As far as I know, both Rz and R1 operations in Q# rotate a qubit about the z-axis. In Q# API reference (https://learn.microsoft.com/en-us/qsharp) I found out that the only difference between them is that R1 also applies rotation about the "PauliI" axis, i.e. changes the global phase. In R operation reference (https://learn.microsoft.com/en-us/qsharp/api/qsharp/microsoft.quantum.intrinsic.r) they also say that "When called with pauli = PauliI, this operation applies a global phase. This phase can be significant when used with the Controlled functor". So the question: can you give an example, how can it be significant?
Rz and R1 are indeed different by a global phase of exp(iθ/2). If you use controlled version of the gate on a state (|0⟩ + |1⟩) ⊗ |0⟩ with the first qubit as control, R1 will not modify the state (since it only affects the |1⟩ state), and Rz will transform the state into (|0⟩ + exp(-iθ/2)|1⟩) ⊗ |0⟩. You can use this effect to distinguish those gates.
You can read more about the solution to this competition task in this blog.
Related
Regarding precedence graphs and conflict serializability, is it considered a cycle when two nodes are connected by a cycle but each arrow refers to a different object? For example:
Thanks!
I am trying to find a way to know what named qubit/quantum register a quantum gate (i.e. labelled Pauli-X gate) would be attached to. The documentation does not have a function nor example that informs me of how to go about doing this. The picture below outlines that I am trying to find qubit n0 from quantum gate U0.
Example quantum circuit
The easiest way to do this would probably be to access the data attribute of your QuantumCircuit object (e.g. circuit.data if your circuit object is named circuit). This will be a list of tuples with the instruction objects (ie the gate instance), the quantum bit arguments for that instruction, and the classical bit arguments for the instruction: (instruction, qargs, cargs). For your example circuit it is simple because there is one only one gate so it'll be the first element in that list. So for that case you can do something like u0_qubits = circuit.data[0][1] and u0_qubits will be a list of the Qubit objects. Trying to do this for larger circuits with possible duplicate gates will obviously be more involved.
I want to create my own C function to interpolate some data points and find an accurate minimum (Overall project is audio frequency tuning, and I'm using the YIN algorithm which is working well). I am implementing this on a digital DSP K22F ARM chip, so I want to minimize floating point multiplies as much as possible to implement within the interrupt while the main function pushes to a display and #/b indicators.
I have gotten to the point where I need to interpolate I have implemented the algorithm, found the integer minimum, and now need to interpolate. Currently I am trying to parabolically interpolate using the 3 points I have. I have found one that works within a small margin of error Most interpolation functions seem to only be made between two points.
It seems as though the secant method on this page would work well for my application. However, I am at a loss for how to combine the 3 points with this 2 point method. Maybe I am going about this the wrong way?
Can someone help me implement the secant method of interpolation?
I found some example code that gets the exact same answer as my code.
Example code:
betterTau = minTau + (fc - fa) / (2 * (2 * fb - fc - fa));
My code:
newpoint = b + ((fa - fb)*(c-b).^2 - (fc - fb)*(b-a)^2) / ...
(2*((fa-fb)*(c-b)+(fc-fb)*(b-a)))
where the x point values are a, b, and c. The values of each point is fa, fb, and fc, respectively
Currently I am just simulating in MATLAB before I put it on the board which is why the syntax isn't C. Mathematically, I am not seeing how these two equations are equivalent.
Can someone explain to me how these two functions are equivalent?
Thank you in advance.
I have been reviewing some material on the representation of an AI plan given the STRIPS format, and found that different people seem to formulate the same problem in different ways.
For instance, Wikipedia has an example regarding the Monkey in the lab problem. The problem states that:
A box is available that will enable the monkey to reach the bananas hanging from the ceiling if he climbs up on it. Initially, the monkey is at A, the bananas at B, and the box at C. The monkey and the box have height Low, but if the monkey climbs onto the box, he will have height High, the same as the bananas. The actions available to the monkey include Go from one place to another, Push an object from one place to another, ClimbUp onto or CLimbDown from an object, and Grasp or UnGrasp an object. Grasping the object results in holding the object if the monkey and the object are in the same place at the same height.
Here is the Wikipedia plan (please note that it is not matched exactly to this problem description, but it is the same problem. It doesn't seem to implement Ungrasp, which is not important for this discussion):
Now nowhere in this plan can I see that the bananas are located at Level(high), so the only way this could actually be divulged from the plan would be to read through the entire set of Actions and deduce from there that the Monkey must be at Level(high) to interact with the bananas, hence they must be at Level(high).
Would it be a good idea to put this information in the Initial State, and have something like:
Monkey(m) & Bananas(ba) & Box(bx) & Level(low) & Level(high) & Position(A) & Position(B) & Position(C) & At(m, A, low) & At(ba, B, high) & At(bx, C, low)
It looks quite verbose like that, but at the same time, it allows the reader to understand the scenario just through reading the Initial State. I've also been told that we should not be using constants anywhere in STRIPS, so I thought declaring the A, B, and C as Positions was a good idea.
