I study a promise problem for an unknown unitary operator acting on an
n-qubit system. The operator is promised to take one of two forms: either it implements a fixed permutation of computational basis states, or it implements the same permutation together with a conditional sign change determined by a designated input qubit. I show that these two cases can be distinguished with certainty using a single query to the unknown operator and a measurement of only one qubit. The procedure requires no ancilla qubits and uses only
n+1 Hadamard gates in addition to the oracle call. The promise is intrinsically quantum, since the two cases differ only in their relative-phase structure and therefore have no direct classical counterpart in the usual black-box model.
The unitary synthesis problem (Aaronson-Kuperberg, CCC 2007) asks whether every
n-qubit unitary
U is computable by efficient quantum circuits relative to some classical oracle
f=fU depending on
U. Recently, Lombardi-Ma-Wright (STOC 2024) proved that Haar-random unitaries cannot be efficiently synthesized by algorithms that make 1 query (or poly
(n) parallel queries) to an arbitrary classical oracle. In this work, we prove several results about the hardness (and easiness!) of variants of unitary synthesis. Our results include: (1) 1-query vs. 2-query unitary synthesis: we prove 1-query lower bounds for synthesizing random permutation unitaries
P∣x⟩=∣π(x)⟩, as well as random alternating-basis phase unitaries
F2⋅H⊗n⋅F1. This gives 1-query lower bounds for "explicit" families of unitaries that have efficient (even 2-query) synthesis algorithms. (2) Upper bound for complex phase unitaries: we also consider complex phase unitaries
∣x⟩↦αx∣x⟩, which have a clean 2-query synthesis algorithm with no obvious 1-query algorithm. In this case, we prove an upper bound: there are 1-query algorithms (relative to binary phase oracles) that constant-approximate these unitaries in diamond distance. In order to prove our lower bounds, we introduce and analyze two new cryptographic games: the oracle state search game and the oracle Choi state game. Compared to prior work, our framework is mathematically simple, more flexible in what it can prove, and more accurately captures the hardness of synthesizing unitaries that are not "fully random". Finally, we also use the search game to prove a new hardness-of-approximation result for quantum programs (synthesizing unitaries relative to quantum advice) for phase unitaries, giving a sharper separation between 1-query unitary synthesis and quantum programs.
Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B.
We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes
PostBQP and
PreciseBQP, the exponentially precise variant of
BQP. Both classes equal
PP in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.
Quantum algorithms are typically understood in terms of the evolution of a multi-qubit quantum system under a prescribed sequence of unitary transformations. The input to the algorithm prescribes some of the unitary transformations in the sequence with others remaining fixed. For oracle query algorithms, the input determines the oracle unitary transformation. Such algorithms can be regarded as devices for discriminating amongst a set of unitary transformations. The question arises: "Given a set of known oracle unitary transformations, to what extent is it possible to discriminate amongst them?" We investigate this for the Deutsch-Jozsa problem. The task of discriminating amongst the admissible oracle unitary transformations results in an exhaustive collection of algorithms which can solve the problem with certainty.
We study minimum-error identification of an unknown single-bit Boolean function given black-box (oracle) access with one allowed query. Rather than stopping at an abstract optimal measurement, we give a fully constructive solution: an explicit state preparation and an explicit measurement unitary whose computational-basis readout achieves the Helstrom-optimal success probability 3/4 for distinguishing the four possible functions. The resulting circuit is low depth, uses a fixed gate set, and (in this smallest setting) requires no entanglement in the input state. Beyond the specific example, the main message is operational. It highlights a regime in which optimal oracle discrimination is not only well-defined but implementably explicit: the optimal POVM collapses to a compact gate-level primitive that can be compiled, verified, and composed inside larger routines. Motivated by this, we discuss a "what if" question that is open in spirit: for fixed (n,m,k), could optimal k-query identification (possibly for large hypothesis classes) admit deterministic, closed-form descriptions of the inter-query unitaries and the final measurement unitary acting on the natural n+m-qubit input--output registers (and, if needed, small work registers)? Even when such descriptions are not compact and do not evade known circuit-complexity barriers for generic Boolean functions, making the optimum constructive at the circuit level would be valuable for theory-to-hardware translation and for clarifying which forms of "oracle access" are physically meaningful.
We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t -query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn . More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G , we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t -query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.