Simultaneous Sylvester Systems & Symmetries

Joshua Maglione

joshmaglione.com

Creative Commons Attribution 4.0 International License

A related quantum problem

A qubit (quantum bit) is modeled by \(q \in\mathbb{C}^2\) via spin of electron:

q = \alpha \cdot (\uparrow) + \beta \cdot (\downarrow) = \alpha | 0 \rangle + \beta | 1 \rangle.

Measuring a qubit:

\begin{aligned} P(q=0) &= |\alpha|^2 &&&&& P(q=1) &= |\beta|^2 &&&\Longrightarrow&&& |\alpha|^2 + |\beta|^2 = 1 \end{aligned}

(simplified!)

Logical gates (e.g. NOT, AND, OR) modeled by unitary group:

\mathrm{U}_2 = \left\{X \in \mathrm{GL}_2(\mathbb{C}) ~\middle|~ X \overline{X}^{\mathrm{t}} = I\right\}

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

| 0 \rangle \overset{H}{\longrightarrow} \frac{1}{\sqrt{2}} (|0\rangle + | 1\rangle)

System of qubits \(q_1,\dots, q_n\in\mathbb{C}^2\) contained in the Hilbert space

q_1\otimes \cdots \otimes q_n \in\mathbb{C}^{2^n}

System \(|\Phi\rangle\in \mathbb{C}^{2^n}\) not entangled if written as \(|\Phi\rangle = q_1\otimes \cdots \otimes q_n\).

\sqrt{2} \cdot |GHZ\rangle = |000\rangle + |111\rangle

\sqrt{3} \cdot |W\rangle = |001\rangle + |010\rangle + |100\rangle

\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}

\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}

\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}

front

back

multi-way array

Problem. Understand when two systems \(|\Phi\rangle, |\Psi\rangle\) equivalent:

LU-equivalent: \(\exists X_1,\dots, X_n\in \mathrm{U}_2\) such that

X_1\otimes\cdots\otimes X_n |\Phi\rangle = \sum_i(X_1q_{i1})\otimes \cdots \otimes (X_nq_{in}) = |\Psi\rangle

SLOCC-equivalent: \(\exists X_1,\dots, X_n\in \mathrm{SL}_2\) such that

(same condition as above)

A nearly identical problem exists in algebra

are LU- and SLOCC-inequivalent

Nilpotent groups & tensors

Nilpotent group \(G\) of class \(c\):

\(G = \gamma_1(G) \supset \gamma_2(G) \supset \cdots \supset \gamma_c(G) \supset \gamma_{c+1}(G) = \{1\}.\)

Important operation: commutator \([\cdot,\cdot]: G\times G \to G\) via

(x,y)\mapsto x^{-1}y^{-1}xy

Recursively define: \(\gamma_1(G) = G\) and for all \(k\geqslant 1\),

\gamma_{k+1}(G) = [\gamma_{k}(G), G] = \langle [x,y] ~|~ x\in\gamma_k(G), y\in G\rangle

Lower central series terminates:

Commutator yields a product:

[\gamma_a(G),\ \gamma_b(G)] \subseteq \gamma_{a+b}(G)

\begin{array}{ccccc} \gamma_a/\gamma_{a+1} & \times & \gamma_b/\gamma_{b+1} & \longrightarrow & \gamma_{a+b}/\gamma_{a+b+1} \\ & & & & \\ & & & & \\ \Z^{r_a} & \times & \Z^{r_b} & \longrightarrow & \Z^{r_{a+b}} \end{array}

\rightsquigarrow

a \(\Z\)-bilinear map

Problem. Given two class \(2\) nilpotent groups \(G,H\):

Isomorphism: \(\exists X_1\in\mathrm{GL}_r(\Z), X_2\in\mathrm{GL}_s(\Z)\) such that

Automorphism: \(\exists X_1\in\mathrm{GL}_r(\Z), X_2\in\mathrm{GL}_s(\Z)\) such that

G \supset \gamma_2(G) \supset \{1_G\}

H \supset \gamma_2(H) \supset \{1_H\}

[\cdot , \cdot]_G,\ [\cdot , \cdot]_H : \Z^r \times \Z^r \longrightarrow \Z^s

(same condition as above with \([\cdot,\cdot]_G\))

Tensor Isomorphism

Both problems special cases of tensor isomorphism

b : \Z^{r_i} \times \Z^{r_j} \longrightarrow \Z^{r_k}

multi-way arrays

bilinear maps

Given two \(K\)-tensors \(s, t : V_1\times \cdots \times V_n \to V_0\) decide if \(s\) equals \(t\) after a change of bases in \(V_i\) for all \(i\).

Goal: see how algebras associated to tensors help.

Groups from elliptic curves

Joint with Mima Stanojkovski (arXiv:2212.03941)

From \(E\) and \(P\) build \(3\times 3\times 3\) tensor:

\left( \begin{bmatrix} 1 & 0 & 0 \\ 0 & x_0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} -x_0 & -y_0 & 0 \\ y_0 & a + x_0^2 & 0 \\ 0 & 0 & -1 \end{bmatrix} \right)

Group scheme

\mathbf{G}_{E, P}

over \(\Z\)

E : y^2 = x^3 + ax + b

Elliptic curve

P = (x_0, y_0)\in\Z^2

linear determinantal

representation

Build a group from \(t : U\times V \to W\), due to Baer (more general).

