other while 1010 and 1100 are correlated, even though each of them gives completely random result taken separately.

  • $\begingroup$ How does your "truth-table" for the Hadamard gate apply in the case of 4-bit strings? And can you clarify how the actual state $|\Psi^-\rangle$ (as opposed to orthogonal pairs of single-qubit states) would be represented? $\endgroup$ – Niel de Beaudrap Nov 26 '13 at 21:59
  • $\begingroup$ @Niel de Beaudrap do you mean 4-bit string or 4-sub-bit bit? $\endgroup$ – Anixx Nov 26 '13 at 22:01
  • $\begingroup$ I mean the quantum states which you associate to 4-bit strings. (Regardless of whether it makes sense to talk about "sub-bits", it certainly does make sense to refer to the mapping from bit-strings to quantum states which is the basis of your model.) $\endgroup$ – Niel de Beaudrap Nov 26 '13 at 22:08
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    $\begingroup$ Do you know who the original proposal was by? Where it was published, or what year? Do you have any clues as to where this was originally meant to have been described? $\endgroup$ – Niel de Beaudrap Nov 28 '13 at 13:16
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    $\begingroup$ I've voted to close as not being research level: it doesn't seem as though any revision of this post is forthcoming which would actually make this supposed description of quantum computing well-defined. $\endgroup$ – Niel de Beaudrap Dec 14 '13 at 0:31

I'm not sure who would suggest that qubits can meaningfully be described this way, or why anyone would do so. There are simply too many missing details, and it falls afoul of no-go theorems for local hidden-variable theories in quantum mechanics. This isn't "an exhaustive description of a digital quantum computer that uses a finite set of pure states to perform its operation" — even for single qubits — and extensions to multiple qubits immediately presents problems.

  1. How does one map binary strings to pure states? Why did you associate $0011 \mapsto \frac1{\sqrt2}(|0\rangle + |1\rangle)$ and $0110 \mapsto \frac1{\sqrt2}(|0\rangle + \mathrm e^{i\pi/3}|1\rangle)$ rather than the other way around? What specific mapping does one use for the boolean strings of length 4, with Hamming weight 1? What is the significance of the overlaps between those states, and the states you give for strings of Hamming weight 2? None of these details are provided, or even implied in any way that I can tell.

  2. What transformations of states are allowed depends crucially on the answers to the previous questions, but also is likely to be somewhat artificial from a physical point of view. Consider an operation mapping $|0\rangle $ to $ \frac1{\sqrt2}(|0\rangle + |1\rangle)$, and $|1\rangle $ to $ \frac1{\sqrt2}(|0\rangle - |1\rangle)$. In quantum computation, there is exactly one operation which does this — the Hadamard gate — and it's self-inverse. What transformation of bit-strings does it correspond to? Unfortunately, none. The strings of Hamming-weight 2 correspond to equally spaced states on the equator of the Bloch sphere (at 60° intervals), so their images in the Hadamard operation must include states at every 60° interval in latitude on the intersection of the XZ plane and the sphere — so your states for 4-bit strings must include $\frac12(\sqrt3|0\rangle \pm |1\rangle)$. This is not a real problem, but then what other states are there associated to strings of Hamming weight 1, and what does the Hadamard gate map them to? Unfortunately there is no solution for four-bit strings.

    In fact, it's not clear that any operations other than the identity operation and bit-flips can be represented, so it isn't clear how states involving the sub-bits would actually be prepared. One could respond that perhaps only the "continuum limit" actually describes the state-space of a non-trivial quantum computation, but then one should also not describe it as a "limit" of these discrete "models".

  3. Related to the above problem: How does one represent an entangled state? Consider the state $$ |\Psi^-\rangle = \mathrm{CNOT} \Bigl[\tfrac1{\sqrt2}(|0\rangle -|1\rangle)\otimes |1\rangle\Bigr].$$ How would you represent this state? The representation has to somehow include the fact that

    • measuring the first (or second) qubit gives a random result;
    • measuring the first bit has the effect of also measuring the second qubit;
    • the outcomes of the two qubits are opposite to each other;
    • the above also holds if you first perform any given reversible operation $U$ to both first and second qubits;
    • this should preferably not require the model to include communication between the qubits any time you measure one of them.

    That is to say: to represent the state $|\Psi^-\rangle $, you would need to come up with a local hidden variable model for quantum mechanics. But it's possible to prove that this is impossible, for instance using Bell's Theorem, which shows that the measurement outcomes that one may obtain from $|\Psi^-\rangle $ using various unitaries $U$ first (as above) are correlated in a way that no local hidden variable model may reproduce.

It seems to me that the idea of trying to represent qubits in terms of "sub-bits" is an attempt to represent the randomness of single-qubit measurement by a hidden-variable model. Indeed, it doesn't seem capable of anything more than this; and unfortunately it doesn't even do that particularly well — it just doesn't provide any insight into quantum computation. And any similar model will succumb to the fact that no local hidden-variable theory can represent quantum mechanics.

  • $\begingroup$ "What is the significance of the overlaps between those states, and the states you give for strings of Hamming weight 2?" - It seems to me that the strings with Hamming weight 2 are on the equator of the Bloch sphere, while those with Hamming weight 1 are not so there cannot be overlap. $\endgroup$ – Anixx Nov 26 '13 at 13:27
  • $\begingroup$ I have added an update to address some of your questions. $\endgroup$ – Anixx Nov 26 '13 at 15:12
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    $\begingroup$ Is this an intuitive mapping, or do you actually have an algorithm for mapping a string to a point on the Bloch sphere? If so, please give it. I don't understand. There is no way of mapping the 16 four-bit binary strings to the surface of the Bloch sphere so that Hamming distance corresponds to spherical distance. $\endgroup$ – Peter Shor Nov 26 '13 at 15:31
  • $\begingroup$ @Peter Shor I currently cannot tell you about the exact algorithm, but as I understand it, with a fixed division depth N points with the same Hamming weight occupy the same latitude on the Bloch sphere, placed at the vertices of a right polygon. If to introduce a value S=Hw/N (Hamming weight devided by division depth), all points with the same S will occupy the same latitude, that is a circle. So both the number of latitudes available, and the points on a latitude depends on the depth N. $\endgroup$ – Anixx Nov 26 '13 at 15:54
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    $\begingroup$ @Anixx: if there isn't an exact algorithm, then the sub-bit model of quantum computation isn't an exact model, and thus is almost totally useless. $\endgroup$ – Peter Shor Nov 26 '13 at 15:55

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