An Axiomatic Framework for Auditing Quantum-Generated Distributions

Auditing Quantum Outputs through Adaptive Stochastic Generation
Abstract: This paper treats the execution logs (transcripts) of quantum computers as adaptively generated probability distributions. By integrating (i) Measure Theory (existence and uniqueness of path measures via Ionescu–Tulcea), (ii) Martingale Analysis (finite sample guarantees via Doob Decomposition and Azuma–Hoeffding), and (iii) Information-Theoretic Lower Bounds (Le Cam-type bounds for adaptive transcripts), we present a unified framework that identifies auditable guarantees (upper bounds) and unavoidable limitations (lower bounds). While this paper does not claim that quantum computing is universally optimal or unique, it axiomatically establishes that quantum computation behaves as a stochastic sampler and provides auditable error guarantees and fundamental lower bounds for its outputs.

1. Measure-Theoretic Construction

Axiom A1
Standard Borel Space The observation space $(\mathcal{X}, \mathcal{B})$ is assumed to be a Standard Borel Space. This ensures the existence of regular conditional probabilities.
Axiom A2
Projectability of Quantum Implementation to Classical Kernels Any observation $X_i$ generated by a quantum implementation (including quantum circuits, channels, and measurements) induces a stochastic kernel conditioned on the history $x_{
Definition 1.1
Adaptive Policy and Stochastic Kernel
  • History: $x_{< i} := (x_1, \dots, x_{i-1}) \in \mathcal{X}^{i-1}$.
  • Policy $\pi$: A sequence of measurable maps $\pi_i: \mathcal{X}^{i-1} \to \Theta$.
  • Stochastic Kernel $K_i$: A map $K_i: \mathcal{X}^{i-1} \times \Theta \to \Delta(\mathcal{X})$ where, for any $A$, $x_{< i} \mapsto K_i(A \mid x_{< i}, \pi_i(x_{< i}))$ is measurable.
Theorem 1
Sampling Representation / Ionescu-Tulcea Given a sequence of stochastic kernels $K_i$ on a Standard Borel Space $(\mathcal{X}, \mathcal{B})$ and an adaptive policy $\pi$, there exists a unique probability measure (Path Measure) $P^\pi$ on the product space $(\mathcal{X}^m, \mathcal{B}^{\otimes m})$ satisfying the following chain rule:
$$P^\pi(A_1 \times \dots \times A_m) = \int_{A_1} \dots \int_{A_m} \prod_{i=1}^m K_i(dx_i \mid x_{< i}, \pi_i(x_{< i}))$$
Proof. The existence and uniqueness of the measure are guaranteed by the Ionescu-Tulcea Theorem. In quantum implementations, since each step induces a classical stochastic kernel $K_i(\cdot\mid x_{■

2. Estimation Tasks and Lipschitz Continuity

Definition U1
TV Distance and Target Total Variation (TV) distance is defined as $d_{TV}(P, Q) := \sup_{A \in \mathcal{B}} |P(A) - Q(A)|$. For a bounded measurable function $h: \mathcal{X} \to \mathbb{R}$ ($|h| \le 1$), the functional is defined as $\mu(P) := \mathbb{E}_{P}[h]$.
Definition U1''
Parametric Representation A reference distribution $P_\theta\in\Delta(\mathcal{X})$ is determined by the parameter $\theta$. The target value is defined as $\mu(\theta):=\mu(P_\theta)=\mathbb{E}_{P_\theta}[h]$.
Lemma U1'
Expectation Bound via TV Distance For any probability measures $P, Q$ and any measurable function $h: \mathcal{X} \to \mathbb{R}$ satisfying $|h| \le 1$:
$$|\mathbb{E}_{P}[h]-\mathbb{E}_{Q}[h]| \le 2\,d_{TV}(P, Q)$$
Proof. This follows from the dual representation of the TV distance: $d_{TV}(P, Q) = \frac{1}{2}\sup_{|f|\le 1}|\mathbb{E}_{P}[f]-\mathbb{E}_{Q}[f]|$. ■

