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arXiv:2610.05389

On State Distinguishing Without Inverse Queries

Emma Wang, Joseph Carolan, Andrew M. Childs

quant-phSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

In the quantum state distinguishing problem, an algorithm is given copies of a state $|ψ\rangle$ or $|φ\rangle$ separated by trace distance $\varepsilon$, and tasked with deciding which is the case. It is a standard result that $Θ(\varepsilon^{-2})$ copies are required, while having access to a unitary state preparation oracle and its inverse improves this to $Θ(\varepsilon^{-1})$ queries. We study the intermediate case with forward-only access, capturing scenarios where the oracle is a physical process that cannot be reversed. Tang and Wright recently showed that distinguishing a known state from an \emph{unknown} close state requires $Ω(\varepsilon^{-2})$ forward queries in high dimension; we strengthen this result by proving the same bound even when both states are known, as in the state distinguishing setting. Moreover, our lower bound uses a direct hybrid argument, avoiding advanced compressed oracle techniques used in the prior work. In contrast, we give a dimension-independent algorithm solving a continuous-time variant of state distinguishing with only $O(\varepsilon^{-1})$ total forward evolution time. The algorithm uses quantum Zeno dynamics to confine evolution to a known two-dimensional subspace. Thus, a quadratic advantage survives without inverse access for a natural continuous-time analogue of state distinguishing.

