Learning at the Edge of Causality: Optimal Learning-Sample Complexity from No-Signaling Constraints

TL;DR

This work investigates how relativistic causality, encoded as the no-signaling principle, regulates the learnability of quantum states in a Grover-like setting. By introducing an amplify–learn architecture that interleaves state learning with amplitude amplification, the authors show that enforcing no-signaling prevents a hypothetical logarithmic-round speedup from materializing, forcing learning-sample and reflection-circuit costs to scale at least as $\Omega(\sqrt{N}/\log N)$ per round. They derive sharp computational bounds for learning circuit-generated pure states, which, when combined with the no-signaling constraints, align to the same scaling as the physical lower bounds, thereby unifying query, gate, and sample complexities in a causality-consistent triangle. In the universal-state-learning regime, the sample cost remains exponential, but the causality-enforced gate costs unlock a restricted corridor where learning remains feasible yet tightly bounded, illustrating physics as a regulator of learnability with broad implications for future quantum algorithms and learning-control architectures.

Abstract

What ultimately fixes the sample cost of quantum learning -- algorithmic ingenuity or physical law? We study this question in an arena where computation, learning, and causality collide. A twist on Grover's search that reflects about an a priori unknown state can collapse the query complexity from $O(\sqrt{N})$ to $O(\log N)$ over a search space $N$, i.e., an exponential speedup. Yet, standard quantum theory forbids such a unknown-state reflection (no-reflection theorem). We therefore build a state-learning-assisted architecture, called ``amplify-learn,'' which alternates the coherent amplitude amplification with state learning. Embedding this amplify-learn into the Bao-Bouland-Jordan no-signaling framework, we show that the logarithmic-round dream would open a super-luminal communication channel unless each round expends the learning-sample and reflection-circuit budgets scaling at least as $Ω(\sqrt{N}/\log N)$. In parallel, we derive tight computational learning-theoretic sample bounds for learning circuit-generated pure states, revealing a state-universal ansatz ``lock'' at order $N$ in the worst case. The dramatic closure is that no-signaling does not merely veto the unphysical primitive, but it fixes the only consistent reflection-circuit complexity, and feeding this causality-enforced complexity into the computational learning bound makes it collapse onto the very same $\sqrt{N}/\log N$ scaling demanded by no-signaling alone. No-signaling thus acts as a regulator of learnability: a constraint that mediates between physics and computation, welding query, gate, and sample complexities into a single causality-compatible triangle.

Figures (3)

• Figure 1: Schematic of the "amplify--learn" protocol. In a certain round $r$, $\hat{A}(\boldsymbol{\theta}_{\psi_r})\hat{R}_0\hat{A}(\boldsymbol{\theta}_{\psi_r})^\dagger$ consists of $\hat{R}_{\psi_r}$ with the learned parameter $\boldsymbol{\theta}_{\psi_r}$; i.e., $\hat{A}(\boldsymbol{\theta}_{\psi_r})\left|0\right> = \left|\psi\right>$. With the target reflection $\hat{R}_\tau$ and $\hat{A}(\boldsymbol{\theta}_{\psi_r})\hat{R}_0\hat{A}(\boldsymbol{\theta}_{\psi_r})^\dagger$ can generate the target-amplitude amplified state $\left|\psi_{r+1}\right>$. Here, note that the state $\left|\psi_{r+1}\right>$ is unknown. We generate the copies of $\left|\psi_{r+1}\right>$ and feed into a state-learning module, in which $\boldsymbol{\theta}_{\psi_r}$ is updated to $\boldsymbol{\theta}_{\psi_{r+1}}$. Concretely, the circuit $\hat{A}(\boldsymbol{\theta}')^\dagger$ (incorporated in the state-learning module with observable $\hat{O}=\left|0\right>\!\!\left<0\right|$) learns $\hat{A}(\boldsymbol{\theta}_{\psi_r})\hat{R}_0\hat{A}(\boldsymbol{\theta}_{\psi_r})^\dagger\hat{R}_\tau$, so that $\hat{A}(\boldsymbol{\theta}')^\dagger\left|\psi_{r+1}\right> = \left|0\right>$, or equivalently, $\hat{A}(\boldsymbol{\theta}')\left|0\right> = \left|\psi_{r+1}\right>$. By identifying $\boldsymbol{\theta}'$ and (re)dialing $\boldsymbol{\theta}_{\psi_r} \to \boldsymbol{\theta}' := \boldsymbol{\theta}_{\psi_{r+1}}$, we can repeat the aforementioned processes.
• Figure 2: Causality triangle and physics-computation concordance Embedding our amplify--learn loop within the Bao--Bouland--Jordan equivalence (super-Grover search $\Leftrightarrow$ superluminal signaling) upgrades no-signaling into an operational complexity principle: $Q_{\rm tot}(N)$ must remain $\Omega(\sqrt{N})$, which forces the per-round costs $M_s$ and $G_{\rm ref}$ to scale at least $\Omega(\sqrt{N}/\log N)$. In parallel, the computational learning theory yields $\tilde{M}_s(n,G,\epsilon,\delta)\ge (c/\epsilon^2)\min\{2^n,G\}$, giving the universal-ansatz tomography lock $\tilde{M}_s=\tilde{\Omega}(N/\epsilon^2)$ for $G=\Theta(2^n)$. The crucial closure is that in amplify--learn the learned reflection circuit is the hypothesis: no-signaling fixes $G=G_{\rm ref}$, and the computational bound collapses to $\tilde{M}_s \ge \Omega((1/\epsilon^2)\sqrt{N}/\log N)$, matching the no-signaling-only-derived bound. Equivalently, the logarithmic number of rounds is exactly compensated by the per-round reflection cost, restoring the Grover $\sqrt{N}$ barrier. No-signaling thus acts as a regulator of learnability, carving out the causality-compatible corridor and aligning query, gate and sample complexities into one triangle.
• Figure S1: Schematic construction of a super-luminal classical channel from a hypothetical super-Grover search algorithm. Bob and Alice are placed at space-like separation. Before they part, they agree on two candidate oracles $O_0$ and $O_1$ acting on Alice's query register, and on a primitive operation $\mathcal{M}$ that, when available to Alice in addition to standard quantum gates, yields an $o(\sqrt{N})$-query super-Grover search. The shared resource is arranged so that Bob's local choice of $b \in \{0,1\}$—implemented by acting only on his subsystem—selects which oracle $O_b$ is effectively realized on Alice's side, without transmitting any physical system between them. After the parties are separated, Bob encodes his bit by choosing $b$ and thereby fixing $O_b$, while Alice, who has no direct access to Bob's choice, runs the super-Grover search that uses $\mathcal{M}$ and treats $O_b$ as the underlying search oracle. If such a primitive $\mathcal{M}$ existed within the standard quantum theory, Alice would recover $b$ in time $o(\sqrt{N})$, i.e., before any light signal from Bob could arrive, thus realizing a super-luminal signaling protocol. For more detailed and self-contained scenario, see Appendix \ref{['append:bao_equiv']} or Ref. bao2016grover.

Theorems & Definitions (55)

• Theorem 1: no-reflection theorem Kumar2011
• Theorem 2: No-signaling enforces a unique $\sqrt{N}/\log N$ scaling
• Theorem 3: Computational learning-sample lower bound for circuit-generated states
• Corollary 1: Universal-ansatz lock at $N=2^n$ samples
• Corollary 2: "Three Bounds" unification into a causality-compatible triangle
• Theorem 4: Standard amplitude amplification
• proof
• Theorem 5: Cubic amplification with previous-output reflections
• proof
• Corollary 3: Imaginary logarithmic-query search