আপনি দক্ষতার সাথে প্রশস্ততাগুলির নিরঙ্কুশ মানগুলি পুনরুদ্ধার করতে পারবেন না , তবে আপনি যদি স্বেচ্ছামূলকভাবে অনেকগুলি নমুনার জন্য অনুমতি দেন তবে আপনি যে পরিমাণ ডিগ্রী পছন্দ করতে পারেন সেগুলি নির্ধারণ করতে পারেন।
আরো নির্দিষ্টভাবে, যদি ইনপুট রাষ্ট্র প্রথম প্রত্যেকটি একটি একক ফোটন হল মোড, এবং এক আউটপুট থেকে নমুনা কোনও একটি সংখ্যার আঁকা করতে ইচ্ছুক, তাহলে এটি নীতিগতভাবে সম্ভব স্থায়ী অনুমান করার জন্য একটি যাই হোক না কেন মাত্রায় যথার্থতাটি পছন্দ করে, প্রথম বারের বিভিন্ন এন আউটপুট পোর্টে এন ইনপুট ফোটনগুলি যে সময় বের হয় তার বিভাজন গণনা করে । যদিও এটি লক্ষণীয় যে বোসনস্যাম্পলিংয়ের সাথে এর আসলে তেমন কোনও সম্পর্ক নেই, কারণ দৃ the়তার ফলাফলটি ফোটনের সংখ্যার চেয়ে অনেক বেশি মোডের সংখ্যা নিয়ন্ত্রণ করে এবং এটি স্যাম্পলিংয়ের দক্ষতা সম্পর্কে।nAnn
BosonSampling
আমি বোসনের স্যাম্পলিং কী তা সম্পর্কে একটি সংক্ষিপ্ত ভূমিকা চেষ্টা করব, তবে এটি লক্ষ করা উচিত যে আমি সম্ভবত এ্যারনসনের চেয়ে ভাল কাজটি আর করতে পারি না, সুতরাং তার সম্পর্কিত ব্লগ পোস্টগুলিতে নজর রাখা সম্ভবত একটি ভাল ধারণা (যেমন ব্লগ /? পি = 473 এবং ব্লগ /? পি = 1177 ) এবং এতে লিঙ্ক রয়েছে।
বোসনস্যাম্পলিং একটি স্যাম্পলিং সমস্যা। এটি কিছুটা বিভ্রান্তিকর হতে পারে যে লোকেরা সাধারণত নির্দিষ্ট উত্তরগুলির সাথে সমস্যাগুলি ভাবতে বেশি ব্যবহৃত হয়। একটি নমুনা সমস্যা বিভিন্ন যে সমস্যার সমাধান হ'ল কিছু সম্ভাবনা বন্টন থেকে আঁকা নমুনার একটি সেট ।
Indeed, the problem a boson sampler solves is that of sampling from a specific probability distribution. More specifically, sampling from the probability distribution of the possible outcome (many-boson) states.
Consider as a simple example a case with 2 photons in 4 modes, and let's say we fix the input state to be (1,1,0,0)≡|1,1,0,0⟩ (that is, a single photon in each of the two first two input modes).
Ignoring the output states with more than one photon in each mode, there are (42)=6 possible output two-photon states:
(1,1,0,0),(1,0,1,0),(1,0,0,1),(0,1,1,0),(0,1,0,1) and (0,0,1,1).
Let us denote for convenience with oi,i=1,.,6 the i-th one (so, for example, o2=(1,0,1,0)).
Then, a possible solution to BosonSampling could be the series of outcomes:
o1,o4,o2,o2,o5.
To make an analogy to a maybe more familiar case, it's like saying that we want to sample from a Gaussian probability distribution.
This means that we want to find a sequence of numbers which, if we draw enough of them and put them into a histogram, will produce something close to a Gaussian.
Computing permanents
It turns out that the probability amplitude of a given input state |r⟩ to a given output state |s⟩ is (proportional to) the permanent of a suitable matrix built out of the unitary matrix characterizing the (single-boson) evolution.
More specifically, if R denotes the mode assignment list(1) associated to |r⟩, S that of |s⟩, and U is the unitary matrix describing the evolution, then the probability amplitude A(r→s) of going from |r⟩ to |s⟩ is given by
A(r→s)=1r!s!−−−√permU[R|S],
with
U[R|S] denoting the matrix built by taking from
U the rows specified by
R and the columns specified by
S.
Thus, considering the fixed input state |r0⟩, the probability distribution of the possible outcomes is given by the probabilities
ps=1r0!s!|permU[R|S]|2.
BosonSampling is the problem of drawing "points" according to this distribution.
This is not the same as computing the probabilities ps, or even computing the permanents themselves.
Indeed, computing the permanents of complex matrices is hard, and it is not expected even for quantum computers to be able to do it efficiently.
The gist of the matter is that sampling from a probability distribution is in general easier than computing the distribution itself.
While a naive way to sample from a distribution is to compute the probabilities (if not already known) and use those to draw the points, there might be smarter ways to do it.
A boson sampler is something that is able to draw points according to a specific probability distribution, even though the probabilities making up the distribution itself are not known (or better said, not efficiently computable).
Furthermore, while it may look like the ability to efficiently sample from a distribution should translate into the ability of efficiently estimating the underlying probabilities, this is not the case as soon as there are exponentially many possible outcomes.
This is indeed the case of boson sampling with uniformly random unitaries (that is, the original setting of BosonSampling), in which there are (mn) possible n-boson in m-modes output states (again, neglecting states with more than one boson in some mode). For m≫n, this number increases exponentially with n.
This means that, in practice, you would need to draw an exponential number of samples to even have a decent chance of seeing a single outcome more than once, let alone estimate with any decent accuracy the probabilities themselves (it is important to note that this is not the core reason for the hardness though, as the exponential number of possible outcomes could be overcome with smarter methods).
In some particular cases, it is possible to efficiently estimate the permanent of matrices using a boson sampling set-up. This will only be feasible if one of the submatrices has a large (i.e. not exponentially small) permanent associated with it, so that the input-output pair associated with it will happen frequently enough for an estimate to be feasible in polynomial time. This is a very atypical situation, and will not arise if you draw unitaries at random. For a trivial example, consider matrices that are very close to identity - the event in which all photons come out in the same modes they came in will correspond to a permanent which can be estimated experimentally. Besides only being feasible for some particular matrices, a careful analysis of the statistical error incurred in evaluating permanents in this way shows that this is not more efficient than known classical algorithms for approximating permanents (technically, within a small additive error) (2).
Columns involved
Let U be the unitary describing the one-boson evolution.
Then, basically by definition, the output amplitudes describing the evolution of a single photon entering in the k-th mode are in the k-th column of U.
The unitary describing the evolution of the many-boson states, however, is not actually U, but a bigger unitary, often denoted by φn(U), whose elements are computed from permanents of matrices built out of U.
Informally speaking though, if the input state has photons in, say, the first n modes, then naturally only the first n columns of U must be necessary (and sufficient) to describe the evolution, as the other columns will describe the evolution of photons entering in modes that we are not actually using.
(1) This is just another way to describe a many-boson state. Instead of characterizing the state as the list of occupation numbers for each mode (that is, number of bosons in first mode, number in second, etc.), we characterize the states by naming the mode occupied by each boson.
So, for example, the state (1,0,1,0) can be equivalently written as (1,3), and these are two equivalent ways to say that there is one boson in the first and one boson in the third mode.
(2): S. Aaronson and T. Hance. "Generalizing and Derandomizing Gurvits's Approximation Algorithm for the Permanent". https://eccc.weizmann.ac.il/report/2012/170/