From a1b8789b175bc210d48119fe3d732c7ba7b4b7ea Mon Sep 17 00:00:00 2001 From: Matthew Weiss Date: Thu, 21 Jan 2021 13:24:29 -0500 Subject: [PATCH] fixed --- README.rst | 2 +- docs/_build/doctrees/environment.pickle | Bin 494013 -> 494013 bytes docs/_build/doctrees/index.doctree | Bin 22266 -> 21794 bytes ...1_Stabilization_via_Symmetrization.doctree | Bin 60009 -> 60009 bytes .../10_Introduction_to_Majorana_Stars.doctree | Bin 2084173 -> 2084173 bytes ...tars_and_Structured_Gaussian_Beams.doctree | Bin 173058 -> 173058 bytes docs/_build/html/index.html | 2 +- ...10.1_Stabilization_via_Symmetrization.html | 6 ++--- .../10_Introduction_to_Majorana_Stars.html | 10 ++++---- ...a_Stars_and_Structured_Gaussian_Beams.html | 24 +++++++++--------- docs/_build/html/searchindex.js | 2 +- spheres_presentation.pdf | Bin 0 -> 3959522 bytes 12 files changed, 23 insertions(+), 23 deletions(-) create mode 100644 spheres_presentation.pdf diff --git a/README.rst b/README.rst index bca434b..b88a366 100644 --- a/README.rst +++ b/README.rst @@ -34,7 +34,7 @@ toolbox for higher spin and symmetrization 3. The "Schwinger oscillator" representation of a spin-j state as the total energy 2j subspace of two quantum harmonic oscillators. Indeed, the full space of the two oscillators furnishes a representation of spin with a variable j value: a superposition of j values. This construction can be interpreted as the "second quantization" of qubit, and is appropriate for implementation of photonic quantum computers. -In addition, we provide many useful tools for dealing with oscillators and spin more generally: coherent state polynomials for oscillators, quantum polyhedra, a little spinorial special relativity, and much more. Everything is accompanied by 3D visualizations thanks to `vpython`, and interfaces to popular quantum computing libraries from `qutip` to `StrawberryFields`. +Everything is accompanied by 3D visualizations thanks to `vpython` (and `matplotlib`) and interfaces to popular quantum computing libraries from `qutip` to `StrawberryFields`. Finally, we provide tools for implementing a form of quantum error correction or "stablization" by harnessing the power of symmetrization. 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m<>^3o`y>C6?Y-}XJ5RfNT7p0l^Yn+_CI>CMv2L>RGspheres|j, m> basis states of a spin-j. Such a representation is naturally useful for simulating spin-j states on a qubit based quantum computer, and we provide circuits for preparing such states.

  • The “Schwinger oscillator” representation of a spin-j state as the total energy 2j subspace of two quantum harmonic oscillators. Indeed, the full space of the two oscillators furnishes a representation of spin with a variable j value: a superposition of j values. This construction can be interpreted as the “second quantization” of qubit, and is appropriate for implementation of photonic quantum computers.

  • -

    In addition, we provide many useful tools for dealing with oscillators and spin more generally: coherent state polynomials for oscillators, quantum polyhedra, a little spinorial special relativity, and much more. Everything is accompanied by 3D visualizations thanks to vpython, and interfaces to popular quantum computing libraries from qutip to StrawberryFields.

    +

    Everything is accompanied by 3D visualizations thanks to vpython (and matplotlib) and interfaces to popular quantum computing libraries from qutip to StrawberryFields.

    Finally, we provide tools for implementing a form of quantum error correction or “stablization” by harnessing the power of symmetrization. We provide automatic generation of circuits which perform a given quantum experiment multiple times in parallel while periodically projecting them all jointly into the symmetric subspace, which in principle increases the reliability of the computation under noisy conditions.

    spheres is a work in progress! Beware!

    Get started:

    diff --git a/docs/_build/html/notebooks/10.1_Stabilization_via_Symmetrization.html b/docs/_build/html/notebooks/10.1_Stabilization_via_Symmetrization.html index 62af0dc..f0c1b6f 100644 --- a/docs/_build/html/notebooks/10.1_Stabilization_via_Symmetrization.html +++ b/docs/_build/html/notebooks/10.1_Stabilization_via_Symmetrization.html @@ -691,7 +691,7 @@

    Stablization via Symmetrizationpytket won’t display its latex output to jupyter notebooks):

    -

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    Each layer consists of some single qubit gates and a CNOT.

    [2]:
    @@ -707,7 +707,7 @@ 

    Stablization via Symmetrization

    +

    9b0e896987a740ac83f46d20918d34b4

    We have two copies of the circuit running in parallel, and we do symmetrization after each of the two layers.

    First, let’s evaluate the original circuit analytically to see what the answers should be.

