Universal extensions of restricted classes of quantum operations

TL;DR

This paper classifies what additional gates are needed to achieve universal quantum operations in three key restricted settings, using group and control theory, for systems with arbitrary particles and dimensions.

Contribution

It provides a complete classification of gate sets needed for universality in bosonic and fermionic linear optics scenarios, extending the understanding of quantum control.

Findings

Classified gate sets for particle-number preserving bosonic linear optics.

Classified gate sets for particle-number preserving fermionic linear optics.

Provided complete solutions for universality in various particle and dimension settings.

Abstract

For numerous applications of quantum theory it is desirable to be able to apply arbitrary unitary operations on a given quantum system. However, in particular situations only a subset of unitary operations is easily accessible. This raises the question of what additional unitary gates should be added to a given gate-set in order to attain physical universality, i.e., to be able to perform arbitrary unitary transformation on the relevant Hilbert space. In this work, we study this problem for three paradigmatic cases of naturally occurring restricted gate-sets: (A) particle-number preserving bosonic linear optics, (B) particle-number preserving fermionic linear optics, and (C) general (not necessarily particle-number preserving) fermionic linear optics. Using tools from group theory and control theory, we classify, in each of these scenarios, what sets of gates are generated, if an additional gate is added to the set of allowed transformations. This allows us to solve the universality problem completely for arbitrary number of particles and for arbitrary dimensions of the single-particle Hilbert space.

Tables

Table 1. Table 1: Notation used throughout the paper.

Symbol	Explanation
$d$	dimension of the local Hilbert space (number of modes)
${\mathscr{H}}_b$	Hilbert space of $N$ bosons in $d$ modes
${\mathscr{H}}_b$	Hilbert space of $N$ fermions in $d$ modes
${\mathscr{H}}_{\mathrm{Fock}}$	fermionic Fock space for $d$ modes
${\mathscr{H}}_{\mathrm{Fock}}^{+}$	positive-parity subspace of fermionic Fock space for $d$ modes
${\mathrm{LO}}_b$	passive bosonic linear optics acting on $N$ bosons in $d$ modes
${\mathrm{LO}}_f$	passive fermionic linear optics acting on $N$ fermions in $d$ modes
$\mathrm{FLO}$	active fermionic linear optics acting on ${\mathscr{H}}_{\mathrm{Fock}}^{+}$ for $d$ modes
$\mathrm{SU}\left(\mathscr{H}\right)$	special unitary group on Hilbert space $\mathscr{H}$
$\mathrm{SO}\left(\mathscr{H}\right)$	special orthogonal group on Hilbert space $\mathscr{H}$
$\mathrm{USp}\left(\mathscr{H}\right)$	unitary symplectic group on Hilbert space $\mathscr{H}$
$\mathrm{SU}\left(d\right)$	special unitary group on ${ℂ}^d$
$\mathrm{Spin}\left(2d\right)$	Spinor group of ${ℝ}^{2d}$
$\mathrm{Lie}\left(K\right)$	Lie algebra of a Lie group $K$
$\Pi ,\pi$	representations of a Lie group and Lie algebra respectively
$\mathrm{𝟙}$	Identity operator on the relevant Hilbert space
$𝕋\left(\mathscr{H}\right)$	Unitary gates proportional to identity on Hilbert space $\mathscr{H}$
$f_i,f_i^{†}$	fermionic anihilation and creation operators
$m_i$	Majorana-fermion operator
$Q$	fermionic parity operator acting in ${\mathscr{H}}_{\mathrm{Fock}}$
$|n_1,n_2,\dots ,n_d⟩$	fermionic Fock state with occupation number $n_k$ in mode $k$
$\left[d\right]$	$d$ -element set $\{1,2,\dots ,d\}$

Equations

$$V_t=\exp \left(-it{\hat{n}}_a{\hat{n}}_b\right),$$

$${\mathrm{e}}^{2it\left[l\left(N-l\right)-k\left(N-k\right)\right]}\ne 1,$$

$$\mathbb{L}_{b}=\mbox{$|\Psi_{b}\rangle\!\langle\Psi_{b}|$},\ \mbox{$|\Psi_{b}\rangle$}=\sum_{k=0}^{N}(-1)^{k}\mbox{$|D_{k}\rangle$}\mbox{$|D_{N-k}\rangle$}\in\mathcal{H}_{b}\otimes\mathcal{H}_{b}\ ,$$

$$⟨{\mathrm{LO}}_b,V⟩=G_b=\{U\in \mathrm{U}\left({\mathcal{H}}_b\right)|\left[U\otimes U,{\mathbb{L}}_b\right]=0\}.$$

$$⟨{\mathrm{LO}}_b,X⟩=G_b=\{U\in \mathrm{U}\left({\mathcal{H}}_b\right)|\left[U\otimes U,{\mathbb{L}}_b\right]=0\}.$$

$$\mathbb{L}_{f}=\mbox{$|\Psi_{f}\rangle\!\langle\Psi_{f}|$},\text{ with }\mbox{$|\Psi_{f}\rangle$}=\mbox{$|1\rangle$}\wedge\mbox{$|2\rangle$}\wedge\ldots\wedge\mbox{$|2N\rangle$}\in\mathcal{H}_{f}\otimes\mathcal{H}_{f}$$

$$⟨{\mathrm{LO}}_f,V⟩=G_f=\{U\in \mathrm{U}\left({\mathcal{H}}_f\right)|[V\otimes V,{\mathbb{L}}_f]=0\}.$$

$$Y=\sum _{j=1}^{d/2-1}$$

$$({\hat{n}}_{2j-1}-{\hat{n}}_{2j+1}{)}^2(f_{2j}^{†}{f}_{2j+2}^{}+f_{2j+2}^{†}{f}_{2j}^{}).$$

$${\mathbb{L}}_{\mathrm{FLO}}=\frac{1}{2^{d(2d-1)}}\prod _{1\le i<j\le 2d}\left(I\otimes I+m_i{m}_j\otimes m_i{m}_j\right).$$

$$⟨\mathrm{FLO},V⟩=G_{\mathrm{FLO}}=\{U\in \mathrm{U}\left({\mathcal{H}}_{\mathrm{Fock}}^{+}\right)|[U\otimes U,{\mathbb{L}}_{\mathrm{FLO}}]=0\}.$$

$$\Pi :K\ni k\mapsto \Pi (k)\in \mathrm{U}(\mathcal{H})\text{satisfying}\Pi (k_1{k}_2)=\Pi (k_1)\Pi (k_2)\text{for all}{k}_1,k_2\in K.$$

$$\mathrm{Lie}\left(K\right)\coloneqq\mbox{$\left\{\left.X\in\mathrm{Herm}\left(\mathcal{H}_{K}\right)\ \right|\ \mathrm{exp}\left(\mathrm{i}X\right)\in K\right\}$}\ .$$

$${\Pi }_{\ast }:\mathrm{Lie}(K)\in X\mapsto \pi (X)\in \mathrm{Herm}(H),\text{defined by}{\Pi }_{\ast }(X):={\frac{d}{dt}|}_{t=0}\Pi \left(\exp (\mathrm{i}tX)\right).$$

$$\mathcal{H}={\oplus }_i{\mathcal{H}}_i,\Pi ={\oplus }_i{\Pi }_i,$$

$$\mathrm{i}[X,\mathcal{I}]\subset \mathcal{I}\text{for all}X\in \mathrm{Lie}(K).$$

$$\mathrm{SU}(d)=\mbox{$\left\{\left.U\in\mathbb{M}_{d\times d}(\mathbb{C})\ \right|\ UU^{\dagger}=\mathbbmss{1},\mathrm{det}(U)=1\right\}$}\ .$$

$$\mathfrak{su}(d)=\mbox{$\left\{\left.X\in\mathbb{M}_{d\times d}(\mathbb{C})\ \right|\ X=X^{\dagger},\operatorname{tr}(X)=0\right\}$}\ .$$

$$\displaystyle\mathrm{SO}(d)=\mbox{$\left\{\left.O\in\mathbb{M}_{d\times d}(\mathbb{R})\ \right|\ OO^{T}=\mathbbmss{1},\mathrm{det}(O)=1\right\}$}\ ,$$

$$\displaystyle\mathfrak{so}(d)=\mbox{$\left\{\left.X\in\mathrm{i}\mathbb{M}_{d\times d}(\mathbb{R})\ \right|\ X=-X^{T},\operatorname{tr}(X)=0\right\}$}\ .$$

$$\mathcal{J}=\bigoplus_{i=1}^{d}\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right)\ ,$$

$$B_{a}(\mbox{$|v\rangle$},\mbox{$|w\rangle$})=v^{T}\mathcal{J}w\ ,$$

$$\displaystyle\mathrm{USp}(2d)=\mbox{$\left\{\left.O\in\mathbb{M}_{2d\times 2d}(\mathbb{C})\ \right|\ UU^{\dagger}=\mathbbmss{1},U^{T}\mathcal{J}U=\mathcal{J}\right\}$}\ ,$$

$$\displaystyle\mathfrak{usp}(2d)=\mbox{$\left\{\left.X\in\mathbb{M}_{2d\times 2d}(\mathbb{C})\ \right|\ X=X^{\dagger},X^{T}\mathcal{J}+\mathcal{J}X=0\right\}$}\ .$$

$$\mathrm{Lie}\left(\mathrm{LO}_{b}\right)=\mbox{$\left\{\left.\left.\left(h\otimes\mathbbmss{1}\otimes\ldots\otimes\mathbbmss{1}+\ldots+\mathbbmss{1}\otimes\ldots\otimes\mathbbmss{1}\otimes h\right)\right|_{\mathcal{H}_{b}}\ \right|\ \ h\in\mathrm{Herm}\left(\mathbb{C}^{d}\right)\right\}$}\ ;$$

$$\mathrm{Lie}\left(\mathrm{LO}_{f}\right)=\mbox{$\left\{\left.\left.\left(h\otimes\mathbbmss{1}\otimes\ldots\otimes\mathbbmss{1}+\ldots+\mathbbmss{1}\otimes\ldots\otimes\mathbbmss{1}\otimes h\right)\right|_{\mathcal{H}_{f}}\ \right|\ \ h\in\mathrm{Herm}\left(\mathbb{C}^{d}\right)\right\}$}e\ ;$$

$$\mathrm{Lie}\left(\mathrm{FLO}\right)=\mbox{$\left\{\left.\left.\mathrm{i}\left(\sum_{kl}h_{kl}m_{k}m_{l}+\alpha Q\right)\right|_{\mathcal{H}^{+}_{\mathrm{Fock}}}\ \right|\ h_{kl}=-h_{lk}\ ,\ h_{kl}\in\mathbb{R},\ \alpha\in\mathbb{R}\right\}$}\ .$$

$$\mathrm{SU}(d)\ni U\mapsto {\Pi }_b(U)={U^{\otimes N}|}_{{\mathcal{H}}_b}\in \mathrm{U}({\mathcal{H}}_b).$$

$$\mathfrak{su}(d)\ni X⟼{\pi }_b(X)={\left(X\otimes \text{mathbbmss}1\otimes \dots \otimes \text{mathbbmss}1+\dots +\text{mathbbmss}1\otimes \dots \otimes \text{mathbbmss}1\otimes X\right)|}_{{\mathcal{H}}_b}.$$

$$\mathrm{SU}(d)\ni U\mapsto {\Pi }_f(U)={U^{\otimes N}|}_{{\mathcal{H}}_f}\in \mathrm{U}({\mathcal{H}}_f),$$

$$\mathfrak{su}(d)\ni X⟼{\pi }_f(X)={\left(X\otimes \text{mathbbmss}1\otimes \dots \otimes \text{mathbbmss}1+\dots +\text{mathbbmss}1\otimes \dots \text{mathbbmss}1\otimes X\right)|}_{{\mathcal{H}}_f}.$$

$$E_{kl}=\mathrm{i}(\mbox{$|k\rangle\!\langle l|$}-\mbox{$|l\rangle\!\langle k|$})\ ,\ 1\leq k<l\leq 2d\ .$$

Full text

In many applications of quantum mechanics it is important to have full control over a quantum system used to perform a desired task or a quantum protocol. This amounts to being able to implement arbitrary unitary operation on the system in question. Perhaps the most well-known example is the circuit model of quantum computing, where the ability to implement arbitrary unitary gates on a system of many distinguishable particles (say, qubits) is a necessary ingredient for performing universal quantum computation Kitaev et al. (2002); Nielsen and Chuang (2010). From the experimental perspective, it is typically very easy to implement single-qubit gates. This collection of gates, however, does not lead to universal quantum computation and to this aim has to be supplemented by an entangling gate Brylinski and Brylinski (2002). Similar situations appears in other physical contexts. Typically, the set of easily accessible unitary gates acting on a given quantum system, does not ensure full controllability.

This work studies the extension problem for gate-sets appearing naturally in systems consisting of non-distinguishable particles: passive linear optics for (A) bosonic and (B) fermionic systems with fixed number of particles, as well as (C) active linear optics acting on fermionic system with fixed number of modes subject to the parity super-selection rule Bravyi and Kitaev (2002). Specifically, for the aforementioned scenarios, we study what unitary transformations on the relevant Hilbert space can be implemented if the restricted class of gates $K$ is supplemented by additional unitary transformations - see Fig. 1. We investigate two variants of this problem:

• (i)

the gate-set $K$ is supplemented with unitaries of the form $\mathrm{exp}\left(-\mathrm{i}tX\right)$ generated by the Hamiltonian $X$;

• (ii)

the gate-set $K$ is supplemented by a single unitary transformation $V$.

Linear optical transformations are relevant in many contexts. Passive bosonic linear optics describes single-particle evolutions of a system of $N$ identical bosons in $d$ modes. Such transformations are natural for quantum optics, when quantum states of light pass through an optical network formed from beam-splitters and phase shifters Carolan et al. (2015). Linear optics underpins the KLM scheme of quantum computing with photons Knill et al. (2001); Kok et al. (2007) and the boson sampling strategy for demonstrating quantum supremacy with linear optical networks Aaronson and Arkhipov (2011). Moreover, this class of transformations is used to manipulate cold bosonic particles in optical traps Milburn et al. (1997); Albiez et al. (2005). Similarly, passive fermionic linear optics describes single-particle evolutions of a system of $N$ identical fermions in $d$ modes Terhal and DiVincenzo (2002); DiVincenzo and Terhal (2005), which can be realized in integer quantum hall effect systems exhibiting edge channels Bocquillon et al. (2014). Passive fermionic linear optics together with particle-number measurements yields classically simulable model of quantum computation DiVincenzo and Terhal (2005). Moreover, this class of transformations has been recently used to study correlation Tichy (2014) and nonlocality Dasenbrook et al. (2016) properties in fermionic systems. Finally, active fermionic linear optics describes free-fermion transformations that are not necessary particle preserving. These fermionic transformations are the basic ingredient of a classically simulable model of quantum computation Bravyi and Kitaev (2002); Terhal and DiVincenzo (2002); DiVincenzo and Terhal (2005). This computational model has been even explored in the presence of noise Bravyi and Koenig (2011); de Melo et al. (2013); Oszmaniec et al. (2014), and can be connected, via the Jordan-Wigner transformation, to the model of computation based on Matchgates Valiant (2002); Knill (2001); Jozsa and Miyake (2008); Brod (2016).

