Restricted Simon’s Problem

• Restricted Simon’s Problem is a variant of Simon’s period-finding problem where limitations on the hidden XOR-mask (e.g., bounded Hamming weight) refine the problem structure.
• It benchmarks quantum versus classical query complexities, demonstrating exponential quantum advantage in regimes with specific constraints.
• The problem informs cryptanalysis and algorithm design, influencing post-quantum security evaluations and scalable quantum computing methodologies.

Restricted Simon’s Problem (RSP) refers to variants of Simon’s original period-finding problem in which the structure or domain of the hidden XOR-mask (period) is subject to additional constraints, most typically constraints on the allowed bit-strings (e.g., bounded Hamming weight), or restrictions placed on the representation or nature of the black-box function. RSP serves both as a theoretical proving ground for the separation between classical and quantum query complexity, and as a benchmark for the practicality, scalability, and noise-resilience of various quantum computing architectures. RSP arises in cryptanalysis and guides the development of optimized algorithms and lower bounds for both quantum and classical computation.

1. Formal Definition and Variants

The canonical Simon’s problem is defined for a function $f : \{0,1\}^n \to \{0,1\}^{n-1}$ with the promise that there exists a unique, nonzero string $s$ (the “period” or xor-mask) such that

$f(x) = f(y) \iff x \oplus y \in \{0^n, s\}.$

“Restricted” versions specialize this promise in one or several ways:

• Period restriction: $s$ is constrained (e.g., $\textrm{wt}(s) \leq w$ where wt is the Hamming weight) (Singkanipa et al., 15 Jan 2024).
• Domain/target restriction: The function $f$ may be over a subgroup or designed to reflect further algebraic or structural properties, such as $f$ being linear or belonging to a certain class (Apeldoorn et al., 2018), or having a specified domain/codomain (e.g., over $\mathbb{Z}_p^n$ (Ye et al., 2019)).
• Representation restriction: The complexity is considered relative to how $f$ is given: as a black-box oracle, a Boolean circuit, or as an ordered binary decision diagram (OBDD) (Zantema, 2022).

In all cases, the essential computational goal is to either find (recover) $s$ or decide whether $f$ is injective.

2. Quantum and Classical Query Complexity in Restricted Regimes

RSP exemplifies a sharp contrast in query complexity between quantum and classical approaches. In the unrestricted (original) setting, Simon's algorithm provably solves the problem in $O(n)$ quantum queries, whereas any classical algorithm requires $\Omega(\sqrt{2^n})$ queries due to the birthday paradox lower bound (Cai et al., 2016, Wu et al., 2019, Ye et al., 2019):

• Quantum query complexity: $Q(n, k) = \Theta(n - k)$. For $k=1$ (RSP), this is $\Theta(n)$.
• Classical deterministic/nonadaptive query complexity: $C(n, k) = \Theta(\sqrt{k2^{n-k}})$. For $k=1$, this yields $\Theta(\sqrt{2^n})$.

When $f$ is guaranteed to be linear, both classical and quantum query complexity reduce to $\Theta(n)$, eliminating the exponential separation (Apeldoorn et al., 2018). With further restrictions, such as small Hamming weight on $s$, the classical search space (and thus query lower bounds) contracts, for example, to $O(n^{w/2})$ for period $s$ of weight at most $w$ (Singkanipa et al., 15 Jan 2024). However, the quantum query complexity remains polylogarithmic (in the number of candidate periods) for sufficiently small $w$, demonstrating exponential advantage.

The table below summarizes these complexities in different restricted regimes:

Restriction	Quantum Queries	Classical Queries
General case ($k=1$)	$\Theta(n)$	$\Theta(\sqrt{2^n})$
Period Hamming weight $\leq w$	$O(n)$ (polylog in $N_w$)	$O(n^{w/2})$
$f$ is linear	$\Theta(n)$	$\Theta(n)$
Black-box oracle ($f$ arbitrary)	$\Theta(n)$	$\Theta(\sqrt{2^n})$
$f$ as OBDD, dimension $d$	poly($d$) for $V(f)$	poly($d$)
$f$ as circuit (unrestricted)	NP-hard to decide $V(f) > 1$	NP-hard

Here $N_w = \sum_{j=1}^w {n \choose j}$ is the number of weight-$\leq w$ periods.

3. Algorithmic Methodologies and Experimental Realizations

Quantum Algorithms for RSP:

The optimal quantum approach generalizes Simon's original algorithm. For restricted period sets, the procedure remains:

• Prepare a uniform superposition,
• Query the oracle,
• Apply the Hadamard transform,
• Measure to obtain bitstrings orthogonal to $s$,
• Use $O(n)$ samples to reconstruct $s$ with high probability (Hen, 2013, Singkanipa et al., 15 Jan 2024).

In RSP with period restriction, classical postprocessing solves a system where $z^{(i)} \cdot s = 0$ for all collected $z^{(i)}$. The exponential speedup persists so long as the region $N_w$ remains exponential in $n$.

