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On N qubits, take a set of operators that form a universal gate set. Further, assume this gate set is given with a finite presentation of the group it generates and this group has a decideable and efficient word problem. The word problem is taken as the problem of deciding if two words written in the gate set generators are equal. Given this word problem is efficient, we can always find the set of smallest words equal to any given word. Any word is a program and so we have an efficient way to turn any program into its least, or shortest algorithm.

Is any of this possible? What do I mean by this? I am asking the following question: given $N$, the number of qubits, is there any universal gate set with an efficient word problem? If so, doesn't this mean that finding the least version of any algorithm is an efficient problem?

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    Aug 29, 2021 at 2:09

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You appear to be reading the language of circuit complexity onto the language of combinatorial group theory, for example onto general word problems or more particularly onto the word problem for groups.

Indeed, from the the Solovay-Kitaev theorem a set of universal gates, in a certain sense, may "efficiently generate" a dense subset of $\mathrm{SU}(2)$.

However I don't believe the statement that:

Given this word problem is efficient, we can always find the set of smallest words equal to any given word. Any word is a program and so we have an efficient way to turn any program into its least, or shortest algorithm.

is corrected as stated. For example, many word problems are formally undecidable, meaning no algorithm, much less even an efficient algorithm, is even available to determine whether two words generate the same group element.

Furthermore even classically and with certain uniformity restrictions, many circuit optimization problems are known to be NP-complete, meaning there is not likely to be an efficient quantum or classical problem to find a shortest circuit to realize a given Boolean function.

In the case of quantum circuits, an equivalent problem might be to identify whether a quantum circuit is roughly equivalent to the identity. This was shown to be QMA-complete by Janzig, Wocjan, and Beth.

Nonetheless you can test whether two circuits are not necessarily equivalent, for example by running the first on the all-zeroes ket, then running the second in reverse, and measuring. If the circuits are equivalent, then the state must return to the all-zeroes ket.

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It is a big "ask" to assume that the word problem for a finitely presented group is decidable, let alone solvable in polynomial time as some function of the generators. It's likely the set of gates that we can exactly synthesize from any finite gate set are not enough for most useful applications and algorithms of quantum computers. Evidence of this is provided by the fact that Clifford circuits can be efficiently simulated classically.

This is why generally, in circuit synthesis (the study of finding quantum circuits), one only cares about the task of efficiently approximating a given element up to some arbitrary precision $\epsilon>0$, and to find such short approximating words efficiently. In fact, the Solovay-Kitaev algorithm shows that this is always possible for $SU(d)$ (the special unitary group of dimension $d$), which are the elements important for qudits.

However, for qubits, we can actually do better than the Solovay-Kitaev theorem. For example, the Clifford + T gate set has the following efficient approximation property. For any $\epsilon>0$ and $2\times 2$ unitary $U$, there is a word $W$ of length $O(\log(1/\epsilon))$ in the generators $\{S,H,T\}$ such that $\|W-U\|\leq \epsilon$. Furthermore, in most cases we can find $W$ in time $O(poly(\log(1/\epsilon))$. It is an outstanding problem in the field to prove there is such an algorithm for approximating all elements of $PU(2)$ (the projective unitary group) in this way (or to find a hole, where we can't).

All the gate sets we know of with this efficient approximation property generate a special type of arithmetic lattices within $PU(2)$ and seem to come from totally definite quaternion algebras. All this gets into pretty deep number theory very fast, but it's very interesting. Check out this article for more state of the art.

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