What Can Quantum Computers Truly Do? Mathematically Determining Practicality via "Probability Distribution Generation" (SLAs & Lower Bounds)

【The Bottom Line (Key Takeaways)】Quantum computers are "probability distribution generators"

1. Quantum practicality is not a function of "speed." It is a function of "Auditability."

2. We use SLAs (Guarantees) to define actionable conditions and Lower Bounds to prune impossible expectations.

3. Only when these two metrics align does quantum computing become contractable and auditable.

【Introduction】Reframing the Quantum Question

"Will quantum computers ever be practical?"

This question, repeated ad nauseam across the globe, is fundamentally misplaced. Quantum computers are not merely "fast calculators" or "universal optimizers."

At their core, quantum computers are devices designed to generate conditional probability distributions.

The Verdict: The practicality of a quantum computer is determined not by its raw speed, but by its SLA (Guarantees) and its Lower Bounds (Limits).

In this article, we set aside the physical implementation and hardware-level noise. Instead, we frame quantum output as an "auditable object." By mathematically fixing what can be proven and what is strictly impossible, we define the "boundary line" of quantum utility from the GhostDrift perspective.

【1-Minute Diagnosis】Is Your "Quantum Project" Ready for Audit?

Before proceeding, determine whether your project is a "Dream" or a "Business."

• Q1: Can your use case be framed as estimating a probability distribution (e.g., Expectation, Quantiles, or Probability Mass)?

• Q2: Can you fix the tolerance $\epsilon$, failure probability $\delta$, and number of trials $m$ as part of a Service Level Agreement (SLA)?

• Q3: Can you fix the deviation $\gamma$ from the ideal distribution (the upper bound of error) as a mandatory audit item?

【The Verdict】

• FAIL: If you cannot fix an SLA, you are in the realm of "Quantum PR," not "Quantum Utility."

• PASS: If you can fix the SLA, you have achieved "Auditable Practicality." You are ready for the real world.

Definition of Audit: PASS means an SLA can be legally written. FAIL means an SLA cannot be defined. Beyond this point, we ignore the hype and focus solely on contractable utility.

Section 1: What Can Quantum Computers Do? (Debunking 3 Myths)

This article evaluates quantum utility based on audit conditions rather than raw computational speed. Let's dismantle the common misconceptions that cloud the industry:

1. Myth: "Quantum is inherently faster than classical."

   • Reality: Quantum advantage manifests only under specific algorithms and precise conditions.

2. Myth: "Quantum will always find the optimal solution."

   • Reality: Quantum computation is inherently stochastic. It is a device for identifying "likely candidates," not a deterministic solver.

3. Myth: "Practicality is TBD until we see more experimental benchmarks."

   • Reality: This is the industry's biggest blind spot. Engineering practicality must be defined by Mathematical Guarantees (SLA) before the first qubit is even fired.

Ending the treatment of quantum as a "magic wand" is the prerequisite for the era of true Quantum Engineering.

Section 2: The GhostDrift Perspective — Identifying the "Black Box Dice"

At GhostDrift Mathematical Institute, our philosophy is the "mathematical anchoring" of ephemeral phenomena. We view a quantum computer effectively as a highly sophisticated dice-making machine.

Every time the button is pressed, a different result (observation) emerges. To the untrained eye, it looks like chaotic "drift." However, this drift follows strict mathematical laws. No matter how complex the quantum interference, the final output is a "Transcript"—classical data. By treating this transcript as a path of adaptive probability kernels, we transform "magic" into "measurable risk."

【Use Case Examples】Where Quantum Becomes "Real"

The following areas are prime candidates for our auditable framework:

• Finance: Estimating Quantiles and Value at Risk (VaR).

• Drug Discovery: Evaluating probability mass for molecular candidate generation.

• Optimization: Auditing the quality of sample distributions (e.g., Expectation and Constraint Violation Probability).

Section 3: Defining Practicality via Mathematics: Guarantees and Lower Bounds

Our framework relies on three mathematical pillars to anchor the "Guarantees and Limits":

① Fixing the Process: Ionescu–Tulcea Theorem

We describe each quantum step as a chain of conditional probability kernels, defining the entire process as a single, auditable path.

② Calculating the Guarantee (Upper Bound): Azuma–Hoeffding Inequality

Confidence is a commodity purchased with trial count $m$:

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

This formula provides the rigorous basis for calculating the "Guarantees" obtainable under finite resources.

③ Identifying the Limit (Lower Bound): Le Cam’s Method

If the information-theoretic distinguishability $\kappa$ is too small, no amount of trials will yield the required precision:

$$m\kappa \ge 2(1-2\delta)^2$$

Knowing this "Limit Line" allows us to discard impossible expectations and focus on solvable problems.

Section 4: Audit and Responsibility in Quantum Business

Until now, quantum evaluation has belonged to "Researchers" or "PR Teams." But industrial implementation requires the perspective of Stakeholders.

【Who suffers, and how?】

• CFOs, Legal, and Compliance: They suffer when responsibility evaporates behind excuses like "the technology is still experimental" or "the result is not reproducible."

• Clients/Partners: They suffer when they cannot distinguish between system error and inherent probabilistic variance.

Auditability = Contractability. The practicality of quantum is determined not by physics, but by the formalization of responsibility through mathematics.

Conclusion: Trade Magic for Engineering

We have utilized mathematical rigor to anchor the "Ghost" of quantum uncertainty into a "Model."

The full framework and axiomatic proofs are available in our technical report:

[👉 Read Technical Report: Axiomatically Auditable Framework for Quantum-Generated Distributions]

We do not seek a dream-like future; we seek mathematical certainty.

Today's Conclusion: The value of quantum is not "speed"—it is the ability to write a set of conditions under which a third party can certify its use.

The value of quantum is not being "fast"; it is being "auditable."