    @@ -1007,7 +1007,7 @@

    Stablization via Symmetrization result3 result4 -

    6026f4fb513740249480ba1801a3ba6c

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    Clearly, it’s not a given that this method of “stablization” will deliver results at least at the moment. It would be interesting to find some set of conditions that would make this algorithm relevant. The difficulties are, of course, errors in the symmetrization procedure itself, and also the scaling of the auxilliary qubits. Unless intermediate measurements are possible, allowing one to reuse auxilliary qubits, the cost of applying the algorithm quickly becomes prohibitive–and there aren’t enough gains at the small scale, it seems, to make it worth using. In the future, however, one could imagine having thousands of qubits, as well as reliable CSWAP gates, so that one could reliably symmetrize over hundreds copies of the circuit, etc. See the attached papers for some discussion of the types of errors that this method is best at dealing with.


    diff --git a/docs/_build/html/notebooks/10_Introduction_to_Majorana_Stars.html b/docs/_build/html/notebooks/10_Introduction_to_Majorana_Stars.html index dc9e331..6c731da 100644 --- a/docs/_build/html/notebooks/10_Introduction_to_Majorana_Stars.html +++ b/docs/_build/html/notebooks/10_Introduction_to_Majorana_Stars.html @@ -653,7 +653,7 @@

    An Introduction to Majorana’s “Stellar” Representation of Spin

    A “star”

    We’re all familiar with the qubit.

    -

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    It’s just about the simplest quantum system that there is.

    To get started, let’s choose some basis states. If we quantize along the \(Z\) axis, we can write the state of a qubit as a complex linear superposition of basis states \(\mid \uparrow \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\mid \downarrow \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\):

    \(\mid \psi \rangle = \alpha\mid \uparrow \rangle + \beta \mid \downarrow \rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\)

    @@ -1665,7 +1665,7 @@

    A “star”\(z = \frac{\beta}{\alpha}\) as being a point on the extended complex plane (the plane plus an extra point at infinity), which is just the sphere. This map is known as the stereographic projection.

    Explicitly, if \(c = x + iy\), then we have \((\frac{2x}{1+x^2+y^2}, \frac{2y}{1+x^2+y^2}, \frac{1-x^2-y^2}{1+x^2+y^2})\) or \((0,0,-1)\) if \(z=\infty\).

    Below, the map is depicted (where the pole of projection is the North Pole):

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    Moreover, we can go in reverse: \(c = \frac{x}{1+z} + i\frac{y}{1+z}\) or \(\infty\) if \(z=-1\).

    [19]:
    @@ -1718,7 +1718,7 @@ 

    A “star”

    +

    cc5d9501c73a49df82f4ae4c66e385db

    We’ve just seen how each these cardinal directions can be associated with qubit states: in fact, the eigenstates of the Pauli matrices. And thus we observe the interesting fact that antipodal points on the sphere become orthogonal vectors in Hilbert space.

    [20]:
    @@ -1782,14 +1782,14 @@ 

    A “constellation”And so on.

    One might wonder if our simple geometrical representation for a qubit generalizes for higher spin. And the answer is yes! In fact, it was first proposed way back in 1932 by Ettore Majorana. The answer is both simple and elegant: a spin-\(j\) state can be represented by a constellation of \(2j\) points on the sphere (up to phase). These points are often poetically termed “stars” in the literature.

    So just as a spin-\(\frac{1}{2}\) can be specified by a point on a sphere, a spin-\(1\) state can be specified by two points on the sphere, a spin-\(\frac{3}{2}\) state by three points on the sphere, and so on (again, up to phase).

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    The basic construction is this.

    Given a spin-\(j\) state in the \(\mid j, m \rangle\) representation, \(\mid \psi \rangle = \begin{pmatrix}a_{0} \\ a_{1} \\ a_{2} \\a_{3} \\ \vdots\end{pmatrix}\), form a complex polynomial in an unknown \(z\):

    \[p(z) = \sum_{m=-j}^{m=j} (-1)^{j+m} \sqrt{\binom{2j}{j-m}} a_{j+m} z^{j-m}\]

    The roots of this polynomial, called the Majorana polynomial, when stereographically projected from the complex plane to the sphere, yield the constellation. If you lose a degree, (i.e. if the coefficients for the highest powers of \(z\) are \(0\)) then you add a root “at infinity” for each one, and so the spin-\(j\) state can always be specified by \(2j\) point on the sphere.

    In other words, by the fundamental theorem of algebra, no less, a spin-\(j\) state factorizes into \(2j\) pieces.

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    As the above image indicates, the \(\mid j, m\rangle\) basis states (assuming we’ve quantized along the \(Z\) direction) consist of (in the case of spin-\(\frac{3}{2}\)) three stars at the North Pole, none at the South Pole; two stars at the North Pole, one at the South Pole; one at the North Pole, two at the South Pole; none at the North Pole, three at the South Pole. Any spin state can be expressed as a complex linear superposition of these basis states/constellations.

    At the same time, thanks to Majorana, we can also interpret the spin state as a polynomial and find its roots. Each corresponds to a little monomial whose product (as opposed to sum) determines the same state. Each root is associated with a direction in 3D, and under rotations, the stars transform rigidly as a whole constellation.

    diff --git a/docs/_build/html/notebooks/9.2_Majorana_Stars_and_Structured_Gaussian_Beams.html b/docs/_build/html/notebooks/9.2_Majorana_Stars_and_Structured_Gaussian_Beams.html index da488f9..159273b 100644 --- a/docs/_build/html/notebooks/9.2_Majorana_Stars_and_Structured_Gaussian_Beams.html +++ b/docs/_build/html/notebooks/9.2_Majorana_Stars_and_Structured_Gaussian_Beams.html @@ -684,10 +684,10 @@

    Majorana Stars and Structured Gaussian Beams

    +

    c40a658b2229473babd2f556b83b8b6e

    On the other hand, it’s less well known, but one can interpret a full spin-\(1\) state of two stars geometrically in terms of polarization. According to Bliokh, Alonso, and Dennis (and Hannay): the unit vector which bisects the two stars picks out a direction normal to a plane. Projecting the two points onto this plane, one obtains the two foci of the polarization ellipse. In other words, a spin-1 state specifies an ellipse oriented in 3D. The direction normal to the plane is taken to be the direction of propagation of the photon. In the degenerate case of the two stars in the same location, we get circular polarization; and beautifully, the \(m=0\) case, when two stars are opposite, never comes up, since the polarization plane rotates along with them, and we get linearly polarized light instead.

