Quantum circuits with more hadamard layers solve strictly harder problems
Oracle Separations in the Fourier Hierarchy
Computational Complexity
Summary
The authors study a way to measure quantum computations based on how many times a special operation called a Hadamard layer is used. They prove that adding more of these layers makes quantum circuits strictly more powerful, at least when given access to certain kinds of black-box problems called oracles. Their work answers an open question about whether each extra Hadamard layer gives a sharper computational advantage, showing that it does. They also show differences depending on how the oracle answers queries, and prove that some problems remain too hard for quantum circuits with limited layers.
What this means in practice
- •For quantum algorithm designers: Identify limits on quantum algorithms when restricted by the number of Hadamard layers and use these limits to structure new algorithms optimally.
- •For quantum computing platform developers: Tailor quantum hardware or software optimizations considering that circuits with more Hadamard layers access strictly richer computational power under oracle models.
A theory result. No direct application yet.
Authors
Atul Mantri
Abstract
The Fourier hierarchy $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\ge2$. We prove that for every constant $k\ge2$ there is an oracle relative to which $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.