Define \(G_t = U\oplus V\oplus W\) with multiplication:

(u,v,w) \cdot (u',v',w') = (u+u',\ v+v',\ w+w' + \frac{1}{2}(t(u, v') - t(u', v)))

Important feature: the commutator tensor of \(G_t\) is

[\cdot,\cdot]_{G_t} : (U\oplus V) \times (U\oplus V) \to W

((u,v),\ (u',v')) \mapsto t(u, v') - t(u', v)

Lots of class 2 nilpotent groups arise this way:

(infinite) finitely generated, torsion-free groups,
(finite) groups \(G\) such that for some prime \(p\), \(g^p=1_G\) for all \(g\in G\).

Problems:

For ring \(R\), when is \(\mathbf{G}_{E, P}(R) \cong \mathbf{G}_{E',P'}(R)\)?

Decide in polynomial time if \(G \cong \mathbf{G}_{E, P}(\mathbb{F}_q)\).

For rings \(R\) with \(2^{-1}\in R\),

\mathbf{G}_{E, P}(R) = G_t

where \(t : R^3\times R^3 \to R^3\) from \((E, P)\).

These boil down to instances of the tensor isomorphism problem.

Use algebras associated to tensors to reduce the complexity.

Adjoint algebra

Suppose \(t : U\times V \to W\) is a \(K\)-tensor.

\mathrm{Adj}(t) = \left\{ (X, Y^{\mathrm{t}}) ~\middle|~ \begin{array}{c} \forall u \in U,\ \forall v\in V \\ t(Xu, v) = t(u, Yv) \end{array} \right\}

t : U \times V \longrightarrow W

t(u, v)

tensor

evaluation

\mathrm{Adj}(t) = \left\{ (X, Y^{\mathrm{t}}) ~\middle|~ \begin{array}{c} \forall u \in U,\ \forall v\in V \\ t(Xu, v) = t(u, Yv) \end{array} \right\}

Equality \(t(Xu, v) = t(u, Yv)\) implies equality on each slice

\(\displaystyle\left\{\phantom{\sum_{k=0}^n}\right.\)

A_i

\(\displaystyle\left\{\phantom{\sum_{k=0}^n}\right.\)

A_i

\(\displaystyle\left\{\phantom{\sum_{k=0}^n}\right.\)

Ignore \(u\) and \(v\), so the linear system for \(\mathrm{Adj}(t)\) is a Sylvester system:

(\forall i)\left(XA_i - A_iY = 0\right)

XA+BX=C

Sylvester equation

What's the point (of adjoints)?

For \(t : U\times V\to W\), the \(\mathrm{Adj}(t)\) contains structural information.

Certain elements of \(\mathrm{Adj}(t)\) yield change of bases into nice forms:

dense tensor

perp-space

decomposition

isotropic

subspaces

Assume \(V=U\) and \(t(v,v) = 0\) for all \(v\in V\)

For group schemes coming from geometry, e.g. our construction:

this "isotropic decomposition"

\mathbf{G}

Group scheme

over \(\Z\)

always exists and can be constructed in polynomial time.

Useful for recognition: is my group one of those groups?

Centroids & Coefficients

Many different algebras associated to tensors.

\mathrm{Cent}(t) = \left\{(X,Y,Z) ~\middle|~ \begin{array}{c} \forall u\in U,\ \forall v\in V \\ t(Xu, v) = t(u, Yv) = Z\cdot t(u,v) \end{array}\right\}

The centroid of \(t : U \times V \to W\) is

Like with the adjoint algebra, centroid gives structural information.

Certain elements in \(\mathrm{Cent}(t)\) can decompose tensor:

Satisfies universal property: largest ring \(C\) such that \(t\) is \(C\)-bilinear.

\mathrm{Cent}(t) = \left\{(X,Y,Z) ~\middle|~ \begin{array}{c} \forall u\in U,\ \forall v\in V \\ t(Xu, v) = t(u, Yv) = Z\cdot t(u,v) \end{array}\right\}

Example: \(t\) starts as \(\mathbb{Q}\)-bilinear, but \(\mathrm{Cent}(t)\) is a field extension.

\left( \begin{bmatrix} 6 & 2 \\ 8 & 3 \end{bmatrix},\ \begin{bmatrix} 7 & 2 \\ 4 & 1 \end{bmatrix} \right)

Let \(t : \mathbb{Q}^2 \times \mathbb{Q}^2 \to \mathbb{Q}^2\) with multi-way array:

\mathrm{Cent}(t) = \left\langle \begin{pmatrix} 0 & 2 \\ -4 & 7 \end{pmatrix},\ \begin{pmatrix} -8 & 16 \\ -4 & 15 \end{pmatrix},\ \begin{pmatrix} 5 & -1 \\ -2 & 2 \end{pmatrix} \right\rangle \cong \mathbb{Q}(\sqrt{17})

Writing \(t\) over \(K = \mathbb{Q}(\sqrt{17})\), we have \(s : K \times K \to K\) given by

\begin{bmatrix} (33 + 7\sqrt{17})/2 \end{bmatrix}

Deciding isomorphism

Back to groups coming from elliptic curves

Previously, it was \(O(n^{\log n})\), where \(n=|G|\) -- this is brute force.

Using just the centroid, there is a \(O(|G|)\)-algorithm.

Goal: a polynomial-time algorithm in \(\log|G|\), size of input.

Theorem (M--Stanojkovski). There exists an algorithm that, given a finite group \(G\), decides if there exist

prime power \(q\),
\(a,b\in\mathbb{F}_q\) such that \(y^2 = x^3 +ax+b\) defines an elliptic curve \(E\),
\(P\in\mathbb{F}_q^2\) on \(E\) (i.e. satisfies the equation),

such that \(G\cong \mathbf{G}_{E, P}(\mathbb{F}_q)\) using \(O(q) = O(|G|^{1/9})\) operations.