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In the quantum state distinguishing problem, an algorithm is given copies of a state ∣ψ⟩|ψ\rangle∣ψ⟩ or ∣φ⟩|φ\rangle∣φ⟩ separated by trace distance ε\varepsilonε, and tasked with deciding which is the case. It is a standard result that Θ(ε−2)Θ(\varepsilon^{-2})Θ(ε−2) copies are required, while having access to a unitary state preparation oracle and its inverse improves this to Θ(ε−1)Θ(\varepsilon^{-1})Θ(ε−1) queries. We study the intermediate case with forward-only access, capturing scenarios where the oracle is a physical process that cannot be reversed. Tang and Wright recently showed that distinguishing a known state from an \emph{unknown} close state requires Ω(ε−2)Ω(\varepsilon^{-2})Ω(ε−2) forward queries in high dimension; we strengthen this result by proving the same bound even when both states are known, as in the state distinguishing setting. Moreover, our lower bound uses a direct hybrid argument, avoiding advanced compressed oracle techniques used in the prior work. In contrast, we give a dimension-independent algorithm solving a continuous-time variant of state distinguishing with only O(ε−1)O(\varepsilon^{-1})O(ε−1) total forward evolution time. The algorithm uses quantum Zeno dynamics to confine evolution to a known two-dimensional subspace. Thus, a quadratic advantage survives without inverse access for a natural continuous-time analogue of state distinguishing.
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\mathsf{PostBQP}PostBQP and PreciseBQP\mathsf{PreciseBQP}PreciseBQP, the exponentially precise variant of BQP\mathsf{BQP}BQP. Both classes equal PP\mathsf{PP}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.
The unitary synthesis problem (Aaronson-Kuperberg, CCC 2007) asks whether every nnn-qubit unitary UUU is computable by efficient quantum circuits relative to some classical oracle f=fUf = f_Uf=fU​ depending on UUU. Recently, Lombardi-Ma-Wright (STOC 2024) proved that Haar-random unitaries cannot be efficiently synthesized by algorithms that make 1 query (or poly(n)(n)(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)⟩P\lvert x\rangle = \lvert π(x)\rangleP∣x⟩=∣π(x)⟩, as well as random alternating-basis phase unitaries F2⋅H⊗n⋅F1F_2 \cdot H^{\otimes n} \cdot F_1F2​⋅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⟩\lvert x\rangle\mapsto α_x \lvert x\rangle∣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.
We study the query complexity of testing quantum juntas. Given an unknown nnn-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most kkk qubits or is ε\varepsilonε-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to UUU, but not to U†U^\daggerU† or controlled-UUU. We give tight bounds for both unitary and channel junta testing. For sufficiently large kkk, n≥2kn \ge 2kn≥2k, and a broad range of ε\varepsilonε, both problems have query complexity Θ(k/(ε2log⁡k))Θ\left(k/(\varepsilon^2\log k)\right)Θ(k/(ε2logk)). Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an O~(k)\widetilde{O}(k)O(k) upper bound and a Ω~(k)\widetildeΩ(\sqrt{k})Ω(k​) lower bound at constant ε\varepsilonε, establishing the optimal Θ(k/log⁡k)Θ(k/\log k)Θ(k/logk) query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant ε\varepsilonε, they also give a separation between Θ(k/log⁡k)Θ(k/\log k)Θ(k/logk) forward queries and O~(k)\widetilde{O}(\sqrt{k})O(k​) queries with inverse access.
We give a natural problem over input quantum oracles UUU which cannot be solved with exponentially many black-box queries to UUU and U†U^\daggerU†, but which can be solved with constant many queries to UUU and U∗U^*U∗, or UUU and UTU^{\mathrm{T}}UT. We also demonstrate a quantum commitment scheme that is secure against adversaries that query only UUU and U†U^\daggerU†, but is insecure if the adversary can query U∗U^*U∗. These results show that conjugate and transpose queries do give more power to quantum algorithms, lending credence to the idea put forth by Zhandry that cryptographic primitives should prove security against these forms of queries. Our key lemma is that any circuit using qqq forward and inverse queries to a state preparation unitary for a state σσσ can be simulated to ε\varepsilonε error with n=O(q2/ε)n = \mathcal{O}(q^2/\varepsilon)n=O(q2/ε) copies of σσσ. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. We also identify a motif, which we call the "acorn trick", where generically strengthening a quantum resource can be possible if the output is allowed to be random, bypassing no-go theorems for deterministic algorithms. We demonstrate this idea for several settings, including controlization and purification.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most qqq oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision εεε requires Θ(1qε2)Θ(\frac{1}{q ε^2})Θ(qε21​) queries; (ii) (Unbiased) Amplitude estimation up to a precision εεε requires Θ(p(1−p)qε2)Θ(\frac{p(1-p)}{q ε^2})Θ(qε2p(1−p)​) queries; (iii) Search among NNN items requires Θ(Nq)Θ(\frac{N}{q})Θ(qN​) queries; (iv) Two level AND-OR tree (AND of mmm ORs, with nnn inputs to each OR) requires Θ(nmq)Θ(\frac{nm}{q})Θ(qnm​) queries. All of these bounds are optimal up to a constant factor, for all qqq from 1, corresponding to the classical complexity, to Q(f)Q(f)Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
Quantum linear-system algorithms access the coefficient matrix through a block-encoding oracle UAU_AUA​ and the right-hand side through a state-preparation oracle UbU_bUb​. The optimal query complexities of these two oracles are known separately: O(κlog⁡(1/ε))O(κ\log(1/\varepsilon))O(κlog(1/ε)) queries to UAU_AUA​ and O(κ/s)O(κ/s)O(κ/s) queries to UbU_bUb​, where s=α∥A−1b∥s=α\|A^{-1}b\|s=α∥A−1b∥. Whether a single algorithm can achieve both bounds simultaneously has remained open. We resolve this question affirmatively. Given a constant-factor estimate of sss, we give a quantum linear-system algorithm that prepares the normalized solution to error ε\varepsilonε using simultaneously $ O\left(κ\log\frac1\varepsilon\right)$ queries to UAU_AUA​ and O(κ/s)O\left(κ/{s}\right)O(κ/s) queries to UbU_bUb​. The algorithm separates state preparation from precision refinement, so only the matrix-query count depends on the target accuracy. We also prove a matching same-instance worst-case lower bound: for every quantum linear-system algorithm and every solution-norm scale, there exists a single Hermitian instance on which that algorithm requires both Ω(κlog⁡(1/ε))Ω(κ\log(1/\varepsilon))Ω(κlog(1/ε)) matrix queries and Ω(κ/s)Ω(κ/s)Ω(κ/s) state-preparation queries. Thus the two oracle complexities can be optimized simultaneously, and the resulting bounds are jointly optimal.