Symmetric Quantum Strategy (SQS)

• Symmetric Quantum Strategy (SQS) is a framework where Boolean functions remain invariant under permutation group actions, depending solely on the orbit structure of inputs.
• It rigorously analyzes quantum speedup limitations by establishing polynomial relations between classical and quantum query complexities using properties like high transitivity and well-shuffling.
• The method leverages simulation, minimax lemmas, and polynomial techniques to show that strong symmetry restricts exponential quantum advantages, impacting graph properties and related testing problems.

A symmetric quantum strategy (SQS) is defined in terms of Boolean functions invariant under the action of a permutation group. Let $G$ be a group of permutations acting on $[n] = \{1,\ldots,n\}$. A (possibly partial) Boolean function $f : \text{Dom}(f) \subseteq \Sigma^n \to \{0,1\}$ is said to be symmetric under $G$ (a $G$-symmetric function) if for every $x \in \text{Dom}(f)$ and $\pi \in G$, it holds that $x \circ \pi \in \text{Dom}(f)$ and $f(x \circ \pi) = f(x)$, with $(x \circ \pi)_i := x_{\pi(i)}$. In essence, $f$ depends only on the $G$-orbit of its input and thus exhibits a form of global invariance. The study of SQS focuses on the extent to which group symmetry restricts the potential for quantum speedup in query complexity frameworks (Ben-David et al., 2020).

1. Symmetry Under Group Actions

A $G$-symmetric function is characterized by invariance under the action of a permutation group $G \leq S_n$. Such functions are stable under relabelings dictated by $G$ and are therefore "blind" to differences within individual orbits of $G$. The formal definition is:

• $f$ is $G$-symmetric if, for all $x \in \text{Dom}(f)$ and all $\pi \in G$, $f(x \circ \pi) = f(x)$ and $x \circ \pi \in \text{Dom}(f)$.

This definition generalizes classical notions of symmetry in computational problems, accommodating both total and partial Boolean functions.

2. Quantum-Intolerant Group Actions

A group action $G$ is called quantum-intolerant if, for every $G$-symmetric Boolean function $f$, there can be no super-polynomial quantum speedup in the query complexity model. Precisely, there exists some constant $a>0$ such that

$Q(f) = \Omega(R(f)^{1/a})$

or, equivalently,

$R(f) = O(Q(f)^a),$

where $Q(f)$ and $R(f)$ denote the bounded-error quantum and randomized query complexities of $f$, respectively.

The well-shuffling property, as formalized in subsequent sections, provides a sufficient condition for quantum intolerance and captures the inability of quantum algorithms to leverage excessive symmetry for exponential speedups.

3. Transitivity and Constraints on Quantum Speedup

The structure of $G$, especially its degree of transitivity, determines the degree of quantum speedup possible. A group action $G$ on $[n]$ is $k$-transitive if, for every pair of $k$-tuples of distinct elements $(i_1,\ldots,i_k)$ and $(j_1,\ldots,j_k)$, there exists $\pi \in G$ with $\pi(i_t) = j_t$ for all $t=1,\ldots,k$.

Theorem (High transitivity forbids exponential speedups):

If $G$ is $n^{\Omega(1)}$-transitive, then every (possibly partial) Boolean $f$ symmetric under $G$ satisfies

$R(f) = O(Q(f)^{O(1)})$

(i.e., $Q(f) = \Omega(R(f)^{1/O(1)})$).

The proof leverages the fact that for highly transitive $G$, any $T$-query quantum algorithm cannot distinguish a uniformly random $\pi \in G$ from a "small-range" function $\alpha : [n] \to [n]$ with $|\operatorname{Im}(\alpha)| = r \ll n$ as long as $r = \Theta(T^3)$, as shown for $G = S_n$. By minimax and simulation arguments, this enables a classical simulation with $O(r)$ queries, yielding only polynomial quantum-classical tradeoffs.

4. Well-Shuffling Group Actions

A well-shuffling group action is defined through indistinguishability from "small-range" analogues.

