TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Generic Circuit Count Lower Bound

Statement

Fix a nonempty finite gate alphabet of $A \geq 1$ gates, each of arity at most $k$ with $1 \leq k \leq 3$ , and count circuits on $n \geq 1$ classical input wires with zero-initialized ancillas and one designated measured output bit. No additional classical acceptance predicate is free. For a gate budget $C \geq n$ , put $W = n + k C + 1$ . After deleting unused ancillas, every such circuit has at most $W$ wires, and there are at most $W \sum_{j = 0}^{C} (A W^{k})^{j} = \exp (O (C \log C))$ output-designated gate-list descriptions. Each description computes at most one Boolean function with success probability at least $2 / 3$ on every input. Hence, for all sufficiently large $n$ , at least $99 %$ of Boolean functions on $n$ bits require $Ω (2^{n} / n)$ gates. If arbitrary classical postprocessing is allowed, its circuit or description cost must be included; otherwise this conclusion need not hold.

Facts & Assumptions

Given: A fixed finite gate alphabet with $A \geq 1$ gates and maximum arity $1 \leq k \leq 3$ , $n \geq 1$ classical input wires, zero-initialized ancillas, and one designated measured output bit. Count circuits with at most $C$ gate occurrences, where $C \geq n$ ; delete unused ancillas, retaining the output wire, so at most $W = n + k C + 1$ wires remain. The acceptance event is that measured bit being $1$ , with no free classical postprocessing.

[A1]

Finite circuit syntax and ordered wire labels gives finite ordered gate supports and chronological layers; Denotational semantics of a finite layered circuit gives the exact product denotation of those layers. Their current draft bodies were inspected. A layered circuit can therefore be serialized in its stated order with the same denotation.

[F1]

Quantum acceptance probability for a finite promise computation defines $p_{C} (x)$ as the probability of the designated accepting output event for the circuit and input.

[F2]

BQP for promise problems in the fixed QC14 model uses acceptance at least $2 / 3$ on YES inputs and at most $1 / 3$ on NO inputs.

[F3]

Finite circuit resource accounting defines circuit size as the number of gate occurrences and width as the number of all wires, including workspace.

Proof

1.1givenA1F3algebra

By [F3], gate budget $C$ counts gate occurrences and $W$ includes every workspace wire. A circuit with $j$ gate occurrences has at most $(A W^{k})^{j}$ ordered gate-list descriptions on $W$ wire labels: there are $A$ gate types and at most $W^{k}$ ordered supports per type. By [A1], flattening each finite layer list in chronological order preserves its denotation, so every layered circuit of at most $C$ gates is represented by one of these lists. Choosing the designated output wire contributes a factor at most $W$ . Thus the number of descriptions is at most $W \sum_{j = 0}^{C} (A W^{k})^{j}$ . Since $A \geq 1$ , $W \geq 1$ , $C \geq n$ , and $k$ is fixed, $W = O (C)$ for positive $C$ and $A W^{k} \geq 1$ . The sum is at most $(C + 1) (A W^{k})^{C}$ , whose logarithm is $O (C \log C)$ . Duplicate denotations only reduce the number of computed functions.

1.2F1F2algebra

By [F1], a fixed description has one acceptance probability $p_{C} (x)$ at each input. If it computed two distinct Boolean functions with success threshold [F2], some input $x$ would require the same number $p_{C} (x)$ to be both at least $2 / 3$ and at most $1 / 3$ , which is impossible. Therefore one description accounts for at most one truth table.

2.1step 1.1step 1.2algebra

There are exactly $2^{2^{n}}$ Boolean functions on $n$ bits. For $C_{n} = ⌊ c 2^{n} / n ⌋$ , the bound in step 1.1 has base-two logarithm at most $K C_{n} \log_{2} C_{n}$ for a constant $K$ depending only on the fixed alphabet and arity. Since $\log_{2} C_{n} \leq n + O (1)$ , choose $c > 0$ so that this logarithm is at most $2^{n - 1}$ for all sufficiently large $n$ . Then the fraction with at most $C_{n}$ gates is at most $2^{2^{n - 1}} / 2^{2^{n}} < 0.01$ eventually. By step 1.2, each description covers at most one function, so at least $99 %$ require more than $C_{n} = Ω (2^{n} / n)$ gates. This is an almost-all lower bound, not a lower bound for each named function.

3.1givenstep 1.2step 2.1∎

The model in the statement charges the measured output-bit event and includes no free classical postprocessing, so the one-description/one-function count applies as stated. If a classical postprocessing circuit is admitted, it must be included in the finite description and gate count before this argument can be reused.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)