Is it that some people do it differently (which I feel would kind of ruin the idea of having a standardized language to represent things), or is it that one of the ways I have presented isn't in the correct format? I am new to STRIPS, so it is entirely possible (and likely) that I am missing some key points.
This is not the greatest wikipedia ever. The description of STRIPS is accurate, but a little outdated.
Generally you don't need to worry about defining all the variables in the initial state because the variables are defined by the domain (the P in the quadruple in the linked article). For an intuition as to why, you have an operator for MONKEY in your initial state, but you're still introducing a free variable m that is not defined anywhere else. You end up with a chicken and egg problem if you try to do it that way, so instead the facts in the system are just propositional variables which are effectively sentinel values that mean something to the users of the system, not the system itself.
You are correct that you need to define the level for each item as part of the initial state, but the initial state of the example actually correct considering the constraints that the bananas are always high, the box is always low and the monkey is the only thing that changes level. I would probably change the example to have the At proposition take into account the object in question instead of using different proposition names for each object but that's just a style choice; the semantics are the same.
Operators in STRIPS are generally represented by 3 distinct components:
preconditions - each variable in the preconditions list must exactly match the corresponding variable in the current state (trues must be true, falses must be falses) but you ignore all other variables not explicit in the preconditions
add effects - when the action is performed, these are the effects that variables that are added to the state
delete effects - when the action is performed, these are the effects that are deleted from the state
and sometimes a 4th cost component when considering cost optimality
The post conditions listed in your example are the union of the add effects and delete effects. The advantage of separating them will come later when you get into delete relaxation abstractions.
In your proposed initial state you have propositions that contain multiple properties for the same object (e.g. At(bx, C, low)). This is typically avoided in favor of having a proposition for each property of each object in the state. Doing it this way makes you end up with a larger state, but makes for a much simpler implementation since you don't have to decompose a state variable in order to identify the value of a specific property of an object in the preconditions list.
I would like to interpolate a 3D scalar function f(x, y, z). I have coded up a 3D linear interpolation algorithm (http://en.wikipedia.org/wiki/Trilinear_interpolation). This was not so bad.
However, I would like something more sophisticated, e.g. 3D cubic splines. Are there any open source, easy-to-use, publicly available code for interpolating a 3D scalar? I would prefer to use C, but Fortran would be OK as well. I would like to stay away from Matlab.
I have seen similar questions asked here:
Interpolating a scalar field in a 3D space
and
What are some good libraries for 3D interpolation?
The second one was OK with Matlab, which I am not.
As for the first one, the main suggestion was Shepard's method. I am curious how accurate Shepard's method is. For instance, in the case of a uniform grid, one can apply Shepard's method only to nearby grid points, and in that case does it tend to be more accurate than linear interpolation or cubic splines? I imagine not, but wasn't 100% sure, and if in fact it is not better, then I would prefer to find code using something like splines if any such codes are available.
Take a look at Geometric Tools for Interpolation:
templated C++ for tricubic, uniform B-splines, and much more.
(einspline, a C library for B-splines in 1d 2d 3d,
seems to be dormant in 2013; the author doesn't answer emails.
Also, it's C; C++ templates would reduce code bloat for interpolating
floats, colors, vecs ...)
I haven't used either of these.
On Inverse distance weighting
a.k.a. Shepard's method, you can take any number of neighbors: in 3d, 2^3 or 3^3 or 4^3 ...
A general problem is "sagging" — see the plot in the link.
"Accuracy" of any interpolation method is really hard to measure: what's "golden",
for what class of data / what noise ?
And you have two measures, error at the data and smoothness, to trade off
— for
photo enlargement
three:
aliasing, blurring and edge halos.
There's some theory on spline interpolation of band-limited functions, but afaik none at all for IDW.
Added:
What about the
bullseye effect ?
IDW is a terrible choice in almost every case.
It assumes that all of your input data points are local minimums or maximums!
Well, IDW can have peaks above nearby data points, if there are high peaks far away.
For example in 1d,
IDW( [0 0] [1 0] [2 y] ) = y/7 at x = 1/2.
But IDW weights ~ 1 / distance may be too spiky, fall off too fast, for some tasks.
Interpolation methods and kernels have to be chosen to fit specific data and noise — an art.
The bspline-fortran library does 2d-6d b-spline interpolation for data on a regular grid. It is written in modern Fortran (there is a basic subroutine interface and also an object-oriented interface).
vspline is a FOSS C++ template library for b-spline processing. It's dimension-agnostic, so you can use it for 3D data. It's focus is on efficiently processing large raster data sets with multithreaded SIMD code. If you're concerned about precision, it can use long doubles for calculations and has extremely precise precomputed constants for maximum fidelity.