• Small-range strings: $$D_{n,r} = \{\alpha \in [n]^n : |\operatorname{Im}(\alpha)| \leq r\}$$.
• Cost of distinguishing: $$\text{cost}(G, r) = Q_{1/3}(1_G : G \cup D_{n,r} \to \{0,1\})$$ is the bounded-error quantum query complexity of deciding whether a string is a member of $G$ or $D_{n,r}$.

A class of actions $\mathcal{G}$ is well-shuffling with power $a$ if there exists $b>0$ such that, for all $G \in \mathcal{G}$ and all $r < n$,

$\text{cost}(G, r) \geq \frac{r^{1/a}}{b}.$

Theorem (Well-shuffling $\Rightarrow$ polynomial speedups):

Let $f$ be $G$-symmetric and $Q(f)$ its quantum query complexity. There exists a universal constant $c$ such that

$$R(f) \leq \min\{r : \text{cost}(G, r) \geq c Q(f) \}.$$

If $\text{cost}(G,r) \geq \Theta(r^{1/a})$, then $R(f) = O(Q(f)^a)$, precluding super-polynomial quantum speedups.

Well-shuffling is preserved under:

• Induced actions on $k$-tuples,
• Restriction to subsets of orbits,
• Direct products,
• Group generation by merging actions.

This compositional closure enables bootstrapping from $S_n$ to broader families of symmetric groups relevant in graph and combinatorial problems.

5. Tight Complexity Trade-offs for Symmetric Functions

Quantitative relations between classical and quantum complexities are established for key group actions:

Group	$\text{cost}(G, r)$	Complexity Relation
$S_n$	$\geq r^{1/3}/C$	$R(f) = O(Q(f)^3)$
$G_k$ (graph relabeling)	$\geq \Omega(r^{1/6})$	$R(f) = O(Q(f)^6)$

In particular, for $S_n$-symmetric $f$, $Q(f) = \Omega(R(f)^{1/3})$; for Boolean functions on adjacency matrices symmetric under vertex relabeling, $Q(f) = \Omega(R(f)^{1/6})$. Consequently, no graph property or property-testing problem (e.g., graph isomorphism, expansion) admits super-polynomial quantum speedups, settling a previously open problem.

6. Core Proof Ingredients

Critical elements of the proofs include:

• Minimax lemma for quantum hardness: There exists a distribution $\mu$ on $\text{Dom}(f)$ such that any quantum algorithm using $< Q_\epsilon(f)$ queries errs on average by more than $\epsilon$ under $\mu$.
• Collision lower bounds: Distinguishing a random permutation from an $r$-to-1 function on $[n]$ requires $\Omega(r^{1/3})$ quantum queries—implying $\text{cost}(S_n, r) = \Omega(r^{1/3})$.
• Polynomial method: A $T$-query quantum distinguisher induces a degree $\leq 2T$ polynomial whose acceptance probability must separate the two distributions. $k$-transitive $G$ "fools" low-degree tests analogously to $S_n$, achieving comparable trade-offs.
• Simulation trick: If $G$ is indistinguishable from $D_{n,r}$ in $T$ queries, then a $T$-query $G$-symmetric quantum algorithm remains successful when the group action is replaced with a small-range function $\alpha$, allowing classical simulation in $r$ queries ($R(f) \leq r$ as soon as $T \leq \text{cost}(G,r)/c$).

These tools synthesize recent advances in quantum query complexity and group theory to formalize and quantify the constraints strong symmetry imposes on quantum strategies.

7. Implications and Closure Properties

The closure of the well-shuffling property under induced actions, restrictions, direct products, and merges implies broad applicability of the main results beyond $S_n$. For example, any class of combinatorial problems whose symmetries subsume $S_n$ is subject to the same polynomial bounds on quantum advantage. A plausible implication is that future quantum speedups in property testing and structure-invariant settings require either finding less symmetric functions or circumventing the well-shuffling barrier through novel frameworks.

For further details regarding omitted constants, extended proofs, and deeper discussion of closure properties, see Ben-David and Podder (2019) (Ben-David et al., 2020).