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    (As the paper points out, however, this construction gets the size of ellipse wrong, and they suggest a slight variant as a remedy.)


    So we have this correspondence between \(X, Y, Z\) directions on the sphere and diagonal/antidiagonal, clockwise/counterclockwise, horizontal/vertical polarization states in the plane.

    This manifests as well in the oscillator representation. As we’ve seen, we can “second quantize” a qubit, introducing a quantum harmonic oscillator to each of its two basis states \(\mid \uparrow \rangle\) and \(\mid \downarrow \rangle\), and interpret the resulting Fock space in terms of a “tower” of spin states of all possible \(j\) values. To wit, the \(\mid 00 \rangle\) state of the two oscillators corresponds to a spin-\(0\) state, the \(\mid 10 \rangle\) and @@ -718,9 +718,9 @@

    Majorana Stars and Structured Gaussian Beams\(Q\)’s and \(P\)’s are the position and momentum operators. The total angular momentum is the sum of the spin and orbital parts.

    Well, the same goes for light! On the one hand, if a light beam is cylindrically symmetric, and one choses the origin at the beam axis, then the orbital angular momentum will be 0. On the other hand, one can consider helical modes of light, such that the beam’s wavefront is corkscrewing around an “optical vortex” at the center.

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    One imagines polarization itself as a kind of corkscrewing, but this is actually a second level kind of corkscrewing. Consider this image:

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    The idea is we’re considering a spatial cross section of a beam, such that the polarization state actually varies across it! In other words, the polarization is a property of “each point” as it were, each photon, but the OAM is a property of the overall beam. This is a great reference.


    Today we’ll be considering “structured Gaussian beams.” They are solutions to the “paraxial wave equation,” in other words, the regime of something like a laser, where pretty much everything that’s going on is going on close to the axis of propagation (the small angle apparoximation). Structured Gaussian beams are “self-similar,” in that their cross-sections all look the same up to scaling. The structured part is going to make things interesting!

    Following Bassa and Konrad, we can turn Maxwell’s equations into scalar wave equations for each component of the electric and magnetic fields:

    @@ -752,31 +752,31 @@

    Majorana Stars and Structured Gaussian Beams

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    Here, \(A\) is some normalization.

    \(w_{0}\) is the “waist radius” or “beam waist”, which measures the size of the beam at the point of focus \(z=0\), and \(w(z)\) is the beam width.

    \(R(z) = z(1 + (\frac{z_{R}}{z})^2)\) is the the radius of curvature of the beam’s wavefronts.

    \(z_{R}\) is the “Rayleigh distance”: \(\frac{\pi w_{0}^2 n}{\lambda}\), where \(\lambda\) is the wavelength and \(n\) is the index of refraction.

    \(\phi(z) = \arctan(\frac{z}{z_{R}})\) is the Gouy phase, which has a great name!

    This picture from wikipedia is illuminating:

    -

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    The intensity profile, or cross section of the beam, at its point of focus, would look like:

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    In either direction along the direction of propagation, it would look the same, only slowly increasing in size.

    So that’s the simplest case!


    We now turn to structured Gaussian beams.

    Given what I said earlier, and knowing that the energy states of the quantum harmonic oscillator can be written in terms of the Hermite polynomials, it would make sense if we could decompose our classical light beam into the so-called Hermite-Gaussian modes, which essentially consist in two Hermite polynomials in x and y multiplied by a Gaussian. Shamelessly ripped from wikipedia:

    -

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    Here \(E_{0}\) hides the normalization and \(\psi(z) = (N+1)\arctan(\frac{z}{z_{R}})\), the Gouy phase. \(N = l + m\), where \(l\) and \(m\) denote which Hermite polynomials we’re using.

    Okay. But morally speaking, it’s just a 2D quantum harmonic oscillator state energy state with some bells and whistles.

    Compare the wavefunction for a 1D quantum harmonic oscillator energy state:

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    The difference is that we’ve got this \(\frac{w_{0}}{w(z)}\) term at the front, we’re dividing things by (powers of) \(w(z)\), and we have these phase terms. The effect is to make sure we’re describing a beam.


    In what follows, however, we’ll be following this reference, and so use instead the Laguerre-Gaussian modes, which are closely related.

    We’ll see that the Hermite-Gaussian modes have horizontal/vertical symmetries in the plane, while the Laguerre-Gaussian modes have circular symmetries. And in fact, by using \(X, Y, Z\) rotations, we’ll be able to rotate Hermite-Gaussian modes into Laguerre-Gaussian modes, and vice versa, and even get modes that are in between (Hermite-Laguerre-Gaussian modes). It will be just the same correspondence that we saw before in the relationship between the sphere of a qubit and the polarization ellipse.

    The Laguerre-Gauss modes are given by:

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    Here \(\textbf{r}\) is understood as \((r, \phi, w)\): we’re working in cylindrical coordinates, where \(w\) is \(w(z)\) from above. The index \(N\) takes integer values and \(l\) runs from \(-N\) to \(N\) in steps of two. They parameterize the generalized Laguerre polynomials. We have some normalization, some phase terms, and a Gaussian falloff.