3. SLA Closure under Finite Resources

Assumption U2-A
Uniform Bound on TV Deviation Fix an ideal (reference) distribution $P_{\theta^\ast}$. Let $Q_i(\cdot) := K_i(\cdot \mid \mathcal{F}_{i-1})$ be the conditional distribution at each step. We assume there exists a constant $\gamma \in [0,1]$ such that:
$$d_{TV}(Q_i, P_{\theta^\ast}) \le \gamma \quad \text{holds a.s. for all } i=1, \dots, m.$$
Theorem U2
Doob Decomposition + Azuma-Hoeffding

Define the estimator as $\hat{\mu}_m := \frac{1}{m}\sum_{i=1}^m h(X_i)$. Let $\mathcal{F}_i = \sigma(X_1, \dots, X_i)$ and define the martingale difference as $\Delta_i := h(X_i) - \mathbb{E}[h(X_i) \mid \mathcal{F}_{i-1}]$. The following Doob decomposition holds:

$$\hat{\mu}_m - \mu(\theta^*) = \frac{1}{m}\sum_{i=1}^m \Delta_i + \frac{1}{m}\sum_{i=1}^m (\mathbb{E}[h(X_i) \mid \mathcal{F}_{i-1}] - \mu(\theta^*))$$

Since $|h| \le 1$, we have $|\Delta_i| \le 2$ (a.s.). Applying the Azuma–Hoeffding inequality for $|\Delta_i| \le 2$, for any $\epsilon > 0$:

$$P\left( \left| \frac{1}{m}\sum_{i=1}^m \Delta_i \right| \ge \epsilon \right) \le 2\exp\left( - \frac{m\epsilon^2}{8} \right)$$

Furthermore, under Assumption U2-A, applying Lemma U1' to $P=Q_i$ and $Q=P_{\theta^\ast}$ yields:

$$\left|\mathbb{E}[h(X_i)\mid\mathcal{F}_{i-1}]-\mu(\theta^\ast)\right| \le 2\,d_{TV}(Q_i, P_{\theta^\ast}) \le 2\gamma$$

Consequently:

$$P\left( |\hat{\mu}_m - \mu(\theta^*)| \ge \epsilon + 2\gamma \right) \le 2\exp\left( - \frac{m\epsilon^2}{8} \right)$$

Thus, the condition for satisfying an SLA $(\epsilon_{total}, \delta)$ is uniquely determined as $\epsilon_{total} > 2\gamma$ and $m \ge \frac{8}{(\epsilon_{total}-2\gamma)^2} \ln \frac{2}{\delta}$.

4. Information-Theoretic Lower Bounds

Theorem U3
Adaptive Lower Bounds (Le Cam + KL Chain Rule)

Let $P^\pi_\theta$ be the distribution of the transcript $X_{1:m}$ generated by an arbitrary policy $\pi$. We denote the stochastic kernel at step $i$ under parameter $\theta$ as $K_{i,\theta}$.

For any two points $\theta_0, \theta_1$, let $\Delta := |\mu(\theta_0) - \mu(\theta_1)|$. For any estimator $\hat{\mu}$, the following holds:

$$ \max_{j\in\{0,1\}} P^\pi_{\theta_j}\!\left(|\hat{\mu}-\mu(\theta_j)|\ge \frac{\Delta}{2}\right) \ge \frac12\left(1-d_{TV}(P^\pi_{\theta_0},P^\pi_{\theta_1})\right). \tag{U3-1}$$

Applying Pinsker's inequality:

$$ d_{TV}(P^\pi_{\theta_0},P^\pi_{\theta_1}) \le \sqrt{\frac12\,D_{KL}(P^\pi_{\theta_0}\|P^\pi_{\theta_1})}. \tag{U3-2}$$

By the KL Chain Rule:

$$ D_{KL}(P^\pi_{\theta_0}\|P^\pi_{\theta_1}) = \mathbb{E}_{P^\pi_{\theta_0}}\!\left[\sum_{i=1}^m D_{KL}(K_{i,\theta_0}(\cdot\mid \mathcal{F}_{i-1})\|K_{i,\theta_1}(\cdot\mid \mathcal{F}_{i-1}))\right] \tag{U3-3}$$

