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Coherent query amplitudes have degree at most the query count

Statement

Let $A$ be a local bit-query algorithm on $N$ input bits whose non-oracle steps are all fixed unitaries, and suppose every execution uses at most $T$ queries. For each computational-basis state $∣ a, b, w ⟩$ of its fixed register, its amplitude after every circuit prefix is represented on the Boolean input cube by a complex polynomial in $X_{0}, \dots, X_{N - 1}$ whose support has no monomial of total degree greater than the number of queries in that prefix (and at the end none has degree greater than $T$ ).

Facts & Assumptions

Given: A finite coherent local bit-query circuit, an input $x \in {0, 1}^{N}$ , and the query convention of Oracle-only bit-query algorithm.

[F1]

Oracle-only bit-query algorithm and Standard bit-flip query convention: non-oracle unitaries and the initial state are independent of $x$ , and the oracle maps a basis address-target pair $(a, b)$ to $(a, b \oplus g_{x} (a))$ .

[F2]

Polynomial rings in finitely many commuting indeterminates by iteration and Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ : a complex polynomial is a finite coefficient sum of monomials; its total degree is the maximum sum of exponents in its support.

[F3]

$C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ : finite addition and multiplication of complex polynomial coefficients obey field arithmetic.

Proof

1.1givenF1baseih

Before any query, every basis amplitude is a coordinate of the fixed initial state, hence is a constant polynomial whose support has no degree above zero. Suppose at a prefix with $t$ queries every amplitude has a representative with no supported monomial of degree greater than $t$ .

2.1F2F3step 1.1algebra

A fixed unitary $U$ sends an amplitude vector $(α_{v} (x))_{v}$ to $α_{u}^{'} (x) = \sum_{v} U_{u v} α_{v} (x)$ . This is a finite complex linear combination by [F2]–[F3], so each new amplitude has no supported monomial of degree greater than $t$ .

2.2F1F2F3step 1.1algebra

At a query, if the address is one-hot $e_{i}$ , the target bit amplitude update is $α_{e_{i}, b, w}^{'} (x) = (1 - x_{i}) α_{e_{i}, b, w} (x) + x_{i} α_{e_{i}, b \oplus 1, w} (x)$ ; at a non-one-hot address the amplitude is unchanged. Each coefficient is affine in $x_{i}$ , so the new amplitude has no supported monomial of degree greater than $t + 1$ . This includes $N = 0$ , when there are no one-hot addresses and the query leaves the target unchanged.

3.1step 2.1step 2.2ihdischarge-induction∎

Traverse the finite circuit in its chronological order. A non-oracle layer preserves the support degree bound by step 2.1; each query raises that bound by at most one by step 2.2. Induction on the finite prefix length therefore gives no supported monomial above the query count at every prefix, and in particular none above $T$ at termination.

Depends on

Used by

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Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025), §21.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (arXiv:1907.09415), §11.2 (standard reference, not scraped)