    The upshot is that any structured Gaussian beam of total order \(N\) can be decomposed in terms of LG modes as:

    @@ -1031,7 +1031,7 @@

    Majorana Stars and Structured Gaussian Beams

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    They start with a spin coherent state at \(\phi = \frac{\pi}{2}, \theta = \frac{\pi}{2}\) in spherical coordiantes, for \(j=3\). This means \(6\) stars at that point on the sphere.

    [10]:
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    Majorana Stars and Structured Gaussian Beams\(j=\frac{3}{2}\). Also, for comparison, an actual picture, grainy due to the quantum nature of the photons:

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    [20]:
     
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:[116,139,145,171,172],calibr:70,call:[170,171,172,174],can:[73,80,94,106,114,155,162,168,169,170,171,172,173,174],canva:[155,170,171,173],cardin:170,care:174,carri:171,cartesian:[4,5,11,14,17,18,19,20,94,97,99,101,104,106,149,155,170,171,174],cascad:174,caus:94,cba:174,cbad:174,cbda:174,cbit_regist:75,cdab:174,cdba:174,center:171,certain:[45,172],certainli:171,chanc:[94,170],chang:[167,173],channel:171,character:170,charg:170,chart:[104,170],check:[168,169,170,171,172,174],chiark:38,choic:[171,174],choos:[169,170,173],chose:171,chosen:[169,171],circ:[169,174],circ_dist:169,circ_error:169,circ_info:[47,53,55,65,73,75,169,174],circ_meas_filt:169,circ_noise_model:169,circl:170,circuit:[42,43,44,46,47,48,49,50,51,52,54,55,56,57,60,62,63,64,65,68,73,75,168,169,172,174],circuit_info:[46,54,62,75],circular:171,circularli:171,classic:[5,46,53,54,171],classicalregist:174,clbit_label:174,clean:169,clean_sym_circ_count:169,cleanli:171,clear:[155,170,171],clear_snapshot:155,clear_trail:155,clearli:[169,170,172,174],clebsch:173,clockwis:171,clone:174,close:171,cnot:169,cntrl_bit:[46,54,75],cntrl_qubit:[46,54,75],code:[171,172,174],coeff:[138,152,173],coeffici:[79,81,82,86,89,90,91,94,101,104,128,170,173],coher:[94,106,155,156,168,171],coincid:[170,171],collect:[173,174],color:[162,170,171,173],colorsi:171,column:172,com:[3,145,171],combin:[79,172],combine_result:174,come:[114,170,171,172,173,174],commun:171,compar:[64,130,131,132,171,173,174],compare_nophas:170,comparison:[171,172],complet:[170,171,173],complex:[3,9,10,11,12,15,18,20,40,88,89,91,92,93,94,99,101,104,105,106,113,114,115,116,117,120,130,141,143,144,147,155,168,170,171,172,173,174],complex_real_symplect:122,compon:[94,101,104,138,143,144,152,168,170,171,174],compris:169,comput:[75,112,145,168,169,171],concaten:174,concentr:[80,94,162,170,171,173],conceptu:174,concern:[172,174],conclud:171,condit:[168,169,170,172,174],configur:171,confin:[170,171],confirm:172,conj:170,conjug:[94,101,104,170],connect:174,consid:[170,171,172,173,174],consist:[169,170,171],consol:170,constant:[40,94,171],constel:[28,80,162,169,171,172,173],construct:[22,23,24,25,27,28,29,31,32,34,35,36,46,48,54,58,62,75,79,80,81,83,84,85,108,109,110,111,112,115,157,168,169,170,171,172,173,174],construct_cnrl_permut:174,construct_permut:174,construct_qft:174,contain:[73,92,95,97,99,156,164,169,173,174],context:[21,107,171,172,174],continu:[73,136],contract:173,contribut:[94,97,168,170],control:[43,44,46,51,52,54,55,64,75,169,172,174],contruct:174,convent:[14,19,171,174],convention:172,convert:[3,5,9,10,12,13,14,15,16,17,19,20,69,71,86,90,94,108,110,113,114,115,116,122,134,170,172,173,174],convinc:174,coodin:4,cool:170,coord:106,coordiant:171,coordin:[4,5,40,92,94,95,97,99,101,104,106,136,146,147,148,149,155,157,170,171,172,173,174],copi:[64,75,155,169,171,174],core:159,corkscrew:171,correct:[75,168,169,171,172,174],correct_antisym:174,correct_dm:174,correct_jrho:174,correct_st:174,correct_sym:174,correspond:[2,7,23,31,40,48,53,79,81,82,84,86,88,89,92,97,98,99,112,124,129,138,142,152,157,169,170,171,172,173,174],cos:[14,172,174],cost:169,could:[67,101,104,169,170,171,173,174],count:[63,68,69,71,112,170,171,172,174],counterclockwis:171,coupl:173,cours:[169,170,171,174],cover:170,cperm:174,creat:174,creation:[28,29,114,116,171,172],creg_siz:174,cross:[171,173],crossov:171,crucial:173,cswap:[169,174],curvatur:171,cut:170,cute:170,cutoff:[22,26,30,31,32,35,36],cutoff_dim:[22,26,30,31,32,35,36,171,172],cylindr:[4,5,171],dabc:174,dacb:174,dag:[94,170,173,174],dagger:[43,44,114,172,174],data:[45,170,173,174],dbac:174,dbca:174,dcab:174,dcba:174,deal:[21,168,169,171,173],decim:[131,132],decompos:[79,171,172,173],decomposit:[81,82,86,93,95,96,162,171,172,173],decreas:171,deepcopi:174,