In this work, we completely solve problems (i) and (ii) for the scenarios A-C. We characterize the unitary transformations that are implementable (maybe approximately) by linear optical gates supplemented with any additional Hamiltonian or a gate. Our characterization is given in terms of explicit algebraic conditions on the Hamiltonian $X$ or the gate $V$ that can be can be tested operationally. The resulting behavior is surprisingly rich and structurally depends on the number of modes and the number of particles. In particular, contrary to what intuition might suggest, it is not true that every non-trivial extra gate or Hamiltonian provides universality in scenarios A-C. Solution of problems (i) and (ii) gives the clear understanding of what resources are necessary to have full physical controllability in the contexts listed above. Moreover, our results can be viewed as a step towards a solution of the general problem of classification of invertible quantum circuits posed recently by Aaronson and co-workers Bouland and Aaronson (2014); Aaronson et al. (2015). On the technical level we use techniques of the theory of Lie groups and Lie algebras, which have recently proved useful in studies on controllability of quantum systems Altafini (2002); Zeier and Schulte-Herbrüggen (2011); Zimborás et al. (2014, 2015) and on universality of gate-sets Sawicki (2016); Sawicki and Karnas (2016, 2016).

Our general results have application to concrete physical examples. First, we consider the problem of extension to universality of passive bosonic linear optics for $d=2$ modes via cross-Kerr interaction Brod and Combes (2016), which is of relevance in quantum metrology with random bosonic states Oszmaniec et al. (2016). We also show that, quite surprisingly, there exist simple nonlinear Hamiltonians that do not lead to universality when added to passive fermionic or bosonic linear optics. Finally, it turns out that a simple quartic (in Majorana operators) Hamiltonian promotes active fermionic linear optics to universality in the positive-parity subspace.

Setting—

Let us start with formally defining the extension problem for (i) an additional Hamiltonian and (ii) an additional gate. In general we say that a unitary gate $U$ can be generated by unitaries from a set $\mathcal{S}$ if it can be approximated with arbitrary precision (in operator norm) by a sequence of products of gates from $\mathcal{S}$. We denote by $\langle K,X\rangle$ the set of unitaries that can be generated form the restricted gate-set $K$ and unitaries of the form $\mathrm{exp}\left(-\mathrm{i}tX\right)$, where $t$ is an arbitrary real number. Likewise, slightly abusing the notation, we denote by $\langle K,V\rangle$ the set of unitaries that can be generated by elements form $K$ and an extra gate $V$. The aim of this work is to characterize sets $\langle K,X\rangle$ and $\langle K,V\rangle$ for different linear optical groups $K$ (acting on the appropriate Hilbert spaces $\mathcal{H}$). If the resulting gate-set $\langle K,X\rangle$ (or $\langle K,V\rangle$) form the full unitary group $\mathrm{U}(\mathcal{H})$, we say that the Hamiltonian $X$ (or the gate $V$) promotes the restricted collection of gates $K$ to universality in $\mathcal{H}$. Since unitaries $U$ and $e^{i\alpha}U$ are physically indistinguishable, we assume, without loss of generality, that gates of the form $\mathrm{exp}(\mathrm{i}\theta)\mathbbmss{1}$ are contained in $K$. In what follows by $\mathbb{T}(\mathcal{H})$ we will denote the unitaries proportional to $\mathbbmss{1}$ on $\mathcal{H}$. We would like to remark that the notion of physical universality in $\mathcal{H}$ introduced above does not imply computational universality in a sense of complexity theory Nielsen and Chuang (2010).

We denote the Hilbert space of $N$ bosons in $d$ modes by $\mathcal{H}_{b}$. We have $\mathcal{H}_{b}=\mathrm{Sym}^{N}\left(\mathbb{C}^{d}\right)$, i.e., in this case the bosonic Hilbert space can be identified with the totally symmetric subspace of the Hilbert space of $N$ distinguishable qudits, $\left(\mathbb{C}^{d}\right)^{\otimes N}$. In this language the group of passive linear optical bosonic transformations, denoted by $\mathrm{LO}_{b}$, can defined as the group of unitaries of the form $U^{\otimes N}$, with $U\in\mathrm{U}(d)$, restricted to the bosonic subspace $\mathcal{H}_{b}$.

The Hilbert space of $N$ (spinless) fermionic particles in $d$ modes ($d\geq N$) is $\mathcal{H}_{f}=\wedge^{N}\left(\mathbb{C}^{d}\right)$, i.e., the totally antisymmetric subspace of $\left(\mathbb{C}^{d}\right)^{\otimes N}$. Similarly to the bosonic case, the group of passive fermionic linear optics $\mathrm{LO}_{f}$ is defined as the group of unitaries $U^{\otimes N}$, with $U\in\mathrm{U}(d)$, restricted to the fermionic subspace $\mathcal{H}_{f}$.

In the case of fermionic system without the restricted number of particles, the relevant Hilbert space is direct sum of the different $N$-particle fermionic Hilbert spaces, i.e. the Fock space $\mathcal{H}_{\mathrm{Fock}}{=}\oplus_{N=0}^{d}\wedge^{N}\mathbb{C}^{d}$. $\mathcal{H}_{\mathrm{Fock}}$ is spanned by the Fock states $|n_{1},\ldots,n_{d}\rangle$, with $n_{k}\in\{0,1\}$. In this space we have fermionic creation and annihilation operators, $f_{k}^{\dagger}$, $f_{k}$ satisfying the canonical anti-commutation relations and one can define the standard number operators $\hat{n}_{k}=f^{\dagger}_{k}f^{\phantom{\dagger}}_{k}$. It is also convenient to introduce Majorana fermion operators $m_{2k-1}{=}f^{\phantom{\dagger}}_{k}{+}f_{k}^{\dagger}$, $m_{2k}{=}i\left(f^{\phantom{\dagger}}_{k}{-}f_{k}^{\dagger}\right)$, for $k{=}1,{\ldots},d$. Majorana operators satisfy the anticommutation relations $\left\{m_{i},\,m_{j}\right\}=2\delta_{ij}\mathbbmss{1}$. In many situations fermionic systems obey the so-called parity superselection rule Bravyi and Kitaev (2002); Zimborás et al. (2014); Bravyi (2004) which states that any physically accessible operations must commute with the total parity operator $Q=(-1)^{\sum_{k=1}^{d}\hat{n}_{k}}$. In this work, we restrict our attention to the positive-parity subspace $\mathcal{H}^{+}_{\mathrm{Fock}}$, i.e., the subspace spanned by Fock states with even number of particles. Results completely analogous to the ones presented here can be obtained also for the negative-parity subspace. Active fermionic linear optics acting in $\mathcal{H}_{\mathrm{Fock}}$ consists of unitaries of the form $\mathrm{exp}(\sum_{l,k=1}^{2d}h_{kl}m_{k}m_{l})$, where $h_{kl}$ is a real antisymmetric $2d\times 2d$ matrix. Since we are interested only in the positive-parity subspace $\mathcal{H}^{+}_{\mathrm{Fock}}$, we formally define the group of active fermionic linear transformations as the group consisting of unitaries $(\mathrm{exp}(\mathrm{i}\phi)\mathbbmss{1})\mathrm{exp}(\sum_{l,k=1}^{2d}h_{kl}m_{k}m_{l})$, restricted to $\mathcal{H}^{+}_{\mathrm{Fock}}$.

The groups introduced above are compact Lie subgroups of the unitary group $\mathrm{U}\left(\mathcal{H}\right)$, where $\mathcal{H}$ is the Hilbert space describing the relevant physical system. For a given Lie group $K\subset\mathrm{U}\left(\mathcal{H}\right)$ it is convenient to work with the corresponding Lie algebra consisting of Hamiltonians that generate (via exponentiation) unitaries belonging to $K$. The groups studied in this paper and the corresponding Lie algebras are closely related to the irreducible representations of simple Lie algebras Hall (2000) in the relevant Hilbert spaces. This observation allowing us to obtain our central results given in Theorems 1-4 below. In particular, we use the classification results by Dynkin Dynkin et al. (2000) concerning the maximal Lie subalgebras of simple Lie algebras. For the sake of clarity of the presentation, we moved technical proofs to the Appendix and focused on the discussion of the physical meaning of our results.

Application—

Before stating our results in full generality, let us present first an exemplary application of our findings. In the reference Oszmaniec et al. (2016), the authors were interested in extending bosonic linear optics to universality in $\mathcal{H}_{b}$ by adding an additional gate. This problem was motivated by the need to construct physically-accessible universal gate-set in $\mathcal{H}_{b}$, which can be used to generate, via construction based on random circuits Brandão et al. (2016), approximate bosonic $t$-designs. The example below proves that a singe gate based on the cross-Kerr nonlinearity suffices to promote bosonic linear optics to universality in $\mathcal{H}_{b}$. It should be mentioned that Kerr-like transformation have been previously used to obtain universal quantum computation in continuous-variable systems Lloyd and Braunstein (1999).

[TABLE]

where $\hat{n}_{a,b}$ are the occupation number operators corresponding to modes $a$ and $b$. Let $\langle\mathrm{LO}_{b},V_{t}\rangle$ be the group of transformations generated by passive bosonic linear optics and $V_{t}$. Then, $\langle\mathrm{LO}_{b},V_{t}\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$ if and only if

[TABLE]

for at least one pair $(k,l)$ , where $k,l=0,\ldots,N$. In particular, the gate $V_{\frac{\pi}{3}}$ promotes passive bosonic linear optics to universality in \mathcal{H}_{b}\ for $d=2$ modes.

The above result follows from Theorem 1 stated below. The explicit computations leading to condition (2) are presented in Appendix D.

Main results—

We start with the presentation of results concerning passive bosonic linear optics.

[TABLE]

where $|D_{k}\rangle$ denote the two-mode Dicke states with $k$-particles being in the first mode. We have the following possibilities:

• (a)

If $d>2$, then $\langle\mathrm{LO}_{b},V\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

• (b)

If $d=2$, $N\neq 6$ and $[V\otimes V,\mathbb{L}_{b}]=0$, then

[TABLE]

• (c)

If $d=2$ and $[V\otimes V,\mathbb{L}_{b}]\neq 0$, then $\langle\mathrm{LO}_{b},V\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

In the above theorem we have situations with $N=1$ particles as for them $\mathrm{LO}_{b}=\mathrm{U}(\mathcal{H}_{b})$. We see that for $d\neq 2$ any additional gate promotes $\mathrm{LO}_{b}$ to universality in the bosonic space $\mathcal{H}_{b}$. For $d=2$ the resulting gate-set $\langle\mathrm{LO}_{b},V\rangle$ depends only on the commutator $[V\otimes V,\mathbb{L}_{b}]$. If it is nonzero, then $V$ again extends $\mathrm{LO}_{b}$ to universality; while if it vanishes (and $N\neq 6$) $V$ extends $\mathrm{LO}_{b}$ to the ”middle group” $G_{b}$. Up to a global phase the group $G_{b}$ consists of unitaries that preserve the bilinear form defined by B(\mbox{|\psi\rangle},\mbox{|\phi\rangle})=\mbox{\langle\Psi_{b}|}(\mbox{|\psi\rangle}\otimes\mbox{|\phi\rangle}). Here, by preservation we mean that B_{b}(U\mbox{|\psi\rangle},U\mbox{|\phi\rangle})=B_{b}(\mbox{|\psi\rangle},\mbox{|\phi\rangle}), for all vectors $|\phi\rangle$,$|\psi\rangle$. If the number of particles $N$ is even then $|\Psi_{b}\rangle$ is a symmetric tensor and defines the real inner product. In this case we have $G_{b}=\langle\mathbb{T}(\mathcal{H}_{b}),\mathrm{SO}(\mathcal{H}_{b})\rangle$, where $SO(\mathcal{H}_{b})$ is the special orthogonal group on $\mathcal{H}_{b}$. When the number of particles $N$ is odd, the vector $|\Psi_{b}\rangle$ is antisymmetric and defines the symplectic structure (i.e. non-degenerate and antisymmetric form) on $\mathcal{H}_{b}$. In this case we have $G_{b}=\langle\mathbb{T}(\mathcal{H}_{b}),\mathrm{USp}(\mathcal{H}_{b})\rangle$, where $\mathrm{USp}(\mathcal{H}_{b})$ is the unitary symplectic group. The two differ considerably as $\mathrm{USp}(\mathcal{H})$ acts transitively on the set of pure states on $\mathcal{H}$ Zimborás et al. (2014); Schirmer et al. (2002); Albertini and D’Alessandro (2003). On the other hand, $\mathrm{SO}(\mathcal{H})$ acts transitively only on ”real” pure states. Thus, for odd number of particles, adding any additional gate gives either the full unitary controllability or the pure-state controllability. In the case of $d=2$ modes and $N=6$ particles if $[V\otimes V,\mathbb{L}_{b}]=0$ the situation complicates due to the presence of additional group (related to the exotic group $\mathrm{G}_{2}$) in between $\mathrm{LO}_{b}$ and $G_{b}$. We leave the description of this exceptional case as an interesting open problem.

From Theorem 1 one can infer the following result concerning the additional Hamiltonian $X$ acting on $\mathcal{H}_{b}$ (intuitively, one can obtain it by setting $V=\mathrm{exp}(\mathrm{i}tX)$ in Theorem 1 and differentiating over $t$).

• (a)

If $d>2$, then $\langle\mathrm{LO}_{b},X\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

• (b)

If $d=2$, $N\neq 6$, and $\left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{b}\right]=0$, then

[TABLE]

• (c)

If $d=2$ and $\left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{b}\right]\neq 0$, then $\langle\mathrm{LO}_{b},X\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

Combining the above result and the discussion below Theorem 1 we see that there might exist physical Hamiltonians that add different controllability properties to $\mathrm{LO}_{b}$, depending on the number of particles $N$. The following example shows that this is indeed the case.

• (a)

For even $N$: $\langle\mathrm{LO}_{b},X_{3}\rangle=\langle\mathbb{T}(\mathcal{H}_{b}),\mathrm{SO}(\mathcal{H}_{b})\rangle$;

• (b)

For odd $N$: $\langle\mathrm{LO}_{b},X_{3}\rangle=\langle\mathbb{T}(\mathcal{H}_{b}),\mathrm{USp}(\mathcal{H}_{b})\rangle$.

In, particular for odd $N$ we have full controllability on the set of pure states on $\mathcal{H}_{b}$, whereas for even $N$ this is not the case.

