Optimal Limit Sequences: Theory & Applications

• Optimal limit sequences are rigorously defined constructs that achieve extremal properties under domain-specific constraints, such as minimal state transfer time and discrepancy minimization.
• They are constructed via explicit algorithms and variational principles, employing techniques like Hilbert velocity integration, Haar function expansion, and randomized rounding.
• These sequences unify approaches across quantum dynamics, irregular distribution theory, scheduling games, and binary pattern analysis, offering practical insights for theoretical and applied research.

Optimal limit sequences have emerged as central objects in diverse areas including quantum dynamics, irregularity of distribution theory, security games, and binary sequence analysis. In each domain, optimal limit sequences encode extremal structure respecting specific constraints—state transfer fidelity and duration in quantum control, discrepancy decay for infinite sequences, quasi-regularity for combinatorial scheduling, and fixed pattern densities in binary strings. These constructions are invariably tied to sharp extremality or minimax theorems, explicit algorithms, and geometric or variational principles.

1. Quantum Speed Limit and Time-Optimal Control Sequences

The quantum speed limit (QSL) constitutes a fundamental lower bound on the time $T$ required to transfer a quantum state $|\psi_i\rangle$ to a target $|\psi_f\rangle$ using a time-dependent control Hamiltonian. Geometrically, the QSL is given by integrating the instantaneous energy variance $\Delta E(t)$ over the trajectory in projective Hilbert space. The bound reads:

$$\int_{0}^{T}\Delta E(t)\,dt \ge \arccos|\langle \psi_i | \psi_f \rangle|$$

If $\Delta E(t)$ is constant, this reduces to the Mandelstam–Tamm form.

Optimal limit sequences in quantum control are the class of control protocols $\mathbf{u}_{\mathrm{opt}}(T;t)$ that achieve the prescribed fidelity $F^*$ in the minimal possible time, respecting the QSL. The trade-off is governed by the direct Hilbert velocity $v_d(t)$, the component of state-change rate directly decreasing the Fubini–Study distance to the target. The fidelity-time law is:

$$F(T) = \cos^2 \Big(\arccos \sqrt{F(0)} - \int_{0}^{T} v_d(t)\,dt \Big)$$

Optimality requires $v_d(t)$ constant in time, enforced by a covariance condition on infinitesimal time redistribution. The optimizability index,

$$\sigma_Q = \frac{\mathrm{Std}(Q)}{\langle Q \rangle_T}$$

satisfies $\sigma_Q\to0$ only for time-optimal controls.

Construction proceeds by iteratively optimizing for target $F^*$, computing $Q_{\mathrm{opt}}(T)$ and $\sigma_Q$, and invoking linear extrapolation to predict the required $T$:

$$T_2 = T_1 + \frac{\arcsin \sqrt{F^*} - \arcsin \sqrt{F_1}}{Q_{\mathrm{opt}}(T_1)}$$

Multiple optimum classes, each representing a locally time-optimal family, are traced by varied OC algorithm initializations; the globally shortest $T_{\rm QSL}$ defines the unique optimal limit sequence for the system (Gajdacz et al., 2014).

2. $L_p$-Discrepancy: Optimal Infinite Sequences

In Quasi-Monte Carlo theory and irregularity of distribution, the $L_p$-discrepancy of an infinite sequence $S$ in $[0,1)^d$ quantifies deviation from equidistribution. For the first $N$ points,

$$L_{p,N}(S) = \| A_N(x) - N x_1 \cdots x_d \|_{L_p([0,1]^d)}$$

Proinov’s lower bound asserts $L_{p,N}(S) \ge c_{p,d} N^{-1} (\log N)^{d/2}$ for any $p>1$.

Order-2 digital $(t,d)$-sequences over $\mathbb{F}_2$, constructed from interlaced generating matrices with prescribed dyadic structure, achieve

$$L_{p,N}(S_d) \ll_{p,d} 2^t \frac{(\log N)^{d/2}}{N}$$

for all finite $p>1$. This matches the lower bound for all $p\in(1,\infty)$—these explicit sequences are thus $L_p$-discrepancy optimal limit sequences.

Proof is via Haar (Walsh) function expansion, precise coefficient bounds exploiting equidistribution properties, and Littlewood–Paley inequalities. The extension to all finite $p$ ensures optimal error bounds not only for Sobolev spaces (via $L_2$) but also for Besov, Triebel–Lizorkin, and exponential Orlicz spaces in QMC integration (Dick et al., 2016).