def:[169,171,173,174],defin:[94,99,106,168,171,173,174],definit:114,degener:171,degre:[89,94,168,170,172,174],deliv:169,demonstr:170,denni:171,denot:[170,171],densiti:[47,49,55,134,138,162,173,174],depend:[46,54,169,171,174],depict:[170,171,172],depth:[64,67,73,169],deriv:173,describ:[171,172,173],desir:[45,171,172,174],destroi:[155,162,165],detail:[171,173],detector:172,determin:[170,172,173],determinist:174,deutsch:174,diag:174,diagon:[24,171,172],diagram:[172,174],dict:[30,34,46,47,49,53,54,55,58,62,63,64,65,68,69,71,73,75,79,80,81,82,84,86,112,124,138,142,152],dictionari:[30,34,46,48,53,54,63,64,65,68,73,75,79,86,112,138,142,152,169,174],did:[169,171],differ:[169,171,172,173],differenti:173,difficulti:169,dim:[33,129,170,173,174],dimens:[22,26,30,31,32,33,35,36,102,109,112,120,121,123,129,165,174],dimension:[26,112,129,170,171,172,173,174],dirac:[171,172],direct:[94,103,106,165,168,169,170,171,172,174],directli:[101,104,172,174],discret:[164,171],discuss:[169,172,174],disguis:172,displac:[114,115,116,122,123,172],displai:[159,160,165,169],display_dist:169,display_sym_dist:169,disregard:[130,131],dissapear:172,dist:[65,169],distanc:171,distinct:171,distribut:[63,68,112,147,170],disturb:169,ditch:173,divid:[73,170,171,172],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sph","spheres.coordinates.spinor_xyz","spheres.coordinates.xyz_c","spheres.coordinates.xyz_sph","spheres.coordinates.xyz_spinor","spheres.oscillators","spheres.oscillators.annihilators","spheres.oscillators.osc_spin","spheres.oscillators.osc_spinblocks","spheres.oscillators.osc_spins","spheres.oscillators.osc_spintower_map","spheres.oscillators.second_quantize_operator","spheres.oscillators.second_quantize_spin_state","spheres.oscillators.second_quantize_state","spheres.oscillators.second_quantized_paulis","spheres.oscillators.spin_osc","spheres.oscillators.spin_osc_map","spheres.oscillators.spin_tower_dimensions","spheres.oscillators.spinj_xyz_osc","spheres.oscillators.spins_osc","spheres.oscillators.vacuum","spheres.polyhedra","spheres.polyhedra.equidistribute_points","spheres.relativity","spheres.relativity.mobius","spheres.spin_circuits","spheres.spin_circuits.pytket","spheres.spin_circuits.pytket.Rk_pytket","spheres.spin_circuits.pytket.Tkj_pytket","spheres.spin_circuits.pytket.postselect_shots","spheres.spin_circuits.pytket.spin_sym_pytket","spheres.spin_circuits.pytket.spin_tomography_pytket","spheres.spin_circuits.pytket.tomography_circuits_pytket","spheres.spin_circuits.pytket.tomography_dm_pytket","spheres.spin_circuits.qiskit","spheres.spin_circuits.qiskit.Rk_qiskit","spheres.spin_circuits.qiskit.Tkj_qiskit","spheres.spin_circuits.qiskit.postselect_results_qiskit","spheres.spin_circuits.qiskit.spin_sym_qiskit","spheres.spin_circuits.qiskit.spin_tomography_qiskit","spheres.spin_circuits.strawberryfields","spheres.spin_circuits.strawberryfields.spin_osc_strawberryfields","spheres.spin_circuits.strawberryfields.spinj_xyz_strawberryfields","spheres.stabilization","spheres.stabilization.pytket","spheres.stabilization.pytket.bitflip_noise_model","spheres.stabilization.pytket.build_pytket_circuit","spheres.stabilization.pytket.eval_pytket_circuit_ibm","spheres.stabilization.pytket.eval_pytket_symmetrization_performance","spheres.stabilization.pytket.eval_symmetrized_pytket_circuit","spheres.stabilization.pytket.ibmq_16_melbourne_noise_model","spheres.stabilization.pytket.plot_symmetrization_performance","spheres.stabilization.pytket.process_symmetrized_pytket_counts","spheres.stabilization.pytket.pytket_qiskit_counts","spheres.stabilization.pytket.qiskit_error_calibration","spheres.stabilization.pytket.qiskit_pytket_counts","spheres.stabilization.pytket.random_pairs","spheres.stabilization.pytket.random_pytket_circuit","spheres.stabilization.pytket.random_unique_pairs","spheres.stabilization.pytket.symmetrize_pytket_circuit","spheres.stabilization.pytket.thermal_noise_model","spheres.stars","spheres.stars.mixed","spheres.stars.mixed.operator_spherical_decomposition","spheres.stars.mixed.operator_spins","spheres.stars.mixed.spherical_decomposition_operator","spheres.stars.mixed.spherical_decomposition_spins","spheres.stars.mixed.spherical_tensor","spheres.stars.mixed.spherical_tensor_basis","spheres.stars.mixed.spins_operator","spheres.stars.mixed.spins_spherical_decomposition","spheres.stars.pure","spheres.stars.pure.c_spin","spheres.stars.pure.poly_roots","spheres.stars.pure.poly_spin","spheres.stars.pure.roots_poly","spheres.stars.pure.sph_spin","spheres.stars.pure.spin