If the conditional KL at each step is uniformly bounded by $\kappa$:

$$D_{KL}(K_{i,\theta_0}(\cdot\mid \mathcal{F}_{i-1})\|K_{i,\theta_1}(\cdot\mid \mathcal{F}_{i-1})) \le \kappa \quad \text{a.s.} \tag{U3-4}$$

Therefore, from (U3-1) and (U3-2):

$$ \max_{j\in\{0,1\}} P^\pi_{\theta_j}\!\left(|\hat{\mu}-\mu(\theta_j)|\ge \frac{\Delta}{2}\right) \ge \frac12\left(1-\sqrt{\frac{m\kappa}{2}}\right). \tag{U3-5}$$

A necessary condition for achieving an error probability less than $\delta < \frac{1}{2}$ is:

$$ m\kappa \;\ge\; 2(1-2\delta)^2 \tag{U3-NEC}$$

This establishes an unavoidable constraint in the auditing of adaptively generated transcripts.

Proof. (U3-1) follows from Le Cam's method. (U3-3) is the chain rule for kernels; assuming (U3-4) implies $D_{KL}(P^\pi_{\theta_0}\|P^\pi_{\theta_1}) \le m\kappa$. Finally, (U3-NEC) ensures the risk bound $\delta$ is achievable. ■
Note U3-Q
Application to Quantum Computing The execution of a quantum computer induces a classical stochastic kernel $K_{i,\theta}(\cdot \mid \mathcal{F}_{i-1})$ at each step by Axiom A2. Thus, Theorem U3 applies directly to quantum transcripts. Measurement processes do not increase information (Data Processing Inequality), permitting the evaluation of $\kappa$ via Quantum Relative Entropy.

5. Conclusion

The above axiomatic construction establishes a framework for treating quantum computer execution logs (transcripts) as sequences of adaptive stochastic kernels under the minimal axioms (A1, A2). This framework provides (i) finite sample error guarantees (U2), and (ii) unavoidable information-theoretic lower bounds (U3) in a unified format.

What we have presented is not a claim of superiority for specific physical implementations, but rather a framework for fixing auditable "guarantees" and "limitations", categorized by their underlying assumptions. Consequently, given input specifications, one can explicitly determine what is mathematically guaranteed and what is fundamentally impossible.

Selected Bibliography

To axiomatically and measure-theoretically fix the proof structure of this audit report, the following primary literature and standard textbooks were adopted as theoretical pillars.
A. Measure Theory & Stochastic Kernels
  • Olav Kallenberg, Foundations of Modern Probability (Springer, 2021). A theoretical pillar for Axiom A1 and Theorem 1. Reference for Borel spaces.
  • D. P. Bertsekas & S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case. Engineering standard for path measures in discrete time.
B. Martingales & Concentration
  • Wassily Hoeffding, “Probability inequalities for sums of bounded random variables” (JASA, 1963). The origin of exponential bounds in Theorem U2.
  • Igal Sason, “On refined versions of the Azuma–Hoeffding inequality ...” Refined versions ensuring the reliability of SLA calculations.
C. Information Theory & Lower Bounds
  • Yury Polyanskiy & Yihong Wu, Information Theory (CUP Draft). Modern reference for KL, TV, and Pinsker's inequality.
  • Lucien Le Cam, Asymptotic Methods in Statistical Decision Theory (Springer, 1986). Orthodox reference for "Le Cam's Lemma."
D. Quantum Foundations
  • E. B. Davies & J. T. Lewis, “An operational approach to quantum probability” (1970). Classic foundatonal literature treating quantum measurement.
  • John Watrous, The Theory of Quantum Information (CUP, 2018). Mathematical formulation of quantum channels.
Nature of this Audit Report This paper is not a purely mathematical pursuit of truth, but rather an engineering audit report based on "Accountability as a system" and "Auditability under finite resources" as evaluation criteria. The mathematical models are constructed to maximize the visibility of operational risks in reality.
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