We now move to the discussion of fermionic linear optics (both passive and active). In the main text we present the results concertinaing the gate extension problem (ii). The corresponding theorems for the Hamiltonian extension problem are given in the Appendix E and their relation with the results presented in the main text is analogous to the connection between Theorem 2 and Theorem 1.

[TABLE]

where $\wedge$ denotes the standard wedge product. We have the following possibilities:

• (a)

If $d\neq 2N$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$.

• (b)

If $d=2N$ and $V=Wk$, for $k\in\mathrm{LO}_{f}$ and $W=\prod_{i=1}^{d}(f_{i}+f^{\dagger}_{i})$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{LO}_{f}\cup\mathrm{LO}_{f}\cdot W$.

• (c)

If $d=2N$, $V\neq gW$, for $g\in\mathrm{LO}_{f}$, and $[V\otimes V,\mathbb{L}_{f}]=0$, then

[TABLE]

• (d)

If $d=2N$ and $[V\otimes V,\mathbb{L}_{f}]\neq 0$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$.

The structure of the above result is similar to the case of passive bosonic linear optics. In the formulation of the theorem we have excluded the non interesting cases $N\in\{0,1,d-1,d\}$ since for them $\mathrm{LO}_{f}$ equals the full unitary group on the respective Hilbert space. When $d\neq 2N$ every gate promotes $\mathrm{LO}_{f}$ to universality. However, in the physically relevant case of half-filling Greiter et al. (1991), a more interesting ”onion” structure appears. In the case (b) addition of an extra gate of the form $kW$, where $k\in\mathrm{LO}_{f}$ and gate $W$ (describing particle-hole transformation in $\mathcal{H}_{f}$) gives the gate-set $\mathrm{LO}_{f}\cup\mathrm{LO}_{f}\cdot W$ (it is crucial here that $W$ commutes with $\mathbb{L}_{f}$ and that conjugation by $W$ leaves $\mathrm{LO}_{f}$ invariant). The further possibilities are described, similarly to the bosonic case, by the commutation properties of $V\otimes V$ with $\mathrm{LO}_{f}$. If $d$ is not divisible by four we have $G_{f}=\langle\mathbb{T}(\mathcal{H}_{f}),\mathrm{SO}(\mathcal{H}_{f})\rangle$. On the other hand, for $d$ divisible by four, $G_{f}=\langle\mathbb{T}(\mathcal{H}_{f}),\mathrm{USp}(\mathcal{H}_{f})\rangle$. The corresponding bilinear forms are defined by inner products with $|\Psi_{f}\rangle$. Using Theorem 3, one can efficiently obtain results on extensions to universality, as the one given by the next example.

Hamiltonians that are not composed of two-mode terms are also often studied. One typical family of these are the so-called correlated hopping Hamiltonians, where the hopping-term between two sites is multiplied with number operators belonging to other sites. For such Hamiltonians universality is not guaranteed:

[TABLE]

acting on $\mathcal{H}_{f}$ for the case of half filling ($d=2N$). Then, we have the following situations

• (a)

for even $N$: $\langle\mathrm{LO}_{f},Y\rangle=\langle\mathbb{T}(\mathcal{H}_{f}),\mathrm{SO}(\mathcal{H}_{f})\rangle$;

• (b)

for odd $N$: $\langle\mathrm{LO}_{f},Y\rangle=\langle\mathbb{T}(\mathcal{H}_{f}),\mathrm{USp}(\mathcal{H}_{f})\rangle$.

For odd $N$ the Hamiltonian $Y$ together with $\mathrm{LO}_{f}$ ensures full controllability on the set of pure states on $\mathcal{H}_{f}$. However, for even $N$ this is not the case. The above statements are even true for each term appearing in sum Eq. (6). The correlated hopping Hamiltonian $Y$ often appears (in a relabeled form) in the literature on extended Hubbard models Dolcini and Montorsi (2013).

Our last result concerns the extension problem for active fermionic linear optics.

[TABLE]

We have the following possibilities:

• (a)

If $d\neq 2k$, then $\langle\mathrm{FLO},V\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$

• (b)

If $d=2k$, and $[V\otimes V,\mathbb{L}_{\mathrm{FLO}}]=0$, then

[TABLE]

• (c)

If $d=2k$ and $[V\otimes V,\mathbb{L}_{\mathrm{FLO}}]\neq 0$, then $\langle\mathrm{FLO},V\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$.

In the above result be have omitted cases $d\leq 3$ as for them $\mathrm{FLO}$ is itself the full unitary group on $\mathcal{H}^{+}_{\mathrm{Fock}}$ de Melo et al. (2013). If the number of modes is not even, than every gate promotes $\mathrm{FLO}$ to universality in $\mathcal{H}^{+}_{\mathrm{Fock}}$. If the number of modes is even, then an additional gate either (c) promotes $\mathrm{FLO}$ to universality or extends it to the group $G_{\mathrm{FLO}}$ (b). Furthermore, analogously to the case of passive fermionic optics, if $d$ is divisible by four $G_{\mathrm{FLO}}=\langle\mathbb{T}(\mathcal{H}^{+}_{\mathrm{Fock}}),\mathrm{SO}(\mathcal{H}^{+}_{\mathrm{Fock}})\rangle$ and if this is not the case $G_{\mathrm{FLO}}=\langle\mathbb{T}(\mathcal{H}^{+}_{\mathrm{Fock}}),\mathrm{USp}(\mathcal{H}^{+}_{\mathrm{Fock}})\rangle$ (a detailed description of the corresponding orthogonal and symplectic forms is given in Appendix C).

Discussion—

In this letter, we presented a comprehensive treatment of the extension problems for various classes of linear optical gates for bosons and fermions. The resulting behavior is surprisingly rich and critically depends on the number of modes and number of particles present in the system. However, there is a number of interesting problems we did not addressed here. First, it would be interesting to analyze which extra gates or Hamiltonians allow for the most efficient control Nielsen et al. (2006) or the efficient approximation of gates from the appropriate unitary group Harrow et al. (2002). Another important problem concerns the robustness of the extra gate or Hamiltonian to the noise that inevitably affects any quantum system. In future works we also plan to use our results to study (computational) universality of classically simulable models of computation supported on fermionic systems Terhal and DiVincenzo (2002) and Machgates Valiant (2002); Knill (2001); Jozsa and Miyake (2008); Brod (2016).

Appendices

In the appendices we provide the proofs of the results stated in the main text. We organize this part as follows. In Appendix A, the necessary prerequisites from the theory of Lie algebras and Lie groups are given. Then, in Appendix B, we present the relation between considered classes of linear optical gates and the irreducible representations of particular (semi-)simple Lie groups. In Appendix C, we collect the auxiliary results needed in the proofs of the theorems and the computations in examples given in the main text. In Appendix D we proceed with the the computations related to the examples. Finally, in Appendix E we give proofs of the main technical theorems.

Before we continue let us us first introduce the notation that will be used in latter parts of the Appendix.

Appendix A Lie groups and Lie algebras

Here we summarize some facts from the representation theory of Lie groups and Lie algebras that are used in the proofs of our results. This topic is a broad and fascinating one - we refer the reader to the textbooks Hall (2000); Fulton and Harris (1991); Humphreys (1972) for a comprehensive introduction to the subject. Readers familiar with representation theory of Lie groups and Lie algebras can safely skip this part.

In this work we use extensively compact Lie groups, i.e., groups that are compact differential manifolds such that group multiplication is compatible with the differential structure. A unitary representation of a group Lie group $K$ in a Hilbert space $\mathcal{H}$ is a smooth mapping

[TABLE]

Compact Lie groups admit faithful (one-to-one) unitary representations on finite-dimensional Hilbert spaces and therefore can be always modelled as matrix subgroups of the unitary group $\mathrm{U}(\mathcal{H}_{K})$, for a suitable Hilbert space $\mathcal{H}_{K}$. After fixing the model of a Lie group $K$ we can define its Lie algebra as a real vector space of Hermitian operators that after exponentiation give elements from $K$,

[TABLE]

Remark*.*

Note that we decided to treat elements of the Lie algebra as Hermitian operators. Therefore for $X,Y\in\mathrm{Lie}(K)$ we have $\mathrm{i}[X,Y]\in\mathrm{Lie}(K)$, instead of the property $[X,Y]\in\mathrm{Lie}(K)$, which occurs when elements of Lie algebra are taken to be skew-Hermitian.

We say that a linear map $\pi:\mathrm{Lie}(K)\rightarrow\mathrm{Herm}(\mathcal{H})$ is a a representation of $\mathrm{Lie}(K)$ if and only if $\pi([X,Y])=[\pi(X),\pi(Y)]$, for all $X,Y\in\mathrm{Lie}(K)$. A unitary representation $\Pi$ of a Lie group $K$ in $\mathcal{H}$ induces the representation $\Pi_{\ast}$ of its Lie algebra $\mathrm{Lie}(K)$, by the map

[TABLE]

A unitary representation $\Pi$ of $K$ in $\mathcal{H}$ we can promoted it to a unitary representation (usually dedonetd by $\Pi\otimes\Pi$) of $K$ on $\mathcal{H}\otimes\mathcal{H}$ by setting $\left[\Pi\otimes\Pi\right](k)\coloneqq\Pi(k)\otimes\Pi(k)$, for $k\in K$. Likewise, one can consider a ”tensor-square” of the representation of the Lie algebra $\pi$ by defining $\left[\pi\otimes\pi\right](X)\coloneqq\pi(X)\otimes\mathbbmss{1}+\mathbbmss{1}\otimes\pi(X)$, for $X\in\mathrm{Lie}(K)$.

A representation $\Pi$ of $K$ is called irreducible if and only if there exist no proper (different thank ${0}$ or $\mathcal{H}$) subspace of $\mathcal{W}$ of $\mathcal{H}$ which is invariant under the action of $\Pi$ i.e. $\Pi(k)\mathcal{W}\subset\mathcal{W}$ for all $k\in K$. Otherwise, a representation $\Pi$ is called reducible. For compact Lie groups any reducible representation decomposes onto a direct sum of irreducible representations i.e.

[TABLE]

where $\Pi_{i}$ is an irreducible unitary representation of $K$ in $\mathcal{H}_{i}$. Analogously, a representation of a Lie algebra is called irreducible if and only if there is no proper invariant subspace $\mathcal{W}$ of the Hilbert space $\mathcal{H}$ such that $\pi(X)\mathcal{W}\subset\mathcal{W}$, for all $X\in\mathrm{Lie}(K)$. An induced representation of a Lie algebra $\pi$ is irreducible if and only if the representation of of the Lie group $K$ is irreducible. Tensor-square representations of irreducible representations (of Lie groups or Lie algebras) are in general reducible.

Finally, we define the concept of simple Lie algebra. We say that a Lie algebra $\mathrm{Lie}(K)$ is simple if and only if there exist no proper invariant subspace $\mathcal{I}$ of $\mathrm{Lie}(K)$ which satisfies

[TABLE]

A Lie algebra is called semisimple if and only if it a direct sum of simple Lie algebras.

Appendix B Relevant Lie groups and Lie algebras

Here we give the definitions of classical simple Lie groups and algebras that are relevant for our considerations. Then we give the relation between considered classes of linear optical gates and the representations of these groups or corresponding Lie algebras.

Classical groups and Lie algebras

The special unitary group $\mathrm{SU}(d)$ consists of unitary matrices on $\mathbb{C}^{d}$ of determinant one,

[TABLE]

Its Lie algebra, denoted by $\mathfrak{su}(d)$ consists of traceless Hermitian matrices on $\mathbb{C}^{d}$,

[TABLE]

In the similar way we define special orthogonal group $\mathrm{SO}(d)$ and its Lie algebra $\mathfrak{so}(d)$

[TABLE]

Lastly, in order to define unitary symplectic group $\mathrm{USp}(d)$ and its Lie algebra $\mathfrak{usp}(d)$ we need to introduce the matrix $\mathcal{J}$

[TABLE]

which induces the symplectic form $B_{a}$ of $\mathbb{C}^{2d}$ by the expression

[TABLE]

where $w,v$ in the right-hand side of above expression are column vectors consisting of components of vectors \mbox{|v\rangle},\mbox{|w\rangle}\in\mathbb{C}^{2d} in the standard basis. Unitary symplectic group $\mathrm{USp}(d)$ consists of unitary matrices in $\mathbb{C}^{2d}$ that preserve this form,

[TABLE]

In what follows we will adopt the notation: $\mathrm{SU}(\mathcal{H})$, $\mathfrak{su}(\mathcal{H})$, $\mathrm{SO}(\mathcal{H})$, $\mathfrak{so}(\mathcal{H})$, $\mathrm{USp}(\mathcal{H})$ and $\mathfrak{usp}(\mathcal{H})$, when we talk about the appropriate groups or Lie algebras on the abstract (finite-dimensional) Hilbert space $\mathcal{H}$.

Linear optical groups and Lie algebras

Lie algebras of linear optical groups discussed in the main text are formally given by

• •

for passive bosonic linear optics $\mathrm{LO}_{b}$

[TABLE]

• •

for passive fermionic linear optics $\mathrm{LO}_{f}$

[TABLE]

• •

for active fermionic linear optics $\mathrm{FLO}$

[TABLE]

We have included the term proportional to $\left.Q\right|_{\mathcal{H}^{+}_{\mathrm{Fock}}}$ as it generates unitary gates proportional to identity on $\mathcal{H}^{+}_{\mathrm{Fock}}$.

The above Lie algebras after removing the center (consisting of hermitian operators proportional to identity on the appropriate Hilbert space) give the irreducible representations of simple Lie algebras. These Lie algebra representations are related to the representations of simply-connected compact simple Lie groups that we describe below (see Chapter 2 of Oszmaniec (2014) for a detailed explanation of how these Lie algebras fit into a general picture of representation theory).

For passive bosonic linear optics $\mathrm{LO}_{b}$ acting on the space of $N$ bosons in $d$ modes, $\mathcal{H}_{b}$, we have the associated representation of $\mathrm{SU}(d)$ which we denote by $\Pi_{b}$. we have

[TABLE]

The induced representation of Lie algebra $\mathfrak{su}(d)$ is denoted by $\pi_{b}$ and is given by

[TABLE]

Remark 1**.**

For $d=2$ and $N$ particles the representation $\Pi_{b}$ corresponds, in the standard physics notation, to the representation of $\mathrm{SU}(2)$ with the total spin $j=\frac{N}{2}$.