_c","spheres.stars.pure.spin_poly","spheres.stars.pure.spin_sph","spheres.stars.pure.spin_spinors","spheres.stars.pure.spin_xyz","spheres.stars.pure.spinors_spin","spheres.stars.pure.xyz_spin","spheres.stars.star_utils","spheres.stars.star_utils.antipodal","spheres.stars.star_utils.basis","spheres.stars.star_utils.pauli_eigenstate","spheres.stars.star_utils.poleflip","spheres.stars.star_utils.spherical_inner","spheres.stars.star_utils.spin_coherent","spheres.symmetrization","spheres.symmetrization.spin_sym","spheres.symmetrization.spin_sym_map","spheres.symmetrization.sym_spin","spheres.symmetrization.symmetrize","spheres.symmetrization.symmetrized_basis","spheres.symplectic","spheres.symplectic.complex_real_symplectic","spheres.symplectic.complex_real_symplectic2","spheres.symplectic.gaussian_complex_symplectic","spheres.symplectic.is_complex_symplectic","spheres.symplectic.is_real_symplectic","spheres.symplectic.make_gaussian_operator","spheres.symplectic.omega_c","spheres.symplectic.omega_r","spheres.symplectic.operator_real_symplectic","spheres.symplectic.random_gaussian_operator","spheres.symplectic.symplectic_xyz","spheres.symplectic.upgrade_single_mode_operator","spheres.symplectic.upgrade_two_mode_operator","spheres.utils","spheres.utils.binomial","spheres.utils.bitstring_basis","spheres.utils.compare_nophase","spheres.utils.compare_spinors","spheres.utils.compare_unordered","spheres.utils.components","spheres.utils.density_to_purevec","spheres.utils.dirac","spheres.utils.fix_stars","spheres.utils.flatten","spheres.utils.from_pauli_basis","spheres.utils.measure","spheres.utils.normalize","spheres.utils.normalize_phase","spheres.utils.pauli_basis","spheres.utils.phase","spheres.utils.phase_angle","spheres.utils.polygon_area","spheres.utils.qubits_xyz","spheres.utils.rand_c","spheres.utils.rand_sph","spheres.utils.rand_xyz","spheres.utils.spinj_xyz","spheres.utils.tensor_upgrade","spheres.utils.to_pauli_basis","spheres.visualization","spheres.visualization.majorana_sphere","spheres.visualization.majorana_sphere.MajoranaSphere","spheres.visualization.majorana_sphere.SphericalWavefunction","spheres.visualization.majorana_sphere.tangent_plane_rotation","spheres.visualization.matplotlib_spheres","spheres.visualization.matplotlib_spheres.animate_spin","spheres.visualization.matplotlib_spheres.viz_spin","spheres.visualization.operator_sphere","spheres.visualization.operator_sphere.OperatorSphere","spheres.visualization.schwinger_spheres","spheres.visualization.schwinger_spheres.OscillatorPlane","spheres.visualization.schwinger_spheres.SchwingerSpheres","spheres.visualization.vp_object","spheres.visualization.vp_object.VObject","Welcome to the documentation for spheres!","Stablization via Symmetrization","An Introduction to Majorana\u2019s \u201cStellar\u201d Representation of Spin","Majorana Stars and Structured Gaussian Beams","How To Prepare a Spin-j State on a Photonic Quantum Computer","Generalizing the Majorana Representation for Mixed States and Operators","How to Prepare a Permutation Symmetric Multiqubit State on an Actual Quantum 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"spheres.coordinates.xyz_c","spheres.coordinates.xyz_sph","spheres.coordinates.xyz_spinor","spheres.oscillators","spheres.oscillators.annihilators","spheres.oscillators.osc_spin","spheres.oscillators.osc_spinblocks","spheres.oscillators.osc_spins","spheres.oscillators.osc_spintower_map","spheres.oscillators.second_quantize_operator","spheres.oscillators.second_quantize_spin_state","spheres.oscillators.second_quantize_state","spheres.oscillators.second_quantized_paulis","spheres.oscillators.spin_osc","spheres.oscillators.spin_osc_map","spheres.oscillators.spin_tower_dimensions","spheres.oscillators.spinj_xyz_osc","spheres.oscillators.spins_osc","spheres.oscillators.vacuum","spheres.polyhedra","spheres.polyhedra.equidistribute_points","spheres.relativity","spheres.relativity.mobius","spheres.spin_circuits","spheres.spin_circuits.pytket","spheres.spin_circuits.pytket.Rk_pytket","spheres.spin_circuits.pytket.Tkj_pytket","spheres.spin_circuits.pytket.postselect_shots","spheres.spin_circuits.pytket.spin_sym_pytket","spheres.spin_circuits.pytket.spin_tomography_pytket","spheres.spin_circuits.pytket.tomography_circuits_pytket","spheres.spin_circuits.pytket.tomography_dm_pytket","spheres.spin_circuits.qiskit","spheres.spin_circuits.qiskit.Rk_qiskit","spheres.spin_circuits.qiskit.Tkj_qiskit","spheres.spin_circuits.qiskit.postselect_results_qiskit","spheres.spin_circuits.