Recursive Fourier Sampling (RFS) was one of the earliest problems to demonstrate a quantum advantage, and is known to lie outside the Merlin-Arthur complexity class. This work contains a new description of quantum algorithms in phase space terminology, demonstrating its use in RFS, and how and why this gives a better understanding of the quantum advantage in RFS. Most importantly, describing the computational process of quantum computation in phase space terminology gives a much better understanding of why uncomputation is necessary when solving RFS: the advantage is present only when phase coordinate garbage is uncomputed. This is the underlying reason for the limitations of the quantum advantage.
We define a canonical version of the recursive Fourier sampling problem from quantum complexity theory, and systematically address a variety of its computational properties, including quantum query complexity, classical randomized query complexity, polynomial degree, and circuit size. Our analysis yields improvements for several of the known upper and lower bounds with regard to these complexity measures.
For quantum query complexity, we define an explicit reduced-query quantum algorithm for the problem, and recursively construct a quantum circuit that implements the algorithm. For classical query complexity, we give new proofs of the known lower bounds that improve the constants, and we give complete proofs of the complexity class separations relative to oracles, that follow from these bounds. For polynomial degree, we construct an exactly representing real polynomial of extremely low degree, and extend it to representations over other Fields. We show that this degree is exactly optimal, even for a 2 weakly-representing polynomial. For circuit size, we construct a small depth-two circuit for the non-recursive version of the problem, and we show that the size is optimal even if the depth is not restricted. We also prove asymptotically matching upper and lower bounds on circuit size for the full recursive version of the problem when the depth is restricted to two.
To prove the tight results for polynomial degree and circuit size we introduce and define a general notion of reduction for both polynomials and circuits that represent boolean functions on a partial domain. We also introduce and define, separately, a new measure of circuit complexity called effort.
The Fourier hierarchy
FH0⊆FH1⊆FH2⊆⋯, 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
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≥2.
We prove that for every constant
k≥2 there is an oracle relative to which
FHk⊊FHk+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
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.
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.