Juegos de puertas universales para SU (3)?

En la computación cuántica, a menudo nos interesan los casos en que el grupo de operadores unitarios especiales, G, para algún sistema d-dimensional proporciona al grupo entero SU (d) exactamente o incluso solo una aproximación proporcionada por una cubierta densa de SU (d).

Un grupo de orden finito, como el grupo Clifford para un sistema d-dimensional C (d), no dará una cobertura densa. Un grupo de orden infinito no dará una cobertura densa si el grupo es abeliano. Sin embargo, mi intuición aproximada es que un número infinito de puertas y operaciones de cambio de base del grupo Clifford deberían ser suficientes para proporcionar una cobertura densa.

Formalmente, mi pregunta es:

Tengo un grupo G que es un subgrupo de SU (d). G tiene un orden infinito y C (d) es un subgrupo de G. ¿Todos esos G proporcionan una cubierta densa de SU (d)?

Tenga en cuenta que estoy particularmente interesado en el caso cuando d> 2.

Considero que el grupo Clifford es como se define aquí: http://arxiv.org/abs/quant-ph/9802007

Esta no es una respuesta completa, pero tal vez sirva para responder la pregunta.

Dado que tiene un orden infinito pero C ( d ) no, entonces G necesariamente contiene una puerta de grupo no Clifford. Sin embargo, G tiene C ( d ) como subgrupo. Pero para d = 2, el grupo Clifford más cualquier otra puerta que no esté en el grupo Clifford es aproximadamente universal (véase, por ejemplo, el Teorema 1 aquí ). Por lo tanto, todos estos G proporcionan una cobertura densa en S U ( 2 n ) .$G$$C(d)$$G$$G$$C(d)$$d=2$$G$$SU(2^n)$

Para el caso donde parece que es posible demostrar que aún tiene una cobertura densa en las siguientes líneas (usando la notación del papel vinculado en la pregunta):$d>2$

1. Como todas las puertas en son unitarias, todos sus valores propios son raíces de la unidad, que por simplicidad parametrizaré por ángulos reales 0 ≤ θ i < 2 π .$G$$0 \leq \theta_i < 2\pi$
2. As $G$ has infinite order, either $G$ contains gates for which at least one value $\theta_k$ is an irrational multiple of $\pi$ or contains an arbitrarily good approximation to such an irrational multiple of $\pi$. Let us designate one such gate $g$.
3. Then there exists an $n$ such that $g^n$ is arbitrarily close to, but not equal to the identity.
4. Since $g^n$ is unitary it can be written as $\exp(-iH)$.
5. Since the Pauli group as defined in quant-ph/9802007 forms a basis for $d \times d$ matrices, you can write $H = \sum_{j,k = 0}^{d-1}\alpha_{jk} X_d^j Z_d^k$, where $\alpha_{jk} \in \mathbb{C}$ and $|\alpha_{jk}|\leq \epsilon$ for any $\epsilon > 0$ (by [3]), with at least one such $\alpha_{ab}$ not equal to zero.
6. We can then choose $C$ an element from the Clifford group which maps $X_d^j Z_d^k$ to $Z_d$ under conjugation. Thus $C g^n C^\dagger = \exp(-iCHC^\dagger) = \exp(-i(\alpha_{ab}Z_d + \sum_{(j, k) \neq (a,b)} \alpha'_{jk} X_d^j Z_d^k))$, where $\alpha'$ is just a permutation of $\alpha$ and $\alpha_{ab} = \alpha'_{01}$.
7. Note that $Z_d$ satisfies $Z_d (X_d^u Z_d^v) = \omega^{u} (X_d^u Z_d^v) Z_d$. Let us define $g_\ell = Z_d^{-\ell} C g^n C^\dagger Z_d^{\ell} = \exp(-i(\alpha_{ab}Z_d + \sum_{(j, k) \neq (a,b)} \omega^{j\ell} \alpha'_{jk} X_d^j Z_d^k))$.
8. By the Baker-Cambel-Hausdorff theorem, since all $\alpha$ have been made arbitrarily close to the identity, we can evaluate the product of $g' = g_1 \times ... \times g_d$ to first order as $\exp(-i(d\times(\sum_{k} \alpha_{0k} Z^k) + (\sum_{\ell = 1}^d \omega^d)\times\sum_{j \neq 0}\sum_k \alpha_{jk}X_d^j Z_d^k))$. Summing over all routes of unity, for $d>1$ yields $g' = \exp(-i(d\times(\sum_{k\neq b} \alpha_{0k} Z^k))$. This is basically a decoupling sequence which decouples the non-diagonal elements.
9. As only diagonal matrices remain in the exponential, $g'$ must be diagonal. Further due to the restrictions on $\alpha'$ it necessarily has eigenvalues which are non-zero but proportional to $\epsilon$.
10. By varying $\epsilon$ and repeating the above process it should be possible to generate $d$ linearly independent gates: $g'_1 ... g'_d$, such that their product results in a diagonal gate with with irrational and incommensurate phases or an arbitrarily close approximation to one.
11. By the reference given in Mark Howard's answer this, together with the Clifford group, should suffice for approximate universality.