qiskit.spin_sym_qiskit","spheres.spin_circuits.qiskit.spin_tomography_qiskit","spheres.spin_circuits.strawberryfields","spheres.spin_circuits.strawberryfields.spin_osc_strawberryfields","spheres.spin_circuits.strawberryfields.spinj_xyz_strawberryfields","spheres.stabilization","spheres.stabilization.pytket","spheres.stabilization.pytket.bitflip_noise_model","spheres.stabilization.pytket.build_pytket_circuit","spheres.stabilization.pytket.eval_pytket_circuit_ibm","spheres.stabilization.pytket.eval_pytket_symmetrization_performance","spheres.stabilization.pytket.eval_symmetrized_pytket_circuit","spheres.stabilization.pytket.ibmq_16_melbourne_noise_model","spheres.stabilization.pytket.plot_symmetrization_performance","spheres.stabilization.pytket.process_symmetrized_pytket_counts","spheres.stabilization.pytket.pytket_qiskit_counts","spheres.stabilization.pytket.qiskit_error_calibration","spheres.stabilization.pytket.qiskit_pytket_counts","spheres.stabilization.pytket.random_pairs","spheres.stabilization.pytket.random_pytket_circuit","spheres.stabilization.pytket.random_unique_pairs","spheres.stabilization.pytket.symmetrize_pytket_circuit","spheres.stabilization.pytket.thermal_noise_model","spheres.stars","spheres.stars.mixed","spheres.stars.mixed.operator_spherical_decomposition","spheres.stars.mixed.operator_spins","spheres.stars.mixed.spherical_decomposition_operator","spheres.stars.mixed.spherical_decomposition_spins","spheres.stars.mixed.spherical_tensor","spheres.stars.mixed.spherical_tensor_basis","spheres.stars.mixed.spins_operator","spheres.stars.mixed.spins_spherical_decomposition","spheres.stars.pure","spheres.stars.pure.c_spin","spheres.stars.pure.poly_roots","spheres.stars.pure.poly_spin","spheres.stars.pure.roots_poly","spheres.stars.pure.sph_spin","spheres.stars.pure.spin_c","spheres.stars.pure.spin_poly","spheres.stars.pure.spin_sph","spheres.stars.pure.spin_spinors","spheres.stars.pure.spin_xyz","spheres.stars.pure.spinors_spin","spheres.stars.pure.xyz_spin","spheres.stars.star_utils","spheres.stars.star_utils.antipodal","spheres.stars.star_utils.basis","spheres.stars.star_utils.pauli_eigenstate","spheres.stars.star_utils.poleflip","spheres.stars.star_utils.spherical_inner","spheres.stars.star_utils.spin_coherent","spheres.symmetrization","spheres.symmetrization.spin_sym","spheres.symmetrization.spin_sym_map","spheres.symmetrization.sym_spin","spheres.symmetrization.symmetrize","spheres.symmetrization.symmetrized_basis","spheres.symplectic","spheres.symplectic.complex_real_symplectic","spheres.symplectic.complex_real_symplectic2","spheres.symplectic.gaussian_complex_symplectic","spheres.symplectic.is_complex_symplectic","spheres.symplectic.is_real_symplectic","spheres.symplectic.make_gaussian_operator","spheres.symplectic.omega_c","spheres.symplectic.omega_r","spheres.symplectic.operator_real_symplectic","spheres.symplectic.random_gaussian_operator","spheres.symplectic.symplectic_xyz","spheres.symplectic.upgrade_single_mode_operator","spheres.symplectic.upgrade_two_mode_operator","spheres.utils","spheres.utils.binomial","spheres.utils.bitstring_basis","spheres.utils.compare_nophase","spheres.utils.compare_spinors","spheres.utils.compare_unordered","spheres.utils.components","spheres.utils.density_to_purevec","spheres.utils.dirac","spheres.utils.fix_stars","spheres.utils.flatten","spheres.utils.from_pauli_basis","spheres.utils.measure","spheres.utils.normalize","spheres.utils.normalize_phase","spheres.utils.pauli_basis","spheres.utils.phase","spheres.utils.phase_angle","spheres.utils.polygon_area","spheres.utils.qubits_xyz","spheres.utils.rand_c","spheres.utils.rand_sph","spheres.utils.rand_xyz","spheres.utils.spinj_xyz","spheres.utils.tensor_upgrade","spheres.utils.to_pauli_basis","spheres.visualization","spheres.visualization.majorana_sphere","spheres.visualization.majorana_sphere.MajoranaSphere","spheres.visualization.majorana_sphere.SphericalWavefunction","spheres.visualization.majorana_sphere.tangent_plane_rotation","spheres.visualization.matplotlib_spheres","spheres.visualization.matplotlib_spheres.animate_spin","spheres.visualization.matplotlib_spheres.viz_spin","spheres.visualization.operator_sphere","spheres.visualization.operator_sphere.OperatorSphere","spheres.visualization.schwinger_spheres","spheres.visualization.schwinger_spheres.OscillatorPlane","spheres.visualization.schwinger_spheres.SchwingerSpheres","spheres.visualization.vp_object","spheres.visualization.vp_object.VObject","Welcome to the documentation for