Analogously, for passive fermionic linear optics $\mathrm{LO}_{f}$ acting on the space of $N$ fermions in $d$ modes, $\mathcal{H}_{f}$, we have the representation of $\mathrm{SU}(d)$ which we denote by $\Pi_{f}$. It is defined by:

[TABLE]

and the induced representation of Lie algebra $\mathfrak{su}(d)$ is denoted by $\pi_{f}$. It follows that

[TABLE]

For the active fermionic linear optics we have an irreducible spinor representation of the Lie algebra $\mathfrak{so}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}$ denoted by $\pi_{\mathrm{FLO}}^{+}$. We start by describing a representation $\pi_{\mathrm{FLO}}$ that acts in the full Fock space $\mathcal{H}_{\mathrm{Fock}}$. We define this representation by its action on the orthogonal (with respect to Hilbert-Schmidt inner product) basis of $\mathfrak{su}(2d)$ of the form (see Chapeter 2 of Oszmaniec (2014) for details)

[TABLE]

We have

[TABLE]

This representation decomposes onto two irreducible components supported on positive and negative parity subspaces of the Fock space. We denote the restriction of $\pi_{\mathrm{FLO}}$ to $\mathcal{H}^{+}_{\mathrm{Fock}}$ by $\pi_{\mathrm{FLO}}^{+}$. Formally, we have

[TABLE]

The corresponding representations of the group $\mathrm{Spin}(2d)$ (double-cover of $\mathrm{SO}(2d)$) in $\mathcal{H}^{+}_{\mathrm{Fock}}$ and $\mathcal{H}_{\mathrm{Fock}}$ are denoted by $\Pi_{\mathrm{FLO}}^{+}$ and $\Pi_{\mathrm{FLO}}$ respectively.

Appendix C Auxiliary technical results

In this part we state and prove a number of auxiliary results that are necessary for the computations concerning the examples (see Appendix D) and in the proofs of the main results (see Appendix E).

Lemma 1** (Projection for the singlet subspace for the doubled bosonic representation of $\mathrm{SU}(2)$)).**

Let $d=2$ and let $\Pi_{b}\otimes\Pi_{b}$ be the representation of $\mathrm{SU}(2)$ in $\mathcal{H}_{b}\otimes\mathcal{H}_{b}$ induced from representation $\Pi_{b}$ on $\mathcal{H}_{b}$. Then, in the decomposition of $\mathcal{H}_{b}\otimes\mathcal{H}_{b}$ onto irreducible components there always appears a a single trivial representation of $\mathrm{SU}(2)$. Moreover, the projection onto this representation is given by

[TABLE]

where

[TABLE]

Proof.

The fact that in the decomposition of $\mathcal{H}_{b}\otimes\mathcal{H}_{b}$ there is always a single trivial representation follows from the standard rules of addition of angular momentum Sakurai and Napolitano (2011). The formula (31) can be derived using techniques involving Young diagrams. However, checking whether $|\Psi_{b}\rangle$ actually belongs to the trivial representation of $\mathrm{SU}(2)$ can be done easily by inspecting the action of $\mathrm{Lie}(\mathrm{SU}(2))$ on it. This action comes from the fact that just like $\mathrm{SU}(2)$, its Lie algebra is represented in $\mathcal{H}_{b}$ via standard angular momentum representation $\pi_{b}$. The standard basis of $\mathrm{SU}(2)$ is represented in $\mathcal{H}_{b}\otimes\mathcal{H}_{b}$ via

[TABLE]

where $J_{x,y,z}$ denote the standard angular momentum operators (for spin $s=\frac{N}{2}$). Now it suffices to show that

[TABLE]

for $\alpha=x,y,z$. Equation (34) can be verified easily using the standard algebraic properties of operators $J_{x,y,z}$ (recall that the Dicke basis is exactly the standard "angular momentum" basis in which $J_{z}$ is diagonal. ∎

Lemma 2**.**

(Projection for the singlet subspace for the doubled fermionic half-filling representation of $\mathrm{SU}(d)$) Let $d=2N$ and let and let $\Pi_{f}\otimes\Pi_{f}$ be the representation of $\mathrm{SU}(d)$ in $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$ induced from representation $\Pi_{f}$ on $\mathcal{H}_{f}$. In the decomposition of $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$ onto irreducible components there always appears a a single trivial representation of $\mathrm{SU}(d)$. Moreover, the projection onto this representation is given by

[TABLE]

where

[TABLE]

Proof.

The decomposition of the tensor square $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$ onto irreducible representations of $\mathrm{SU}(d)$ (for arbitrary number of particles $N$) can be found in Macdonald (1998) on page 331. From the results given there it follows that for $d=2N$ in the decomposition $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$ there exist only one trivial one dimensional representation $\mathcal{H}^{0}_{f}$. In other words in the case of half filling we have

[TABLE]

where $\tilde{\mathcal{H}}_{f}$ the orthogonal complement of $\mathcal{H}^{0}_{f}$ in $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$. Since $\mathcal{H}^{0}_{f}$ is one dimensional, the projection onto this space is specified by a single vector belonging to it. To find this vector we first embed $\mathcal{H}_{f}$ into the tensor power representation of $\mathrm{SU}(d)$, that is we note that

[TABLE]

and

[TABLE]

Under this inclusion we have $\mathcal{H}_{f}\otimes\mathcal{H}_{f}\subset\left(\mathbb{C}^{d}\right)^{\otimes 2N}$ and consequently

[TABLE]

Now, $\mathcal{H}^{0}_{f}$ can be treated as a subspace of $\left(\mathbb{C}^{2N}\right)^{\otimes 2N}$ (recall that we consider the case $2N=d$). Moreover, we see that for the case of half-filling the vector $|\Psi_{f}\rangle$ given in (36) belongs to $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$. This follows form the fact that it is a totally antisymmetric vector on $\left(\mathbb{C}^{2N}\right)^{\otimes 2N}$. What is more, \mbox{|\Psi_{f}\rangle}\in\mathcal{H}^{0}_{f} which is a result of (40) and the standard properties of the wedge product: For $U\in\mathrm{SU}(d)$ we have \left[\Pi_{f}\otimes\Pi_{f}(U)\right]\mbox{|\Psi_{f}\rangle}=U^{\otimes 2N}\mbox{|\Psi_{f}\rangle}=\mathrm{det}(U)\mbox{|\Psi_{f}\rangle}=\mbox{|\Psi_{f}\rangle}, where in the last equality we have used the fact that $\mathrm{det}(U)=1$ for $U\in\mathrm{SU}(d)$. ∎

Let us introduce some auxiliary notation. For a subset $X$ of $d$-element set $[d]$ we define its indicator function by

[TABLE]

We denote by $\bar{X}$ the complement of $X$ in $[d]$. Moreover, for the case of half-filling ($2N=d$) for a set $X=\{x_{1},x_{2},\ldots,x_{N}\}$ we define $\mathrm{sgn}(X)=\mathrm{sgn}(\omega_{X})$, where $\mathrm{sgn}(\omega_{X})$ is the sign of the permutation $\omega_{X}:[2N]\rightarrow[2N]$ such that

• •

$\omega_{X}(i)$ ($i=1,\ldots,N$) are non-decreasingly ordered elements of $X$;

• •

$\omega_{X}(i)$ ($i=N+1,\ldots,2N$) are non-decreasingly ordered elements of $\bar{X}$.

Lemma 3** (Convenient form of the projection onto the trivial representation of $\mathrm{SU}(d)$ in $\mathcal{H}_{f}\otimes\mathcal{H}_{f}$ for $2N=d$).**

For the case of half-filling ($N=2d$) the invariant vector \mbox{|\Psi_{f}\rangle}\in\mathcal{H}_{f}\otimes\mathcal{H}_{f} can be written in the following way

[TABLE]

where \mbox{|X\rangle}=\mbox{|X(1),X(2),\ldots,X(2N)\rangle}.

Proof.

Using Lemma 2 we know that \mbox{|\Psi_{f}\rangle}=\mbox{|1\rangle}\wedge\mbox{|2\rangle}\wedge\ldots\wedge\mbox{|2N\rangle}. Expanding \mbox{|1\rangle}\wedge\mbox{|2\rangle}\wedge\ldots\wedge\mbox{|2N\rangle} in the first quantisation picture gives

[TABLE]

where $P([2N]$ denotes the permutation group of the set $[2N]$ and \{\mbox{|i\rangle}\}_{i=1}^{d} is the fixed orthonormal basis of $\mathbb{C}^{d}$. Let $\mathbb{P}^{N}_{\mathrm{as}}:(\mathbb{C}^{2N})^{\otimes N}\rightarrow(\mathbb{C}^{2N})^{\otimes N}$ be the projection onto the totally antisymmetric subspace of $(\mathbb{C}^{2N})^{\otimes N}$. From the asymmetry of \mbox{|1\rangle}\wedge\mbox{|2\rangle}\wedge\ldots\wedge\mbox{|2N\rangle} we have \mathbb{P}^{N}_{\mathrm{as}}\otimes\mathbb{P}^{N}_{\mathrm{as}}\mbox{|\Psi_{f}\rangle}=\mbox{|\Psi_{f}\rangle}. Using this and (43) we obtain

[TABLE]

In order to simplify this expression we note we have a decomposition

[TABLE]

where

• •

$\omega_{\sigma([N])}$ has been defined above the formulation of the Lemma ($\sigma([N])$ is a particular subset of $[2N]$) ;

• •

$\tau_{I}(\sigma)\in P([2N])$ is the unique permutation that maps the non-increasingly ordered elements of $\sigma([N])$ to the ordered tuple $(\sigma(1),\sigma(2),\ldots,\sigma(N)$ and acts as identity on $\sigma(\{{N+1,\ldots,2N\})}$;

• •

$\tau_{I}(\sigma)\in P([2N])$ is the unique permutation that maps the non-increasingly ordered elements of $\sigma(\{{N+1,\ldots,2N\}})$ to the ordered tuple $(\sigma(N+1),\sigma(N+2),\ldots,\sigma(2N))$ and acts as identity on $\sigma(\{{1,\ldots,N\})}$ .

Furthermore, we note that due to the antisymmetry of $\mathbb{P}^{N}_{\mathrm{as}}$

[TABLE]

However, from the definition of the Slater determinant we have

[TABLE]

and similarly for the complement set

[TABLE]

Combining together the above identities in the expression (44) we obtain

[TABLE]

The above can be further simplified by exploiting (45) and the homomorphism property of the $\mathrm{sgn}$ function,

[TABLE]

where in the last equality we have a changed a sum over permutations to the sum over $N$-element subsets of $[2N]$, which resulted in the binomial coefficient in front of the sum in (52).

∎

Remark 2**.**

It can be showed that the function $\mathrm{sgn}(X)$ defined above Lemma 3 is given explicitly via

[TABLE]

Using that $\mathrm{sgn}(X)\mathrm{sgn}(\bar{X})=(-1)^{N}$, we obtain

[TABLE]

Lemma 4** (Projection onto the trivial representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$ for even $d$).**

Assume that $d=2k$ and let and let $\Pi_{\mathrm{FLO}}\otimes\Pi_{\mathrm{FLO}}$ be the representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$ induced from representation $\Pi_{\mathrm{FLO}}$ on $\mathcal{H}^{+}_{\mathrm{Fock}}$. In the decomposition of $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$ onto irreducible components there always appears a a single trivial representation of $\mathrm{Spin}(2d)$. Moreover, the projection onto this representation is given by

[TABLE]

where

[TABLE]

and $\mathbb{P}^{+}=\frac{1}{2}(\mathbbmss{1}+Q)$ is the orthonormal projection onto $\mathcal{H}^{+}_{\mathrm{Fock}}\subset\mathcal{H}_{\mathrm{Fock}}$.

Proof.

It will be convenient for us to work first in the full fermionic Fock space $\mathcal{H}_{\mathrm{Fock}}$, that carries a (reducible) representation of $\mathrm{Spin}(2d)$ denoted by $\Pi_{\mathrm{FLO}}$ (see the previous section of the Appendix). We can promote this representation to the representation $\Pi_{\mathrm{FLO}}\otimes\Pi_{\mathrm{FLO}}$ of $\mathrm{Spin}(2d)$ in $\mathcal{H}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}$. We can now identify the trivial representation of $\mathrm{Spin}(2d)$ (or equivalently $\mathfrak{so}(2d)$) in $\mathcal{H}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}$ with the zero eigenspace of the second order Casimir Fulton and Harris (1991); Oszmaniec (2014) of $\mathfrak{so}(2d)$ represented in $\mathcal{H}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}$. The Casimir operator, denoted by $\mathcal{C}_{2}$ is given by

[TABLE]

where in the computation above we have used (29) and the identity $(\mathrm{i}m_{k}m_{l})^{2}=\mathbbmss{1}$, valid for all $k\neq l$. From Eq. (58) we see that $\mathcal{C}_{2}$ is a sum of $2d(2d-1)/2$ commuting positive operators $F_{kl}=\mathbbmss{1}\otimes\mathbbmss{1}-m_{k}m_{l}\otimes m_{k}m_{l}$ having eigenvalues [math] and $1$. The projection onto the zero eigenspace of $F_{kl}$ in is given by

[TABLE]

Finally, the projection onto the zero eigenspace of $\mathcal{C}_{2}$ is the product of all these projections

[TABLE]

By restricting $\mathbb{L}_{FLO}$ to the subspace $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}\subset\mathcal{H}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}$ we finally get that $\mathbb{P}^{+}_{\mathrm{FLO}}$ is the projection onto a trivial representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$. Note however that

[TABLE]

It is easy to see that the operator $\mathbb{A}_{\mathrm{FLO}}$ has support on $\mathcal{H}_{\mathrm{Fock}}^{+}\otimes\mathcal{H}_{\mathrm{Fock}}^{-}\oplus\mathcal{H}_{\mathrm{Fock}}^{-}\otimes\mathcal{H}_{\mathrm{Fock}}^{+}$ for odd $d$ and on $\mathcal{H}_{\mathrm{Fock}}^{+}\otimes\mathcal{H}_{\mathrm{Fock}}^{+}\oplus\mathcal{H}_{\mathrm{Fock}}^{-}\otimes\mathcal{H}_{\mathrm{Fock}}^{-}$ . We thus see that $\mathbb{P}^{+}_{\mathrm{FLO}}=0$ unless $d$ is even. The fact that $\mathbb{P}^{+}_{\mathrm{FLO}}\neq 0$ for even $d$ follows form the fact that in this case positive-parity spinor representations of $\mathrm{Spin}(2d)$ are self-dual - see the remark below Lemma 5 below. ∎

Lemma 5** (Another form of the projection onto the trivial representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$, for even $d$).**

The projection $\mathbb{P}_{\mathrm{FLO}}^{+}$ onto the trivial representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$ can be expressed via a simpler expression

[TABLE]

with

[TABLE]

In the second bracket in above expression we used the convention $m_{2d+1}\equiv m_{1}$.

Proof.