3. Quasi-Regular Sequences in Scheduling Games

In continuous-time security games, the defender’s optimal schedule is reduced to the existence and construction of infinite sequences over a finite alphabet $\Sigma = \{1, \ldots, n\}$ with prescribed symbol frequencies $p_i$ and gap regularity parameter $K$. A sequence is $K$-quasi-regular if, for each $i$, the ratio of the longest to shortest interval between consecutive $i$ symbols never exceeds $K$ in the long run.

Kempe–Schulman–Tamuz show that randomized $2$-quasi-regular sequences are sufficient for minimax optimality. Explicit randomized algorithms produce such sequences efficiently, with each $i$'s visit gap sizes constrained to $[b,2b]$. Deterministic $3$-quasi-regular sequences are constructed via irrational rotations (Golden Ratio schedules), with gap support restricted to three consecutive Fibonacci numbers; the ratio $f_{k+3}/f_{k+1} \le 3$ always holds.

For randomized sequences, $K=2$ is sharp: certain $p$ cannot be matched with lower $K$. Deterministic $2$-quasi-regular sequences remain open for full generality, though for small $p_i$ $(O(\varepsilon/\sqrt{n\log M}))$, matching/periodicity constructions yield $(1+\varepsilon)$ approximations.

The translation to continuous time is via random shift invariant schedules; defender minimax optimality is proven for K=2 (Kempe et al., 2016).

4. Pattern-Density Optimization in Binary Sequences

Kenyon’s work defines the density of a pattern $w$ in a binary sequence $X$ as:

$\rho_w(X) = \frac{N_w(X)}{\binom{n}{m}}$

where $N_w(X)$ counts the (not necessarily consecutive) occurrences of $w$. In the limit, the empirical measure converges to a step-function $f(x)$, and the density is expressed as

$$\rho_w(\mu) = m! \int_{0 \le x_1 < \cdots < x_m \le 1} \prod_{j=1}^m g_w(x_j)\, dx_1 \cdots dx_m$$

Optimization problems include characterization of feasible pattern densities, explicit calculation of extremal densities under constraints, and identification of entropy-maximizing limit sequences.

For a pattern $\tau$ with $m$ ones and $n$ zeros, the feasible region with fixed $\rho_1 = \rho$ is:

$0 \leq \rho_\tau \leq C_\tau \rho^m (1 - \rho)^n$

For $\tau = 1010$, maximal density is

$$\rho_{1010}^{\max} = \frac{12}{e^2} \rho^2 (1-\rho)^2$$

achieved by a unique piecewise-defined $f(x)$, with explicit formula involving non-analytic points at $x=\rho/e$ and $x=1-(1-\rho)/e$. Variational principles for patterns $1^k0$ yield characterizations of optimizers as inverse-distribution functions $H'(y) = 1 / [1-e^{p(y)}]$, parameterized by real polynomials $p(y)$.

Typical limit sequences maximizing entropy under pattern constraints are fully described via the Lagrangian formalism and Euler–Lagrange equations; uniqueness is implied by non-degenerate Jacobian relations established through Vandermonde integrals (Kenyon, 7 Jan 2026).

5. Explicit Constructions and Algorithmic Methods

Domain	Limit Sequence Class	Construction Principles
Quantum control	Time-optimal pulse shapes	Hilbert-velocity/covariance, OC algs
Discrepancy theory	Order-2 digital $(t,d)$-sequences	Matrix interlacing, dyadic expansion
Scheduling games	$K$-quasi-regular sequences	Randomized rounding, rotations
Pattern densities	Entropy-maximizing sublebesgue $f$	Variational calculus, step functions

In quantum control, sequences are generated by iterative control optimization and extrapolation. For $L_p$-discrepancy, interlacings of digital sequence matrices produce explicit order-optimal constructions. Scheduling games employ randomized dependent rounding and ergodic rotations for quasi-regularity. Pattern density problems are resolved by analytical construction of piecewise and polynomial-exponential limit measures.

6. Theoretical Implications and Open Problems

The existence and uniqueness of optimal limit sequences unify extremal principles across several mathematical and physical theories. In each setting, the analytical structure is rigid: optimality is often attained by unique objects defined by variational conditions or metric bounds.

Notable open problems include:

• For $L_\infty$ star-discrepancy, the exact minimal order in $d>1$ remains unresolved, even as the $L_p$ case is closed for all $1 < p < \infty$ (Dick et al., 2016).
• Complete deterministic construction for $2$-quasi-regular sequences for all probability vectors is still unsettled (Kempe et al., 2016).
• Pattern-density optimization in more general or non-binary alphabets, and for more intricate combinatorial patterns, awaits further development (Kenyon, 7 Jan 2026).

These lines of inquiry highlight the structural and algorithmic roles played by optimal limit sequences in both applied and theoretical contexts, especially in the presence of sharp trade-offs, invariance properties, and extremality.