spheres!","Stablization via Symmetrization","An Introduction to Majorana\u2019s \u201cStellar\u201d Representation of Spin","Majorana Stars and Structured Gaussian Beams","How To Prepare a Spin-j State on a Photonic Quantum Computer","Generalizing the Majorana Representation for Mixed States and Operators","How to Prepare a Permutation Symmetric Multiqubit State on an Actual Quantum Computer"],titleterms:{actual:174,animate_spin:159,animate_spin_beam:2,annihil:22,antipod:101,basi:102,beam:[1,2,3,4,5,6,7,171],bibliographi:[169,170,172,173,174],binomi:128,bitflip_noise_model:61,bitstring_basi:129,build_pytket_circuit:62,c_sph:9,c_spin:88,c_spinor:10,c_xyz:11,coher:170,color:3,compare_nophas:130,compare_spinor:131,compare_unord:132,complex_real_symplect:114,complex_real_symplectic2:115,compon:133,comput:[172,174],constel:170,coordin:[8,9,10,11,12,13,14,15,16,17,18,19,20],density_to_purevec:134,dirac:135,document:168,equidistribute_point:38,eval_pytket_circuit_ibm:63,eval_pytket_symmetrization_perform:64,eval_symmetrized_pytket_circuit:65,fix_star:136,flatten:137,from_pauli_basi:138,gaussian:171,gaussian_complex_symplect:116,gener:173,how:[172,174],ibmq_16_melbourne_noise_model:66,indic:168,introduct:170,is_complex_symplect:117,is_real_symplect:118,laguerre_gauss_mod:4,majorana:[170,171,173],majorana_spher:[154,155,156,157],majoranaspher:155,make_gaussian_oper:119,matplotlib_spher:[158,159,160],measur:139,mix:[78,79,80,81,82,83,84,85,86,173],mobiu:40,multiqubit:174,normal:140,normalize_phas:141,omega_c:120,omega_r:121,oper:173,operator_real_symplect:122,operator_spher:[161,162],operator_spherical_decomposit:79,operator_spin:80,operatorspher:162,osc_spin:[23,25],osc_spinblock:24,osc_spintower_map:26,oscil:[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],oscillatorplan:164,pauli_basi:142,pauli_eigenst:103,permut:174,phase:143,phase_angl:144,photon:172,plot_symmetrization_perform:67,poleflip:104,poly_root:89,poly_spin:90,polygon_area:145,polyhedra:[37,38],postselect_results_qiskit:53,postselect_shot:45,prepar:[172,174],process_symmetrized_pytket_count:68,pure:[87,88,89,90,91,92,93,94,95,96,97,98,99],pytket:[42,43,44,45,46,47,48,49,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76],pytket_qiskit_count:69,qiskit:[50,51,52,53,54,55],qiskit_error_calibr:70,qiskit_pytket_count:71,quantum:[172,174],qubits_xyz:146,rand_c:147,rand_sph:148,rand_xyz:149,random_gaussian_oper:123,random_pair:72,random_pytket_circuit:73,random_unique_pair:74,refer:171,rel:[39,40],represent:[170,173],rk_pytket:43,rk_qiskit:51,roots_poli:91,schwinger_spher:[163,164,165],schwingerspher:165,second_quantize_oper:27,second_quantize_spin_st:28,second_quantize_st:29,second_quantized_pauli:30,sph_c:12,sph_spin:92,sph_spinor:13,sph_xyz:14,sphere:[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168],spherical_decomposition_oper:81,spherical_decomposition_spin:82,spherical_inn:105,spherical_tensor:83,spherical_tensor_basi:84,sphericalwavefunct:156,spin:[170,172],spin_beam:5,spin_c:93,spin_circuit:[41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58],spin_coher:106,spin_osc:31,spin_osc_map:32,spin_osc_strawberryfield:57,spin_poli:94,spin_sph:95,spin_spinor:96,spin_sym:108,spin_sym_map:109,spin_sym_pytket:46,spin_sym_qiskit:54,spin_tomography_pytket:47,spin_tomography_qiskit:55,spin_tower_dimens:33,spin_xyz:97,spinj_xyz:150,spinj_xyz_osc:34,spinj_xyz_strawberryfield:58,spinor_c:15,spinor_sph:16,spinor_xyz:17,spinors_spin:98,spins_oper:85,spins_osc:35,spins_spherical_decomposit:86,stabil:[59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76],stabliz:169,star:[77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,170,171],star_util:[100,101,102,103,104,105,106],state:[170,172,173,174],stellar:170,strawberryfield:[56,57,58],structur:171,sym_spin:110,symmetr:[107,108,109,110,111,112,169,174],symmetrize_pytket_circuit:75,symmetrized_basi:112,symplect:[113,114,115,116,117,118,119,120,121,122,123,124,125,126],symplectic_xyz:124,tabl:168,tangent_plane_rot:157,tensor_upgrad:151,thermal_noise_model:76,tkj_pytket:44,tkj_qiskit:52,to_pauli_basi:152,tomography_circuits_pytket:48,tomography_dm_pytket:49,upgrade_single_mode_oper:125,upgrade_two_mode_oper:126,util:[127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152],vacuum:36,via:169,visual:[153,154,155,156,157,158,159,160,161,162,163,164,165,166,167],viz_beam:6,viz_spin:160,viz_spin_beam:7,vobject:167,vp_object:[166,167],welcom:168,xyz_c:18,xyz_sph:19,xyz_spin:99,xyz_spinor:20}}) 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