We first note that the operators $Q_{k,l}=m_{k}m_{l}\otimes m_{k}m_{l}$ (recall that $1\leq k<l\leq 2d$) can be always obtained as a product of ”nearest-neighbour” operators $Q_{i,i+1}$ ($i=1,\ldots,2d-1$). Indeed, we have

[TABLE]

and therefore the joint +1 eigenspace of $Q_{k,l}$ ($1\leq k<l\leq 2d$) is exactly the joint +1 eigenspace of $Q_{i,i+1}$, for $i=1,\ldots,2d-1$. Consequently, we have $\mathbb{L}_{\mathrm{FLO}}=\mathbb{L}^{\prime}_{\mathrm{FLO}}$ and equation (62) follows (note that in (63) we have incorporated, for the sake of symmetry of the expression, a redundant term projecting onto the +1 eigenspace of $Q_{1,2d}$). ∎

Remark 3**.**

From expressions (62) and (63) it follows that $\mathbb{P}_{\mathrm{FLO}}^{+}\neq 0$ if and only if $d$ is even. To show this we first observe that $\mathbb{L}^{\prime}_{\mathrm{FLO}}$ commutes with operators $Q\otimes\mathbbmss{1}$ and $\mathbbmss{1}\otimes Q$ and hence it commutes with the projection $\mathbb{P}^{+}\otimes\mathbb{P}^{+}$. Therefore in order to prove $\mathbb{P}^{+}\otimes\mathbb{P}^{+}\mathbb{L}^{\prime}_{\mathrm{FLO}}\mathbb{P}^{+}\otimes\mathbb{P}^{+}$ it suffices to show $\operatorname{tr}(\mathbb{P}^{+}\otimes\mathbb{P}^{+}\mathbb{L}^{\prime}_{\mathrm{FLO}})\neq 0$. To prove this we first expand the products inside the brackets of (63) and obtain

[TABLE]

where $L_{1}$ and $L_{2}$ contain sums of operators $m_{X}\otimes m_{X}$, with $m_{X}=\prod_{i\in X}m_{i}$ and $X$ is a proper subset of $[2d]=\{1,\ldots,2d\}$. Moreover, from the expansion of the brackets in (63) we see that for every $m_{X}\otimes m_{X}$ appearing in $L_{1}$ there exist no $m_{Y}\otimes m_{Y}$ in $L_{1}$ such that $X\cup Y=[2d]$. From this discussion and the fact that products of Majorana fermion operators are traceless we finally obtain

[TABLE]

where we have used the identities $Q^{2}=\mathbbmss{1}$, $\operatorname{tr}(\mathbbmss{1}\otimes\mathbbmss{1})=2^{2d}$ and $\operatorname{tr}(Q\otimes Q)=0$. Analogous computations show that

[TABLE]

where $\mathbb{P}^{-}=\frac{1}{2}(\mathbbmss{1}-Q)$. Similarly, we have

[TABLE]

and thus we conclude that the trivial representation appears in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}^{-}$ (or in $\mathcal{H}_{\mathrm{Fock}}^{-}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$) only when $d$ is odd.

We close the section of the auxiliary results by giving the explicit form of vector $|\Psi_{\mathrm{FLO}}\rangle$.

Lemma 6** (Explicit form of the vector spanning the trivial representation of $\mathrm{Spin}(2d)$ in $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$ for even $d$.).**

For a subset $X$ of $d$-element set $[d]$ we define $N(X)=\sum_{i\in X}i$. Under this notation we have

[TABLE]

where \mbox{|X\rangle}=\mbox{|X(1),X(2),\ldots,X(d)\rangle} and we have used the notation introduced above Lemma 3.

Proof.

From the relation $\mathbb{L}_{\mathrm{FLO}}\leq\mathbb{A}_{\mathrm{FLO}}$ (see Eq.(61)) we can deduce the following observations

• (i)

Operator $\mathbb{P}^{+}_{\mathrm{FLO}}$ is supported on the subspace $\mathcal{H}_{\mathrm{Fock}}\otimes\mathcal{H}_{\mathrm{Fock}}$ spanned by linearly-independent vectors \mbox{|X\rangle}\otimes\mbox{|\bar{X}\rangle} for even-element subset $X\subset[d]$;

• (ii)

Operator $\mathbb{P}^{+}_{\mathrm{FLO}}$ must necessary project onto one dimensional subspace (this follows form the basic character theory for the compact groups Zeier and Zimborás (2015));

• (iii)

$\mathbb{P}^{+}_{\mathrm{FLO}}$ must be a projection onto a maximally entangled vector (denoted by $|\Psi_{\mathrm{FLO}}\rangle$) in the space $\mathcal{H}^{+}_{\mathrm{Fock}}\otimes\mathcal{H}^{+}_{\mathrm{Fock}}$. If this was not the case then the invariant form B(\mbox{|\psi\rangle},\mbox{|\psi\rangle})=\mbox{\langle\Psi|}\mbox{|\psi\rangle}\otimes\mbox{|\phi\rangle}, for \mbox{|\psi\rangle},\mbox{|\phi\rangle}\in\mathcal{H}^{+}_{\mathrm{Fock}} would have been degenerate which would contradict the irreducibility of the representation $\Pi^{+}_{\mathrm{FLO}}$ in $\mathcal{H}^{+}_{\mathrm{Fock}}$.

Combining these observations we get

[TABLE]

The global phase of $|\Psi_{\mathrm{FLO}}\rangle$ can be set arbitrary way. In what follows we set without the loss of generality $\theta_{\emptyset}=0$. As we will see this choice fixes the values of all remaining phase factors to

[TABLE]

In order to prove this let us study how the unitary operator

[TABLE]

acts on the vector \mbox{|\emptyset\rangle}\otimes\mbox{|[d]\rangle}. Explicit computation gives

[TABLE]

Defining analogously the operator $E_{X}$, where $X\subset[d]$ is even-element subset of $[d]$ we get

[TABLE]

The crucial observation now is that operators $F_{X}$ are in $\Pi^{+}_{\mathrm{FLO}}\otimes\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$ and therefore

[TABLE]

Moreover, we note that

[TABLE]

where $X+Y=X\cup Y\setminus X\cap Y$ denotes the symmetric sum of $X$ and $Y$. Using the above relation together with (75), assumption $\theta_{\emptyset}=0$, and (70) we finally obtain (69).

∎

Remark 4**.**

Using the positive-parity constrain we get $N(\bar{X})+N(X)=(-1)^{d/2}$. Consequently we have

[TABLE]

Remark 5**.**

It is possible to prove that for $d=4m$ we have

[TABLE]

where $|\Psi_{f}\rangle$ is the passive-FLO invariant vector (see Eq. (42)) and $\mathbb{P}_{d/2}:\mathcal{H}^{+}_{\mathrm{Fock}}\rightarrow\mathcal{H}^{+}_{\mathrm{Fock}}$ is the projection onto the half-filling subspace $\bigwedge^{d/2}\left(\mathbb{C}^{d}\right)\subset\mathcal{H}^{+}_{\mathrm{Fock}}$. We suspect that analogous relation holds also for the case of even $d$ which is not divisible by $4$. We leave proofs of these statements to the interested reader.

Appendix D Detailed computations concerning examples from the main text

In this section we present explicit computations concerning examples given in the main text.

D.1 Example 1

The result given in Example 1 follows directly from Theorem 1, or more specifically from the situation described in the possiblity (c). It states that the group generated by passive bosonic linear optics and $V\in\mathrm{U}\left(\mathcal{H}_{b}\right)$ is the full unitary group $\mathrm{U}\left(\mathcal{H}_{b}\right)$ if and only if $\left[V\otimes V,\mathbb{L}_{b}\right]\neq 0$, where $\mathbb{L}_{b}$ is given by Eq. (3). Let $V_{t}=\mathrm{exp}\left(-it\hat{n}_{a}\hat{n}_{b}\right)$. Straightforward computation gives

[TABLE]

where

[TABLE]

From the above expressions it follows that $\left[V_{t}\otimes V_{t},\mathbb{L}_{b}\right]=0$ if and only if $f_{kl}=0$ for all $k,l=0,\ldots,N$ which is equivalent to (2). Solving (80) is an interesting problem in itself. We limit ourselves to proving that for time $t=\frac{\pi}{3}$ the gate $V_{t}$ promotes $\mathrm{LO}_{b}$ to universality. The reason for that is the following: number $3$ does not simultaneously divide integers of the form $n_{l}=l\left(N-l\right)$ ($l=0,\ldots,N$). To see this we first observe that for $N=1,\ldots,5$ the sequence $n_{l}$ takes values

• •

$N=1:$ $n_{l}=(0,0)$;

• •

$N=2:$ $n_{l}=(0,1,0)$;

• •

$N=3:$ $n_{l}=(0,2,2,0)$;

• •

$N=4:$ $n_{l}=(0,3,4,3,0)$;

• •

$N=5:$ $n_{l}=(0,4,8,9,8,4,0)$.

For $N>5$ in the sequence $n_{l}$ we have numbers $\left(N-2\right)2$ and $\left(N-4\right)4$ which cannot be both simultaneously divisible by $3$. Consequently, we have $\left[V_{\frac{\pi}{3}}\otimes V_{\frac{\pi}{3}},\mathbb{L}_{b}\right]\neq 0$.

D.2 Example 2

In order to prove the content of the Example 2 we refer to the point (b) of Theorem 2. Form there it follows that $\langle\mathrm{LO}_{b},X_{3}\rangle=G_{b}$ is and only if $[X_{3}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X_{3},\mathbb{L}_{b}]=0$. Recall that \mathbb{L}_{b}\propto\mbox{|\Psi_{b}\rangle!\langle\Psi_{b}|}, where \mbox{|\Psi_{b}\rangle}=\sum_{k=0}^{N}(-1)^{k}\mbox{|D_{k}\rangle}\mbox{|D_{N-k}\rangle}. Using $X_{3}=\hat{n}^{3}_{a}-\hat{n}^{3}_{b}$ we obtain

[TABLE]

Therefore \left(X_{3}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X_{3}\right)\mbox{|\Psi_{b}\rangle}=0 and consequently $[X_{3}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X_{3},\mathbb{L}_{b}]=0$. As a result we have $\langle\mathrm{LO}_{b},X_{3}\rangle=G_{b}$. The specific cases (a) and (b) discussed in the Example 2 follow immediately from the discussion below Theorem 1.

D.3 Example 3

Any Hamiltonian $H_{\text{tm}}$ acting on $\mathcal{H}_{f}$ that contains only two-mode terms has the following general form:

[TABLE]

where $\alpha_{k,l}=\overline{\alpha}_{l,k}$ and the $\beta_{k,l}$ coefficients are real (moreover, at least one of the $\beta_{k,l}$ coefficients must be different from zero as the Hamiltonian is assumed to be non-trivial and not quadratic). Let us first note that $\sum_{k,l}a_{kl}f^{\dagger}_{k}f_{l}\in\mathrm{Lie}(\mathrm{LO}_{f})$. Moreover, we have

[TABLE]

where $X^{\prime}\in\mathrm{Lie}(\mathrm{LO}_{f})$. Therefore we obtain

[TABLE]

According to Theorem 5 (fermionic analogue of Theorem 2 given in the main text) for $d\neq 2N$ we automatically get $\langle\mathrm{LO}_{f},H^{\prime}_{\text{tm}}\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$. From the same theorem it follows that for $d=2N$ the universality is equivalent to the condition

[TABLE]

which is satisfied if and only if $|\Psi_{f}\rangle$ (see Eq. (42)) is not an eigenvector of $H^{\prime}_{\text{tm}}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes H^{\prime}_{\text{tm}}$. For arbitrary Slater determinant $|X\rangle$ in $d/2$ particle sector, we have

[TABLE]

Moreover, explicit computation using (84) gives $\lambda_{X}=\lambda_{\bar{X}}$, for all $X$ of cardinality $d/2$. Consequently, we obtain

[TABLE]

Thus, by using Eq. (42), we conclude that $|\Psi_{f}\rangle$ is an eigenvector of $H^{\prime}_{\text{tm}}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes H^{\prime}_{\text{tm}}$ iff $\lambda_{X}=\lambda$, for all $d/2$-element subsets of $[d]$. However, we assumed that $H_{\text{tm}}$ (and thus also $H^{\prime}_{\text{tm}}$) is non-trivial, there must exist two Slater determinants $|X_{1}\rangle$ and $|X_{2}\rangle$ with $\lambda_{X_{1}}\neq\lambda_{X_{2}}$ (only the multiple of the identity in $\mathcal{H}_{f}$ can have the same eigenvalue for all Slater determinants). Hence, Eq. (LABEL:eq:HtmX) cannot be satisfied, and consequently $\langle\mathrm{LO}_{f},H_{\text{tm}}\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$ also in the $d=2N$ case.

D.4 Example 4

For Example 4, we proceed somewhat similarly to the proof of the previous example. To prove that the correlated hopping Hamiltonian $Y$ does not promote passive Fermionic Linear Optics to universality in the half-filling case, we have to show that for gates of the form $V=\mathrm{exp}(itY)$ (with arbitrary real $t$) the condition $[V\otimes V,\mathbb{L}_{f}]=0$ holds. As discussed in the previous example, this condition is equivalent to

[TABLE]

If the above holds, then Theorem 3 guarantees that $\langle\mathrm{LO}_{f},X\rangle=G_{f}$, which is a proper subgroup of $\mathrm{U}\left(\mathcal{H}_{f}\right)$ (see also Theorem 5 given in the next part of the Appendix).

We will start by showing that for $E_{1}=(\hat{n}_{2}+\hat{n}_{4}-\hat{n}_{2}\hat{n}_{4})\,f^{\dagger}_{1}f^{\phantom{\dagger}}_{3}$ the equality (E_{1}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes E_{1})\mbox{|\Psi_{f}\rangle}=0 is satisfied. Let us note that for any Slater determinant \mbox{|X\rangle}=\mbox{|X(1),X(2),\ldots,X(d)\rangle} , we have that E_{1}\mbox{|X\rangle}\neq 0 iff $X(1)+X(3)=1$ and $X(2)+X(4)=1$. Now, using the form of $|\Psi_{f}\rangle$ given in Eq. (52), we obtain that

[TABLE]

where we introduced the notation $\widetilde{\mathrm{sgn}}(X^{\prime})=\mathrm{sgn}(\{1,2\}\cup X^{\prime})$ and used that from Eq. (53) it trivailly follows that $\mathrm{sgn}(\{1,2\}\cup X^{\prime})=\mathrm{sgn}(\{3,4\}\cup X^{\prime})=\mathrm{sgn}(\{1,4\}\cup X^{\prime})=\mathrm{sgn}(\{2,3\}\cup X^{\prime})$ for all $X^{\prime}\subset\{5,\ldots,2N\}$.

Next, by straightforward calculations, one obtains

[TABLE]

which immediately implies that (E_{1}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes E_{1})\mbox{|\Psi_{f}\rangle}=0. Analogous computations show that (E^{\dagger}_{1}\otimes\mathbbmss{1}+\mathbbmss{1}\otimes E^{\dagger}_{1})\mbox{|\Psi_{f}\rangle}=0. Now, after defining

[TABLE]

and performing the completely analogous calculations one shows that

[TABLE]

Finally, using $Y=\sum_{j=1}^{d/2}(E^{\phantom{\dagger}}_{2j-1}+E^{\dagger}_{2j-1}+F^{\phantom{\dagger}}_{2j}+F^{\dagger}_{2j})$, we get that (Y\otimes\mathbbmss{1}+\mathbbmss{1}\otimes Y)\mbox{|\Psi_{f}\rangle}=0, which implies (88).

