We obtain a query lower bound for quantum algorithms solving the phase estimation problem. Our analysis generalizes existing lower-bound approaches to the case where the oracle $Q$ is given by controlled powers ${Q}^{p}$ of $Q$, as it is, for example, in Shor's order-finding algorithm. In this setting we will prove a $\ensuremath{\Omega}(\mathrm{log}\phantom{\rule{0.2em}{0ex}}1∕ϵ)$ lower bound for the number of applications of ${Q}^{{p}_{1}}$, ${Q}^{{p}_{2}},\dots{}$. This bound is tight due to a matching upper bound. We obtain the lower bound using a technique based on frequency analysis.
We obtain a query lower bound for quantum algorithms solving the phase estimation problem. Our analysis generalizes existing lower-bound approaches to the case where the oracle Q is given by controlled powers Qp of Q, as it is, for example, in Shor's order-finding algorithm. In this setting we will prove a \ensuremathΩ(log1∕ϵ) lower bound for the number of applications of Qp1, Qp2,…. This bound is tight due to a matching upper bound. We obtain the lower bound using a technique based on frequency analysis.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most q 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 ε requires Θ(qε21) queries;
(ii) (Unbiased) Amplitude estimation up to a precision ε requires Θ(qε2p(1−p)) queries;
(iii) Search among N items requires Θ(qN) queries;
(iv) Two level AND-OR tree (AND of m ORs, with n inputs to each OR) requires Θ(qnm) queries.
All of these bounds are optimal up to a constant factor, for all q from 1, corresponding to the classical complexity, to 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.
In this work, we study the phase estimation problem. We show an alternative, simpler and self-contained proof of query lower bounds. Technically, compared to the previous proofs [NW99, Bes05], our proof is considerably elementary. Specifically, our proof consists of basic linear algebra without using the knowledge of Boolean function analysis and adversary methods. Qualitatively, our bound is tight in the low success probability regime and offers a more fine-grained trade-off. In particular, we prove that for any ε>0,p≥0, every algorithm requires at least Ω(p/ε) queries to obtain an ε-approximation for the phase with probability at least p. However, the existing bounds hold only when p>1/2. Quantitatively, our bound is tight since it matches the well-known phase estimation algorithm of Cleve, Ekert, