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In their original paper (last part of the paper), Abrams and Lloyd present a quantum algorithm that could potentially efficiently solve NP complete problems (in linear time). Their algorithm, robust to small errors, depends on the possible implementation of a nonlinear OR gate. A short presentation can be found here .

Classical irreversible logical  gates can be simulated by unitary gates that involve some ancillary qubits, as can be seen here and here.

Question. What exactly is the reason why the Abrams - Lloyd algorithm does not allow implementation using unitary gates and ancillary qubits?

Note that these ancillary qubits can be set and measured when necessary, during the operation of the algorithm, and these operations introduce nonlinearity into the state evolution.

Martin Vesely
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Cristian Dumitrescu
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  • After some effort, I found Abrams' PhD Thesis (partial, freely available on the Internet ) in which there is an answer to my question, pages 79-80 https://dspace.mit.edu/bitstream/handle/1721.1/85313/43918004-MIT.pdf?sequence=2 – Cristian Dumitrescu May 26 '20 at 06:22
  • If you use higher dimensional Hilbert spaces in order to imbed the nonlinear gate into higher dimensional unitary gates, then you cannot measure the garbage ancillary qubits at intermediate stages of the algorithm because you need to make sure the states can interfere with each other in future iterations. – Cristian Dumitrescu May 26 '20 at 06:47
  • That means that you have to track the relevant qubits that carry the information you are interested in, and measure the qubits (including ancillary qubits) that you are interested in only in the final stages of the algorithm. I don't know whether that's possible, probably not, because the computational path you're interested in would have very small probability. – Cristian Dumitrescu May 26 '20 at 07:34
  • Unless you can find a way to guarantee that all the other computational paths are innocuous (or at least with high probability ), that is they don't drastically decrease the number of states with a $\vert 1 \rangle$ for the final qubit. – Cristian Dumitrescu May 26 '20 at 07:47
  • You get to something like a "superposition of computational paths", admittedly a poor characterization on my part, but intuitive. – Cristian Dumitrescu May 26 '20 at 08:05
  • You can answer your own question then. – M. Stern May 26 '20 at 16:59
  • Thank you for your suggestion @MStern .My answer would not be as good as Abrams' observations in the link above, the interested reader can read his thesis (or the relevant pages). I wonder though whether quantum gate teleportation, the possibility to apply a quantum gate to an unknown state during teleportation could solve this problem. https://quantumcomputing.stackexchange.com/a/1810/10110 Basically, each iteration would be represented by such a quantum gate teleportation (back and forth ) between two systems A and B. I don't know if this solves the problem, but it's worth a look. – Cristian Dumitrescu May 27 '20 at 06:45
  • I like the fact that you can keep a record of the corrections and apply them later. – Cristian Dumitrescu May 27 '20 at 06:57
  • Comments can be deleted later, and the question will show up as unanswered. To name two reasons why you should not answer in a comment. – M. Stern May 27 '20 at 17:59
  • The purpose of my question was to enhance my understanding (and anyone else reading it) of this problem . My comments indicate the direction of my own research (by this I only mean relevant references that I found ) in this matter (which may or may not be relevant to the reader ). Abrams ' observations seem to indicate that an implementation is impossible. My opinion is that quantum gate teleportation might offer a glimmer of hope, but I am not sure (I am not an expert). I am still waiting for an expert level and definite answer. – Cristian Dumitrescu May 28 '20 at 07:59

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Since my question has no answer yet (for about two weeks now) I thought it would be appropriate to include my own conclusions (at the present time ) in the form of an answer.

In Abrams' PhD Thesis (partial, freely available on the Internet  )  there is a partial  answer to this question, pages 79-80  

The Abrams  - Lloyd algorithm depends on the possible implementation of a nonlinear OR gate. This nonlinear OR gate can be simulated (imbedded/embedded  into,  if you like ) by a higher dimensional unitary gate , by setting the extra ancillary qubit(s) to a definite state (that's equivalent to measuring the ancillary qubit(s)). The problem is that the output qubits are entangled,  so at the next iteration if I try to set the ancillary input qubits to a definite state (in order to simulate again the nonlinear OR gate) then I collapse the whole system to an eigenstate, I destroy the superposition.  

However you could increase the dimensionality of the embedding unitary gate for each iteration (that means increasing the number of input/output qubits , including ancillary qubits ). In other words for the first iteration the nonlinear OR gate will be simulated by a unitary gate $U_1 $ of dimension $n+k$ (where $k$ is a constant), with some ancillary qubits set on input to state $\vert 0 \rangle$ for example. All the $n+k$ output qubits of $U_1$ are entangled.  At the next iteration,  I simulate $U_1$ using a unitary $U_2$ gate of dimension $n+2k$ , by setting some extra ancillary qubits to $\vert 0 \rangle$.  At the next iteration,  I simulate $U_2$ using a unitary gate $U_3$ of dimension $n+3k$ , by setting some extra ancillary qubits to $\vert 0 \rangle$.  And so on. This way I can simulate the nonlinear OR gate for each iteration and I do not collapse the whole system to an eigenstate at any intermediate stage of the algorithm. We just have to track the right qubits to measure at the end of the algorithm. 

In other words, there might be some hope related to the implementation of this algorithm, but I would not make a large bet on it, because simulating $U_1$ $U_2$ , $U_3$ , ..... is not exactly the same thing as simulating the nonlinear OR gate (even if $U_1$ with some ancillary qubits set to $\vert 0 \rangle$ does indeed simulate the nonlinear OR gate). When you set the ancillary qubits of $U_2$ to $\vert 0 \rangle$ you need $U_2$ to simulate $U_1$ when its own ancillary qubits are set to $\vert 0 \rangle $. (in order to simulate the nonlinear OR gate). When you set the ancillary qubits of $U_3$ to $\vert 0 \rangle$ you need $U_3$ to simulate $U_2$ when its own ancillary qubits are set to $\vert 0 \rangle$ , and so on. The problem requires a careful analysis, and at this point I don't know if it allows a solution.

Cristian Dumitrescu
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