D.5 Example 5

We prove the assertion given in Example 5 by directly using Theorem 6. From the point (a) of the aforementioned theorem it follows that for $d\neq 2m$ the Hamiltonian $H_{\mathrm{in}}=m_{1}m_{2}m_{3}m_{4}$ always promotes FLO to universality in $\mathcal{H}^{+}_{\mathrm{Fock}}$ (clearly, it does not belong to $\mathrm{Lie}(\mathrm{FLO})$). On the other hand, from the point (c) of the same Theorem it suffices to show

[TABLE]

in order to prove $\langle\mathrm{FLO},H_{\mathrm{in}}\rangle=\mathrm{U}(\mathcal{H}^{+}_{\mathrm{Fock}})$. In what follows we use the alternative representation $\mathbb{L}^{\prime}_{\mathrm{FLO}}$ of the operator $\mathbb{L}_{\mathrm{FLO}}$ given in (63). Explicit computation involving the commutation rules of Majorana-fermion operators give

[TABLE]

The expansion of the product in (98) shows that the whole expression is manifestly nonzero and therefore the condition (95) is satisfied.

Appendix E Proofs of technical theorems

In what follows we present the proofs of the results from the main text. We first prove theorems concerning Hamiltonian extension of various classes of linear optics. Later we prove results on the gate extension problem for these settings.

Lemma 7** (Connected Lie group generated by a matrix Lie algebra Hofmann and Morris (2013)).**

] For a Lie algebra $\mathfrak{k}$ consisting linear operators on $\mathcal{H}$, $\mathfrak{k}\subset\mathfrak{su}(\mathcal{H})$, we denote by $\langle\mathrm{exp}(\mathrm{i}\mathfrak{k})\rangle$ a subgroup of $\mathrm{SU}(\mathcal{H})$ generated by elements of the form $\mathrm{exp}(\mathrm{i}X)$, for $X\in\mathfrak{k}$. Then $\langle\mathrm{exp}(\mathrm{i}\mathfrak{k})\rangle$ is a connected compact Lie subgroup of $\mathrm{SU}(\mathcal{H})$ with the Lie algebra $\mathfrak{k}$, i.e. $\mathrm{Lie}\left(\langle\mathrm{exp}(\mathrm{i}\mathfrak{k})\rangle\right)=\mathfrak{k}$.

E.1 Hamiltonian extensions of linear optics

Theorem 2** (Extensions of Passive Bosonic Linear Optics via an additional Hamiltonian).**

Let $X\notin\mathrm{Lie}\left(\mathrm{LO}_{b}\right)$ be a Hamiltonian acting on Hilbert space $\mathcal{H}_{b}$ of $N$ bosons in $d$ modes. Let $\langle\mathrm{LO}_{b},X\rangle$ be the group of transformations generated by passive bosonic linear optics and $X$ in $\mathcal{H}_{b}$. We have the following possibilities:

• (a)

If $d>2$, then $\langle\mathrm{LO}_{b},X\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

• (b)

If $d=2$, $N\neq 6$, and $\left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{b}\right]=0$, then

[TABLE]

• (c)

If $\left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{b}\right]\neq 0$, then $\langle\mathrm{LO}_{b},X\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right).$

Proof of Theorem 2.

Let us first remark that since diagonal Hamiltonians are contained in $\mathrm{Lie}(\mathrm{LO}_{b})$ (see Eq. (21)), we can assume without loss of generality that $X$ is traceless, i.e., belongs to the Lie algebra $\mathfrak{su}(\mathcal{H}_{b})$ of the special unitary group $\mathrm{SU}(\mathcal{H}_{b})$ (see Appendix B). In what follows we describe the possible forms of the group $\mathcal{G}=\langle\Pi_{b}(\mathrm{SU}(d)),X\rangle$ generated by linear optical gates $\Pi_{b}(\mathrm{SU}(d))$ of determinant one and unitary evolutions of the form $\mathrm{exp}(\mathrm{i}tX)$. The results stated in the formulation of Theorem 2 will be then obtained by using the relation

[TABLE]

where $\mathbb{T}(\mathcal{H}_{b})$ is the group of unitaries on $\mathcal{H}_{b}$ of the form $\mathrm{exp}(\mathrm{i}\theta)\mathbbmss{1}$, where $\theta\in\mathbb{R}$. The structure of $\mathcal{G}$ relies on the classification of maximal simple Lie subalgebras of classical simple compact Lie algebras due to Dynkin Dynkin et al. (2000). Recall that $\mathrm{Lie}\left(\Pi_{b}(\mathrm{SU}(d))\right)=\pi_{b}(\mathfrak{su}(d))$ is the image of a representation of a compact simple Lie algebra $\mathfrak{su}(d)$. By definition (see Lemma 7) the group $\mathcal{G}$ is a connected compact Lie group contained in $\mathrm{SU}(\mathcal{H}_{b})$, and we denote its Lie algebra by $\mathfrak{g}\subset\mathfrak{su}\left(\mathcal{H}_{b}\right)$. Because $X\notin\pi_{b}(\mathfrak{su}(d))$ and $X\in\mathfrak{su}(\mathcal{H}_{b})$, we have the following inclusions of compact connected Lie groups together with the inclusions of the corresponding Lie algebras

[TABLE]

In Dynkin et al. (2000) it was proven that for $d\,{>}\,2$ the Lie algebra $\pi_{b}(\mathfrak{su}(d))$ is a maximal Lie subalgebra in $\mathfrak{su}\left(\mathcal{H}_{b}\right)$. Therefore, by the assumption that $X\notin\pi_{b}(\mathfrak{su}(d))$, we have $\mathfrak{g}=\mathfrak{su}\left(\mathcal{H}_{b}\right)$. Since $\mathcal{G}$ is the unique simply-connected Lie group with Lie algebra $\mathfrak{g}$ we have $\mathcal{G}=\mathrm{SU}(\mathcal{H}_{b})$. This together with (100) concludes the proof of the case (a) since $\langle\mathrm{SU}(\mathcal{H}_{b}),\mathbb{T}(\mathcal{H}_{b})\rangle=\mathrm{U}(\mathcal{H}_{b})$.

We will now treat cases (b) and (c), focusing first on the situations with $d=2$ modes for $N\neq 6$. For such a situation we have the following maximal inclusions of Lie subalgebras Dynkin et al. (2000)

[TABLE]

for even $N$, and

[TABLE]

for odd $N$. By "maximal inclusions" in the previous sentence we mean that

• (i)

there are no other possible Lie algebras between $\pi_{b}(\mathfrak{su}(2))$ and $\mathfrak{so}\left(\mathcal{H}_{b}\right)/\mathfrak{usp}\left(\mathcal{H}_{b}\right)$;

• (ii)

there are no other possible Lie subalgebras between $\mathfrak{so}\left(\mathcal{H}_{b}\right)/\mathfrak{usp}\left(\mathcal{H}_{b}\right)$ and $\mathfrak{su}(\mathcal{H}_{b})$.

Moreover, from Table 5 in Dynkin et al. (2000) it follows that for $N\neq 6$ there is no other Lie algebras between $\pi_{b}(\mathfrak{su}(2))$ and $\mathfrak{su}(\mathcal{H}_{b})$. The Lie groups $\mathrm{SO}\left(\mathcal{H}_{b}\right)$ and $\mathrm{USp}\left(\mathcal{H}_{b}\right)$ are defined as subgroups of $\mathrm{SU}\left(\mathcal{H}_{b}\right)$ that preserve, respectively, the symmetric and the antisymmetric non-degenerate real bilinear form $B$ on $\mathcal{H}_{b}$ (see Lemma 1)

[TABLE]

From the above discussion we can deduce, repeating the reasoning given in the proof of (a), that there are three possibilities:

• (i)

$\mathcal{G}=\mathrm{SO}(\mathcal{H}_{b})$ (for even $N\neq 6$);

• (ii)

$\mathcal{G}=\mathrm{USp}(\mathcal{H}_{b})$ (for odd $N$);

• (iii)

$\mathcal{G}=\mathrm{SU}(\mathcal{H}_{b})$ (for $N\neq 6$).

The cases (i) and (ii) occur if and only if unitaries $V_{t}=\mathrm{exp}(\mathrm{i}tX)$ preserve the form $B$ (for suitable $N$). The preservation of $B$ is equivalent to V_{t}\otimes V_{t}\mbox{|\Psi_{b}\rangle}=\mbox{|\Psi_{b}\rangle}, which on the Lie algebra level translates to

[TABLE]

Under the assumption $\operatorname{tr}(X)=0$ the above condition equivalent to

[TABLE]

We have therefore proved point (b) of the Theorem. If the commutator (107) does not vanish we know that unitaries $V$ do not preserve $B$ and therefore we are in the case (iii). Using (100) we can therefore conclude the proof of the point (c) of the Theorem for $N\neq 6$.

The situation for $N\,{=}\,6$ bosons in $d\,{=}\,2$ modes is more complicated as in this case $\pi_{b}(\mathfrak{su}(2))$ is not maximally embedded in $\mathfrak{so}(\mathcal{H}_{b})$, instead it is maximal in the seven dimensional representation of the Lie algebra of the exotic simple Lie algebra $\mathfrak{g}_{2}$, which itself is a maximal Lie subalgebra of $\mathfrak{so}(\mathcal{H}_{b})$,

[TABLE]

The corresponding inclusions on the Lie group level have the form

[TABLE]

First, we observe that $\mathcal{G}$ is a connected compact Lie group (this follows form Lemma 7). Moreover, its Lie algebra $\mathfrak{g}$ must be simple (this is because $\mathfrak{g}$ contains $\pi_{b}(\mathfrak{su}(2))$, whose elements commute only with the identity on $\mathcal{H}_{b}$, due to the irreducibility of $\pi_{b}$). On the other hand form the results of Dynkin (see Table 5 of Dynkin et al. (2000) ) it follows that all simple Lie subalgebras of $\mathfrak{su}(\mathcal{H}_{b})$ (for $d=2$ and $N=6$ dimension of $\mathcal{H}_{b}$ is 7) are the ones appearing in the sequence of inclusions (108). Since $\mathrm{G}_{2}\subset\mathrm{SO}(\mathcal{H}_{b})$ we can use the reasoning analogous to the one given in the proof of (c) for $N\neq 6$ to conclude that \left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mbox{|\Psi_{b}\rangle!\langle\Psi_{b}|}\right]\neq implies $\mathcal{G}=\mathrm{SU}(\mathcal{H}_{b})$. Using this and (100) we finish the proof. ∎

Theorem 5** (Extensions of Passive Fermionic Linear Optics via additional Hamiltonian).**

Let $X\notin\mathrm{Lie}\left(\mathrm{LO}_{f}\right)$ be a Hamiltonian acting on Hilbert space of $N$ fermions in $d$ modes $\mathcal{H}_{f}$, where $N\notin\{0,1,d-1,d\}$. Let $\langle\mathrm{LO}_{f},X\rangle$ be the group generated by passive fermionic linear optics and $X$ in $\mathcal{H}_{f}$. For $d=2N$ (half-filling) let $L_{f}\in\mathrm{Herm}\left(\mathcal{H}_{f}\otimes\mathcal{H}_{f}\right)$ be defined as in Eq. (5). We have the following possibilities:

• (a)

If $d\neq 2N$, then $\langle\mathrm{LO}_{f},X\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$ .

• (b)

If $d=2N$ and $\left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{f}\right]=0$, then

[TABLE]

• (c)

If $d=2N$ and $[V\otimes V,\mathbb{L}_{f}]\neq 0$, then $\langle\mathrm{LO}_{f},X\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$.

Theorem 6** (Extensions of Active Fermionic Linear Optics via additional Hamiltonian).**

Let $X\notin\mathrm{Lie}\left(\mathrm{FLO}\right)$ be a Hamiltonian acting on positive-parity subspace of fermionic $d$-mode Fock space $\mathcal{H}^{+}_{\mathrm{Fock}}$ with $d>3$. Let $\langle\mathrm{FLO},X\rangle$ be the group of transformations generated by active fermionic linear optics and $X$ in $\mathcal{H}^{+}_{\mathrm{Fock}}$. For $d=2m$ let $\mathbb{L}_{\mathrm{FLO}}\in\mathrm{Herm}\left(\mathcal{H}_{f}\otimes\mathcal{H}_{f}\right)$ be defined as in Eq.(7). We have the following possibilities:

• (a)

If $d\neq 2k$, then $\langle\mathrm{FLO},X\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$ .

• (b)

If $d=2k$ and $[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{\mathrm{FLO}}]=0$, then

[TABLE]

• (d)

If $d=2k$ and $[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mathbb{L}_{\mathrm{FLO}}]\neq 0$, then $\langle\mathrm{FLO},X\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$.

Proofs of Theorems 5 and 6.

The proofs of both theorems are very similar and analogous to the proof for the bosonic passive linear optics. For this reason we present them together. In what follows we will use $K$ to denote passive ($\mathrm{LO}_{f}$) or active ($\mathrm{FLO}$) fermionic linear optical transformations. By $K_{s}$ we will denote the "simple" part of $K$ i.e. a subgroup of $K$ consisting of operators having unit determinant on the appropriate fermionic Hilbert space $\mathcal{H}$ (equal to $\mathcal{H}_{f}$ or $\mathcal{H}^{+}_{\mathrm{Fock}}$ respectively). The symbols $\mathfrak{k}$ and $\mathfrak{k}_{s}$ will denote the Lie algebras of $K$ and $K_{s}$ respectively. The main steps are analogous to the ones form the proof Theorem 2.

Without the loss of generality we can assume that $\operatorname{tr}(X)=0$. After setting $\mathcal{G}=\langle K_{s},X\rangle$ we have

[TABLE] 2. 2.

For the cases specified in points (a) of Theorems 5 and 6 we have that $\mathfrak{k}_{s}\subset\mathfrak{su}(\mathcal{H})$ are maximal simple subalgebras of $\mathfrak{su}(\mathcal{H})$ (see Table 5 of Dynkin et al. (2000)). Therefore, by repeating the reasoning analogous to the bosonic case, we get that under the assumptions given in point (a) we have $\langle K,X\rangle=\mathrm{U}(\mathcal{H})$. 3. 3.

Analogously to the case of $\mathrm{LO}_{b}$, for situations specified in the assumptions of points (b) and (c), the appropriate representations $\Pi_{f}$ and $\Pi_{\mathrm{FLO}}^{+}$ preserve bilinear forms given by B\left(\mbox{|\psi\rangle},\mbox{|\phi\rangle}\right)=\mbox{\langle\Psi|}\mbox{|\psi\rangle}\otimes\mbox{|\phi\rangle}, where, depending on the context, \mbox{|\Psi\rangle}=\mbox{|\Psi_{f}\rangle} or \mbox{|\Psi\rangle}=\mbox{|\Psi_{\mathrm{FLO}}\rangle} (vectors \mbox{|\Psi_{f}\rangle},\mbox{|\Psi_{\mathrm{FLO}}\rangle} have been given explicitly in Lemma 3 and Lemma 6). 4. 4.