I believe the answer to original question is probably yes, but unfortunately, I can't say that definitively. I can help answer Peter's extended question, however.

In math/0001038, by Nebe, Rains, and Sloane, they show that the Clifford group is a maximal finite subgroup of U(2^n). Solovay has also shown this in unpublished work that "uses essentially the classification of finite simple groups." The Nebe et al. paper also shows that the qudit Clifford group is a maximal finite subgroup for prime p, also using the classification of finite groups. This means that the Clifford group plus any gate is an infinite group, which makes one of the assumptions of the original question redundant.

Now, both Rains and Solovay told me that the next step, showing that an infinite group containing the Clifford group is universal, is relatively straightforward. However, I don't know how that step actually works. And more importantly for the original question, I don't know if they were only considering the qubit case or also the qudit case.

Actually, I might add that I don't understand the Nebe, Rains, and Sloane proof either, but would like to.

It's not clear to me whether you're asking about SU(3) or SU(3$^{n}$) acting on a tensor product of qudits. I'll assume you're asking about SU(3). It's not clear to me (despite what I said in a previous version of my answer) that the statement for SU(3) implies the statement for SU(3$^n$).

As long as the set of gates doesn't lie in a subgroup of SU(3), it will generate a dense cover of SU(3). So you need to check whether any of the infinite subgroups of SU(3) contains the Clifford group. I am fairly sure they don't, but I can't say for sure. Here is a math overflow question giving all the Lie subgroups of SU(3).

I thought I should update this thread before the site is frozen forever.

Daniel's answer is on the right lines. This "next step" that he mentions appears in Nebe, Rains and Sloane's later book, "Self-Dual Codes and Invariant Theory".

The answer to this question is therefore "Yes" - and it follows directly from Corollary 6.8.2 in Nebe, Rains and Sloane's book.

I am grateful to Vadym Kliuchnikov who pointed this out to me while I was visiting Waterloo.

I think the following paper may contain the relevant constructions for proving qudit universality

http://dx.doi.org/10.1088/0305-4470/39/11/010

In particular, the comment at the end of section $4$ says that Controlled-phase $CZ$, Fourier transform $F$, and a diagonal gate $D$ with irrational and incommensurate phases gives approximate universality. (This is a sufficient condition on $D$ but I'm pretty sure it is not a necessary condition.)

If your $G$ is of the correct form (and diagonal gates would seem a natural choice) then the result applies

An alternative approach would be to create the ancilla states required for implementation of the qudit Toffoli, or directly using $G$ along with Cliffords to implement the Toffoli. It's hard to say whether this is possible without knowing more about $G$.