Advanced Techniques:

• Amplitude amplification is used to make Simon’s quantum algorithm exact by selectively boosting measurement outcomes that produce novel independent equations for $s$ (Cai et al., 2016).
• Distributed quantum algorithms partition the function $f$ among multiple quantum nodes, employing techniques such as quantum parallelism, sorting operators, or unitary “routing” operators to reconstruct period information with less qubit overhead (Tan et al., 2022, Li et al., 2023, Li et al., 13 Apr 2025).
• Measurement-based (one-way) quantum computing has implemented RSP on cluster states photonic platforms, with resource requirements scaling as $n^2 + n + 1$ qubits for the $n$-qubit RSP instance (Tame et al., 2014).
• Adiabatic quantum algorithms encode the period-finding problem into the ground state of a Hamiltonian that evolves adiabatically, achieving exponential speedup and providing evidence for strong complexity equivalence between adiabatic and circuit-based models (Hen, 2013).

Experimental Realizations and Noise Robustness:

• Quantum speedup for RSP with bounded Hamming weight has been demonstrated on IBM’s 127-qubit devices, up to 58 qubits, with further improvements via dynamical decoupling and measurement error mitigation (Singkanipa et al., 15 Jan 2024).
• NISQ-era experiments reveal sensitivity to device architecture: on platforms with limited connectivity such as IBM superconducting chips, error rates rise rapidly with increasing circuit size, especially for circuits with high two-qubit gate counts (complex oracle realizations). On trapped-ion devices (IonQ), all-to-all connectivity yields more robust performance (Robertson et al., 17 Jun 2024).

4. Impact of Oracle and Function Representation

The computational model and representation of $f$ fundamentally impact the classical complexity of RSP:

• When $f$ is provided as a black-box, finding the hidden period is exponentially hard classically (Zantema, 2022).
• If $f$ is given as a Boolean circuit, the problem of deciding whether a nontrivial period exists (i.e., $V(f) > 1$) is NP-hard due to a reduction from circuit unsatisfiability (Zantema, 2022).
• If $f$ is represented as an ordered BDD (OBDD), the entire symmetry vector space $V(f)$ can be computed in polynomial time in the BDD size (Zantema, 2022). This demonstrates that the exponential classical hardness is not universal but depends on function representation structure, and that BDDs offer an example of an efficiently analyzable “restricted” classical setting.

5. Applications and Cryptanalytic Relevance

Restricted Simon’s Problem directly informs the quantum security analysis of cryptographic schemes. Notable applications:

• Distinguisher and forgery attacks: Simon’s algorithm can be leveraged to distinguish certain block ciphers (3-round Feistel) from random permutations and to forge tags in CBC-MAC and OTR authenticated encryption schemes by exploiting hidden periodicity in their internal structure (Santoli et al., 2016, Liu et al., 2023). Quantum query complexity is $O(n)$, far below the best possible classical complexity.
• Post-quantum cryptography: These findings underline the necessity of reanalyzing symmetric primitives and highlight that classical proofs may no longer guarantee quantum resistance when attackers are allowed quantum superposition queries. RSP thus functions as a prototypical “quantum cryptanalysis” oracle.

6. Connections to the Hidden Subgroup Problem and Quantum Complexity Theory

RSP is a special case of the abelian hidden subgroup problem (HSP), foundational for Shor’s factoring and discrete logarithm algorithms. The close correspondence between period-finding in the circuit and adiabatic models suggests a strong equivalence of computational power, not just up to polynomial factors but possibly exactly, if problem Hamiltonians are suitably engineered (Hen, 2013).

Variants of RSP (e.g., with a restricted set of candidate periods or over more general group domains) provide calibrated benchmarks for quantum speedup, clarifying the interplay between group structure, function restriction, and quantum advantage (Wu et al., 2019, Ye et al., 2019).

7. Algorithm Design, Scalability, and Resource Requirements

Scalability of quantum methods in RSP depends on both hardware and algorithmic choices:

• Resource scaling: In measurement-based one-way quantum computing, the number of qubits and entangling edges grows polynomially ($n^2 + n + 1$ qubits and $2n^2-2n+2$ edges) for the $n$-bit case (Tame et al., 2014).
• Distributed algorithms: Recent advances eliminate the need for sorting operators, reducing the per-node qubit requirement—crucial for NISQ-era feasibility (Li et al., 13 Apr 2025).
• Adiabatic/annealing realizations: QUBO embeddings allow RSP instances to be run on existing quantum annealers, but for tested instances, the required number of successful annealing samples can scale exponentially—rendering classical QUBO solvers more efficient for large $n$ (Robertson et al., 15 Apr 2025).

Overall, RSP remains a central benchmark in quantum algorithmics, cryptanalysis, and complexity theory, both for its conceptual tractability and for its diagnostic value in evaluating current and near-term quantum devices. Its paper continues to yield insight into the limits of classical simulation, the need for structural restrictions in problems to modulate quantum advantage, and the fine-structure of quantum-classical separations under various restrictions and computational representations.