Depending on the value of $d$, bilinear forms $B$ are either symmetric or antisymmetric, giving rise to special orthogonal or unitary symplectic groups ($\mathrm{SO}(\mathcal{H})$ or $\mathrm{USp}(\mathcal{H})$) on $\mathcal{H}$. However, in the considered situations it is always the case (see again Table 5 from Dynkin et al. (2000)) that $\mathfrak{k}_{0}$ is the maximal subalgebra in $\mathfrak{so}(\mathcal{H})$ or $\mathfrak{usp}(\mathcal{H})$ and there are no other Lie subalegbras of $\mathfrak{su}(\mathcal{H})$ containing $\mathfrak{k}_{s}$. This allows us to repeat essentially the whole reasoning from the bosonic scenario. Once once again the commutator \left[X\otimes\mathbbmss{1}+\mathbbmss{1}\otimes X,\mbox{|\Psi_{b}\rangle!\langle\Psi_{b}|}\right] plays the crucial role in deciding of the structure of $\mathcal{G}$ and therefore $\langle K,X\rangle$. The resulting conditions are given precisely in points (b) and (c) of Theorems 5 and 6.

∎

E.2 Gate extensions of linear optical gates

It turns out that most o the results concerning the Hamiltonian extensions of linear optical gates carry over to the gate- extension problems. Before we prove the main theorems we first give a number of auxiliary lemmas concerning the structure of the gates normalising linear optical transformations or related gate-sets. It turns out that the normalising gates play a crucial role in the proofs of results concerning gate extension problems.

Lemma 8** (Normalising gates for orthogonal and unitary symplectic gates).**

Let $K=\mathrm{SO}(\mathcal{H})$ or $K=\mathrm{USp}(\mathcal{H})$ (for even dimensional $\mathcal{H}$). Let $V\in\mathrm{U}(\mathcal{H})$ be the gate satisfying $VKV^{\dagger}=K$, that is for all $U\in K$ we have $VUV^{\dagger}=U^{\prime}$, where $U^{\prime}\in K$. Then, we have $V=\mathrm{exp}(\mathrm{i}\theta)V^{\prime}$, for $\theta\in\mathbb{R}$ and $V^{\prime}\in K$.

Proof.

The preservation of the invariant orthogonal or symplectic structure corresponds to preservation of the fixed vector \mbox{|\Psi\rangle}\in\mathcal{H}\otimes\mathcal{H} (whose symmetry decides whether we are dealing with orthogonal or symplectic group). In other words $\mathrm{U}\in K$ if and only if U\otimes U\mbox{|\Psi\rangle}=\mbox{|\Psi\rangle}. Moreover, the vector $|\Psi\rangle$ is defined uniquely up to a global phase as the inly vector from $\mathcal{H}\otimes\mathcal{H}$ satisfying U\otimes U\mbox{|\Psi\rangle}=\mbox{|\Psi\rangle} for all $U\in K$. Now, consider a gate $V$ that normalises $K$ and an arbitrary $U\in K$. We have

[TABLE]

where we have used the fact that $VUV^{\dagger}\in K$. Since $U$ was arbitrary we conclude that V\otimes V\mbox{|\Psi\rangle}=\mathrm{exp}(\mathrm{i}\theta)\mbox{|\Psi\rangle} and therefore $V=\mathrm{exp}(\mathrm{i}\theta^{\prime})U^{\prime}$, for suitable $\theta^{\prime}\in\mathbb{R}$ and $U^{\prime}\in K$. ∎

Remark 6**.**

From the character theory it follows that compact groups irreducibly represented on $\mathcal{H}$ have at most one trivial representation in $\mathcal{H}\otimes\mathcal{H}$. Therefore, if a given irreducible representation $\Pi$ preserves orthogonal or symplectic structure (defining the corresponding group $K$), the corresponding invariant form is also uniquely defined. By a slight modification of the above argument we get that if an irreducible representation $\Pi$ of a compact group $G$ is real or quaternionic and if for a unitary $V$ we have $V\Pi(G)V^{\dagger}=\Pi(G)$, then $V=\mathrm{exp}(\mathrm{i}\theta)W$ , where $W\in K$ and $\theta\in\mathbb{R}$.

Lemma 9** (Normalising gates for passive Bosonic linear optics).**

Let $V\in\mathrm{U}(\mathcal{H}_{b})$ be the gate satisfying $V\Pi_{b}(\mathrm{SU}(d))V^{\dagger}=\Pi_{b}(\mathrm{SU}(d))$, that is for all $U\in\Pi_{b}(\mathrm{SU}(d))$ we have $VUV^{\dagger}=U^{\prime}$, where $U^{\prime}\in\Pi_{b}(\mathrm{SU}(d)$. Then, we have $V=\mathrm{exp}(\mathrm{i}\theta)V^{\prime}$, for $\theta\in\mathbb{R}$ and $V^{\prime}\in\Pi_{b}(\mathrm{SU}(d))$.

Lemma 10** (Normalising gates form passive Fermionic linear optics: generic case).**

Let $V\in\mathrm{U}(\mathcal{H}_{f})$ be the gate satisfying $V\Pi_{f}(\mathrm{SU}(d))V^{\dagger}=\Pi_{f}(\mathrm{SU}(d))$, that is for all $U\in\Pi_{f}(\mathrm{SU}(d))$ we have $VUV^{\dagger}=U^{\prime}$, where $U^{\prime}\in\Pi_{f}(\mathrm{SU}(d)$. Moreover, assume that $2N\neq d$. Then, we have $V=\mathrm{exp}(\mathrm{i}\theta)V^{\prime}$, for $\theta\in\mathbb{R}$ and $V^{\prime}\in\Pi_{f}(\mathrm{SU}(d))$.

Lemma 11** (Normalising gates form passive Fermionic linear optics: half-filling case).**

Let $V\in\mathrm{U}(\mathcal{H}_{f})$ be the gate satisfying $V\Pi_{f}(\mathrm{SU}(d))V^{\dagger}=\Pi_{f}(\mathrm{SU}(d))$, that is for all $U\in\Pi_{f}(\mathrm{SU}(d))$ we have $VUV^{\dagger}=U^{\prime}$, where $U^{\prime}\in\Pi_{f}(\mathrm{SU}(d)$. Moreover, assume that $2N=d$. Then, we have $V=\mathrm{exp}(\mathrm{i}\theta)WV^{\prime}$ or $V=\mathrm{exp}(\mathrm{i}\theta)V^{\prime}$ for $\theta\in\mathbb{R}$, $V^{\prime}\in\Pi_{f}(\mathrm{SU}(d))$ and $W=\prod_{i=1}^{d}(f_{i}+f^{\dagger}_{i})$.

Lemma 12** (Normalising gates form active Fermionic linear optics).**

Let $V\in\mathrm{U}(\mathcal{H}^{+}_{\mathrm{Fock}})$ be the gate satisfying $V\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))V^{\dagger}=\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$, that is for all $U\in\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$ we have $VUV^{\dagger}=U^{\prime}$, where $U^{\prime}\in\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$. Moreover, assume that $d\geq 4$. Then, we have $V=\mathrm{exp}(\mathrm{i}\theta)V^{\prime}$, for $\theta\in\mathbb{R}$ and $V^{\prime}\in\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$.

Proofs of the above lemmas are to large extent similar and we present them together.

Proof of Lemmas 9, 10, 11 and 12.

Let $V\in\mathrm{U}(\mathcal{H})$ normalize an irreducible representation of a compact Lie group $\Pi(K)\subset\mathrm{U}(\mathcal{H})$. Since any one-parameter subgroup of this group is mapped to another one-parameter subgroup of $\Pi(K)$, one obtains

[TABLE]

where $\mathfrak{k}=\mathrm{Lie}(K)$ and $\pi$ is the induced representation of of the $\mathfrak{k}$ in $\mathcal{H}$. Let us moreover assume that the representation $\pi$ is the faithful representation of $\mathfrak{k}$ (this is the case for the considered groups representations). We can thus infer that for each $X\in\mathfrak{k}$ there exists a unique $X^{\prime}\in\mathfrak{k}$ such that

[TABLE]

Therefore, one can define a unique mapping $\tau_{V}:\mathfrak{k}\to\mathfrak{k}$ with the property that for all $X\in\mathfrak{k}$ $V\pi(X)V^{\dagger}=\pi(\tau_{V}(X))$. Due to the invertibility of $V$, the map $\tau$ is bijective. It is also by definition linear and, from the identity $V\pi([X_{1},X_{2}])V^{\dagger}=[V\pi(X_{1})V^{\dagger},V\,\pi(X_{2})V^{\dagger}]$, one can infer that $[\tau_{V}(X_{1}),\tau_{V}(X_{2})]=\tau_{V}([X_{1},X_{2}])$. Thus, $\tau_{V}$ is an automorphism of $\mathfrak{k}$. We will use this information to prove statements given in Lemmas 9, 10, 11 and 12 (in almost all cases appearing in these lemmas 111The exception is the case $\mathfrak{k}=so(4)$ appearing for $\mathrm{FLO}$ for $d=2$ modes. For this case we have $\pi^{+}_{\mathrm{FLO}}(\mathfrak{so}(4))\approx\mathfrak{su}(\mathcal{H}^{+}_{\mathrm{Fock}})$. transformations for which in this lemmas we have irreducible faithful representation of simple Lie algebras). The fallowing arguments are based on the structure of the automorphism groups of classical simple Lie algebras which can be found in Section 16.5 of Humphreys (1972).

• (I)

Bosonic representation of $\mathrm{SU}(d)$ - Lemma 9. We first note that for the case of of $\mathfrak{su}(2)$ the group of Lie algebra automorphisms $\mathrm{Aut}(\mathfrak{su}(2))$ consists only of inner automorphisms i.e.

[TABLE]

Therefore, we have $\tau_{V}=\mathrm{Ad}_{U}$ for some $U\in\mathrm{SU}(2)$. Consequently, we have that for all $X\in\mathfrak{su}(2)$ the following chan of inequalities hold

[TABLE]

where in the last equality we used the standard relation between the representation of Lie group $\Pi$ and the induced representation of its Lie lagebra $\pi$. Comparing expressions on the both sides of (117) we obtain

[TABLE]

Consequently, by the virtue of Shur’s lemma Fulton and Harris (1991) (a representation $\pi_{b}$ is irreducible) , we have $\Pi_{b}(U)^{\dagger}V=\mathrm{exp}(\mathrm{i}\theta)\mathbbmss{1}$, for $\theta\in\mathbb{R}$. This completes the proof in this case.

If the number of modes is greater then two ($d>2$), the automorphisms group consists of two disjoint parts

[TABLE]

where \mathrm{Inn}(\mathfrak{su}(d))=\mbox{\left{\left.\mathrm{Ad}{U}\ \right|\ \mathrm{Ad}{U}(X)=UXU^{\dagger},\ U\in\mathrm{SU}(d)\right}} and $\alpha(X)=-X$ is a non -inner automorphism. It is now important to know that for every representation $\pi$ we have $\pi\circ\alpha\approx\bar{\pi}$, where $\bar{\pi}$ denotes the dual representation of $\mathfrak{su}(d)$ and $"\approx"$ denotes the equivalence of representations. Now, we need to consider two cases (i) $\tau_{V}\in\mathrm{Inn}(\mathfrak{su}(d))$ and (ii) $\tau_{V}\in\mathrm{Inn}(\mathfrak{su}(d))\cdot\{\alpha\}$. For the case (i) we can essentially repeat the reasoning given for the case of $\mathfrak{su}(2)$ and conclude that $V=\mathrm{exp}(\mathrm{i}\theta)\Pi_{b}(U)$, for some $\theta\in\mathbb{R}$ and $U\in\mathrm{SU}(d)$. On the other hand, the situation (ii) is impossible since this would imply that $\pi_{b}\approx\bar{\pi}_{b}$, which happens only when $d=2$.

• (II)

Fermionic representation of $\mathrm{SU}(d)$: generic case - Lemma 10. For this case the proof is easy since we can adopt again the strategy given in point (I). Concretely, we must have that $\tau_{V}=\mathrm{Ad}_{U}$, for some $U\in\mathrm{SU}(d)$. If it was not the case then we would have $\pi_{f}\approx\bar{\pi}_{f}$, which happens for the case of half filling. Knowing that $\tau_{V}=\mathrm{Ad}_{U}$, for some $U\in\mathrm{SU}(d)$, we conclude that $V=\mathrm{exp}(\mathrm{i}\theta)\Pi_{f}(U^{\prime})$, for $\theta\in\mathbb{R}$ and $U^{\prime}\in\mathrm{SU}(d)$.

• (III)

Fermionic representation of $\mathrm{SU}(d)$: half-filling case - Lemma 11. For this situation we actually have $\pi_{f}\approx\bar{\pi}_{f}$ and therefore we cannot exclude the possibility that $\tau_{V}=\mathrm{Ad}_{U}\alpha$, for $U\in\mathrm{SU}(d)$. Let us assume first that we have another $V^{\prime}\in\mathrm{U}(\mathcal{H}_{f})$ normalising $\Pi_{f}(\mathrm{SU}(d))$ and satisfying $\tau_{V^{\prime}}=\mathrm{Ad}_{U^{\prime}}\alpha$. We then have $V^{\prime}=\mathrm{exp}(\mathrm{i}\theta)\Pi_{f}(U^{\prime\prime})V$ and thus the non-trivial normalising gate $V$ is essentially unique 222In order to prove this assertion we notice that from $\tau_{V^{\prime}}=\mathrm{Ad}_{U^{\prime}}\alpha$ and $\mathrm{Ad}_{U}\alpha$ it follows that $\mathrm{Ad}_{U^{\prime\dagger}}\tau_{V^{\prime}}=\mathrm{Ad}_{U^{\dagger}}\tau_{V}$. We can now use the reasoning analogous to the one given in point (I) to prove that $V^{\prime}=\mathrm{exp}(\mathrm{i}\theta)\Pi_{f}(UU^{\prime\dagger})V$. (up to a global phase and the action of $\Pi_{f}(U^{\prime\prime})$, for $U^{\prime\prime}\in\mathrm{SU}(d)$). Therefore we have two possibilities: (i) $\tau_{V}=\mathrm{Ad}_{U}$ or (ii) $\tau_{V}=\mathrm{Ad}_{U}\cdot\alpha$. The first possibility again results in $V=\mathrm{exp}(\mathrm{i}\theta)\Pi_{f}(U)$. On the other the explicit computation using the fact that we are in the half-filling scenario $2N=d$) shows that the particle-whole duality $W=\prod_{i=1}^{d}(f_{i}+f_{i}^{\dagger})$ effectively realises the non-trivial automorphism i.e.

[TABLE]

This observation, together with the uniqueness of the normalising gate (up to the action of elements form $\Pi_{f}(\mathrm{SU}(d))$ and a global phase factor) finishes the proof.

• (IV)

Positive-parity spinor representation of $\mathrm{Spin}(2d)$ - Lemma 12. If the number of modes is greater then four ($d>4$), the automorphisms group of $\mathfrak{so}(2d)$ consists of two disjoint parts

[TABLE]

where \mathrm{Inn}(\mathfrak{so}(2d))=\mbox{\left{\left.\mathrm{Ad}{U}\ \right|\ \mathrm{Ad}{U}(X)=UXU^{\dagger}\ ,U\in\mathrm{Spin}(2d)\right}}. Thus we again have two possibilities: either (i) $\tau_{V}=\mathrm{Ad}_{U}$, for $U\in\mathrm{Spin}(2d)$ or (ii) $\tau_{V}=\mathrm{Ad}_{U}\cdot\beta$ for a non-inner automorphism $\beta$ of a Lie algebra $\mathfrak{so}(2d)$. The case (i) implies again (see reasoning given in point (I) above) that $V=\mathrm{exp}(\mathrm{i}\theta)\Pi_{\mathrm{FLO}}^{+}(U)$, for $\mathrm{U}\in\mathrm{Spin}(2d)$ and $\theta\in\mathbb{R}$. Let us now settle the possibility (ii). For $d>4$ the outher automorphism $\beta$ of $\mathfrak{so}(2d)$ can be taken to be of the form

[TABLE]

where $E_{ij}$ ($i<j$) are the basis elements of $\mathfrak{so}(2d)$ given in (28). For the positive parity representation $\pi_{\mathrm{FLO}}^{+}$ we have the identity

[TABLE]

Conjugating the above equation by $V$ and using the definition of $\tau_{V}$ we obtain

[TABLE]

Assuming that $\tau_{V}=\mathrm{Ad}_{U}\cdot\beta$ and using (122) we obtain

[TABLE]

which contradicts (124). We then necessarily must have $\tau_{V}=\mathrm{Ad}_{U}$ for some $U\in\mathrm{Spin}(2d)$.

The result for the positive-parity spinor representantion of $\mathrm{Spin}(8)$ for $d=4$ modes follows easily from the Lemma 8 as for this case we have $\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(8))\approx\mathrm{SO}(8)$ Oszmaniec and Kuś (2012); Oszmaniec et al. (2014).

∎

In the following proofs we will need the three well-known technical statements.

Lemma 13** (Extension of a Lie subgroup Hofmann and Morris (2013)).**

Let $K\subset\mathrm{U}(\mathcal{H})$ be a compact Lie group. Le $V\in\mathrm{U}(\mathcal{H})$ be a unitary operator that does not belong to $K$ i.e. $V\notin K$. Then, the group generated by $K$ and $V$, $\langle K,V\rangle$, is the smallest compact Lie subgroup of $\mathrm{U}(\mathcal{H})$ containing both $K$ and $V$.

Lemma 14** (Structure of compact Lie groups Hofmann and Morris (2013)).**

Let $K\subset\mathrm{U}(\mathcal{H})$ be a compact Lie group. Then $K$ has the following structure $K=K_{0}\rtimes D$, where $K_{0}$ is the maximal connected component of $K$ containing $\mathbbmss{1}$ and $D$ is a finite discrete group. Moreover, $K_{0}$ is a normal subgroup of $K$.

Lemma 15** (Extension by a non-normalising element).**

Let $K\subset\mathrm{U}(\mathcal{H})$ be a connected compact Lie subgroup of $\mathrm{U}(H)$. Let $V\in\mathrm{U}(\mathcal{H})$ be a unitary gate that does not normalise $K$ and let $\mathcal{G}=\langle K,V\rangle$ be the Lie group generated by the $K$ and $V$ (see Lemma 14). Then, we have a strict inclusion

[TABLE]

where $\mathcal{G}_{0}$ is a connected component of $\mathcal{G}$. In other words we have $\dim(\mathrm{Lie}(K))<\dim(\mathcal{G}_{0})$.

Proof.

We have $VKV^{\dagger}\subset\mathcal{G}$ but $K\neq VKV^{\dagger}$. Moreover, $VKV^{\dagger}$ is a subset of $\mathcal{G}_{0}$, the connected component of $\mathcal{G}$. Since the mapping $U\rightarrow VUV^{\dagger}$ is a diffeomorphism in $\mathrm{U}(\mathcal{H})$, $\mathrm{Lie}(K)$ is mapped bijectively to $\mathrm{Lie}(VKV^{\dagger})$. However, we also have $\mathrm{Lie}(K)\neq\mathrm{Lie}(VKV^{\dagger})$, since, because $K$ is connected, these Lie algebras uniquely specify the groups $K$ and $VKV^{\dagger}$. Consequently,

[TABLE]

and therefore Lie algebra of $\mathcal{G}_{0}$ must be strictly bigger then $\mathrm{Lie}(K)$. ∎

Theorem 1** (Extensions of Passive Bosonic Linear Optics with an additional gate).**

Let $V\notin\mathrm{LO}_{b}$ be a gate acting on the Hilbert space $\mathcal{H}_{b}$ of $N>1$ bosons in $d$ modes; and let $\langle\mathrm{LO}_{b},V\rangle$ be the group of transformations generated by passive bosonic linear optics and $V$ in $\mathcal{H}_{b}$. We have the following possibilities:

• (a)

If $d>2$, then $\langle\mathrm{LO}_{b},V\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

• (b)

If $d=2$, $N\neq 6$ and $[V\otimes V,\mathbb{L}_{b}]=0$, then $\langle\mathrm{LO}_{b},V\rangle=G_{b}=\{\left.U\in\mathrm{U}\left(\mathcal{H}_{b}\right)\right|\left[U\otimes U,\mathbb{L}_{b}\right]=0\ \}$ .

• (c)

If $d=2$ and $[V\otimes V,\mathbb{L}_{b}]\neq 0$, then $\langle\mathrm{LO}_{b},V\rangle=\mathrm{U}\left(\mathcal{H}_{b}\right)$.

Proof.

Similarly to the case of the Hamiltonian extension we note that $\mathrm{LO}_{b}=\langle\Pi_{b}(\mathrm{SU}(d)),\mathbb{T}(\mathcal{H}_{b})\rangle$ and we can assume without the loss of generality that $V\in\mathrm{SU}(\mathcal{H}_{b})$. We then have

[TABLE]

For this reason in what follows we will focus on the properties of $\mathcal{G}$. From Lemma 9 we see that $V$ cannot normalise $\Pi_{b}(\mathrm{SU}(d))$. By by the virtue of Lemma 15 we have a strict inclusion $\pi_{b}(\mathfrak{su}(d))\subset\mathrm{Lie}(\mathcal{G}_{0})$. We now give the specific proofs of different cases given in the theorem.

• (a)

From the proof of Theorem 2 we have that for $d>2$ a Lie algebra $\pi_{b}(\mathfrak{su}(d))$ is a maximal Lie sub algebra of $\mathfrak{su}(\mathcal{H})$. Therefore $\mathrm{Lie}(\mathcal{G}_{0})=\mathfrak{su}(\mathcal{H}_{b})$ and consequently $\mathcal{G}_{0}=\mathcal{G}=\mathrm{SU}(\mathcal{H}_{b})$. Form this an (128) the point (a) follows.

• (b)

From the proof of point (b) of Theorem 2 we know that for $d=2$ and $N\neq 6$ the Lie algebra $\pi_{b}(\mathfrak{su}(2))$ is a maximal Lie subalgebra in $\mathfrak{so}(\mathcal{H}_{b})$ or $\mathfrak{usp}(\mathcal{H})$ (depending whether $N$ is odd or even). The condition $[V\otimes V,\mathbb{L}_{b}$ is then equivalent to $\mathcal{G}\subset\mathrm{SO}(\mathcal{H}_{b})$ (for even $N$) or $\mathcal{G}\subset\mathrm{USp}(\mathcal{H}_{b})$ (for odd $N$). Because of the maximality of the inclusions of $\pi_{b}(\mathfrak{su}(2))$ in the Lie algebras of these groups we conclude $\mathcal{G}=\mathrm{SO}(\mathcal{H}_{b})$ (for even $N$) and $\mathcal{G}=\mathrm{USp}(\mathcal{H}_{b})$ (for odd $N$). Using (128) we conclude the proof of (b).

• (c)

From Lemma 14 it follows that $\mathcal{G}=\mathcal{G}_{0}\rtimes D$, where $\mathcal{G}_{0}$ is simply-connected Le group and $D$ is a discrete group that normalizes $\mathcal{G}_{0}$. In the proof of Theorem 2 we saw that the only Lie algebras in $\mathfrak{su}(\mathcal{H}_{b})$ containing $\pi_{b}(\mathfrak{su}(2))$ are (i) $\mathfrak{so}(\mathcal{H}_{b})$ (for even $N$), (ii) $\mathfrak{usp}(\mathcal{H}_{b})$ (for odd $N$)), (iii) $\mathfrak{g}_{2}$ (contained in $\mathfrak{so}(\mathcal{H}_{b})$, appearing only for $N=6$, and (iv) $\mathfrak{su}(\mathcal{H}_{b})$ itself. Condition $[V\otimes V,\mathbb{L}_{b}]\neq 0$ implies that $V$ is not contained in any of the connected subgroups of $\mathrm{SU}(\mathcal{H}_{b})$: $\mathrm{SO}(\mathcal{H}_{b})$, $\mathrm{USp}(\mathcal{H}_{b})$, or $\mathrm{G}_{2}$. Moreover, by the virtue of Lemma 8 and Remark 6 $V$ cannot normalise any of these groups unless $V=\mathrm{exp}(\mathrm{i}\theta)\mathbbmss{1}$ or $V=\mathrm{exp}(\mathrm{i}\theta)W$, with $W\in\mathrm{SO}(\mathcal{H}_{b})$ (for $N=6)$). However, these two possibilities contradict the assumption $[V\otimes V,\mathbb{L}_{b}]\neq 0$. We therefore conclude that $\mathcal{G}_{0}=\mathrm{SU}(\mathcal{H})$, which together with (128) finishes the proof.

∎

Theorem 3** (Extensions of Passive Fermionic Linear Optics with an additional gate).**

Let $V\notin\mathrm{LO}_{f}$ be a gate acting on Hilbert space of $N$ fermions in $d$ modes $\mathcal{H}_{f}$, where $N\notin\{0,1,d-1,d\}$. Let $\langle\mathrm{LO}_{f},V\rangle$ be the group of transformations generated by passive fermionic linear optics and $V$ in $\mathcal{H}_{f}$. We -have the following possibilities:

• (a)

If $d\neq 2N$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$.

• (b)

If $d=2N$ and $V=Wk$, for $k\in\mathrm{LO}_{f}$ and $W=\prod_{i=1}^{d}(f_{i}+f^{\dagger}_{i})$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{LO}_{f}\cup\mathrm{LO}_{f}\cdot W$.

• (c)

If $d=2N$, $V\neq gW$, for $g\in\mathrm{LO}_{f}$, and $[V\otimes V,\mathbb{L}_{f}]=0$, then $\langle\mathrm{LO}_{f},V\rangle=G_{f}=\{\left.U\in\mathrm{U}\left(\mathcal{H}_{f}\right)\right|\ [V\otimes V,\mathbb{L}_{f}]=0\ \}$.

• (d)

If $d=2N$ and $[V\otimes V,\mathbb{L}_{f}]\neq 0$, then $\langle\mathrm{LO}_{f},V\rangle=\mathrm{U}\left(\mathcal{H}_{f}\right)$.

Proof.

Except for the case (b) the strategy of the proof for $\mathrm{LO}_{f}$ is analogous to the proof of the bosonic case. Just like in that scenario we have $\mathrm{LO}_{f}=\langle\Pi_{f}(\mathrm{SU}(d)),\mathbb{T}(\mathcal{H}_{f})\rangle$ and we can assume without the loss of generality that $V\in\mathrm{SU}(\mathcal{H}_{f})$. We thus obtain,

[TABLE]

By the virtue of Lemma 10 for the cases (a), (c) and (d) the normaliser of $\Pi_{f}(\mathrm{SU}(d))$ is either trivial of $V$ does not normalise $\Pi_{f}(\mathrm{SU}(d))$. Furthermore, one can prove points (a), (c) and (d) by essentially mimicking the resining given in the proofs points (a), (b) and (c) of Theorem 1.

In order to prove the case (b) of the current theorem we observe that the gate $W$ normalises $\Pi_{f}(\mathrm{SU}(d))$ (and therefore also $\mathrm{LO}_{f}$). Moreover, a direct computation shows that $W^{2}\propto\mathbbmss{1}$ and therefore $W^{2}\in\mathbb{T}(\mathcal{H}_{f})$. Thus any gate form $\langle\mathrm{LO}_{f},W\rangle$ can be presented as element form $\mathrm{LO}_{f}\cup\mathrm{LO}_{f}\cdot W$.

∎

Theorem 4** (Extensions of active Fermionic Linear Optics by an additional gate).**

Let $\langle\mathrm{FLO},V\rangle$ be the group of transformations generated by active linear optics and $V$ acting in positive-parity Fock subspace $\mathcal{H}^{+}_{\mathrm{Fock}}$ with $d>3$ modes. We have the following possibilities:

• (a)

If $d\neq 2k$, then $\langle\mathrm{FLO},V\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$

• (b)

If $d=2k$, and $[V\otimes V,\mathbb{L}_{\mathrm{FLO}}]=0$, then $\langle\mathrm{FLO},V\rangle=G_{\mathrm{FLO}}=\{\left.U\in\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)\right|[U\otimes U,\mathbb{L}_{\mathrm{FLO}}]=0\ \}$.

• (c)

If $d=2k$ and $[V\otimes V,\mathbb{L}_{\mathrm{FLO}}]\neq 0$, then $\langle\mathrm{FLO},V\rangle=\mathrm{U}\left(\mathcal{H}^{+}_{\mathrm{Fock}}\right)$.

Proof.

The proof follows essentially the same steps as the one of Theorem 1. Similarly to the bosonic case we have $\mathrm{FLO}=\langle\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d)),\mathbb{T}(\mathcal{H}^{+}_{\mathrm{Fock}})\rangle$ and we can assume without the loss of generality that $V\in\mathrm{SU}(\mathcal{H}^{+}_{\mathrm{Fock}})$. Then, we have

[TABLE]

Because of Lemma 12 we see that the normaliser of $\Pi^{+}_{\mathrm{FLO}}(\mathrm{Spin}(2d))$ consists of elements form $\mathbb{T}(\mathcal{H}^{+}_{\mathrm{Fock}})$. The proofs for cases (a), (b), (c) are then exactly analogous to their bosonic counterparts. ∎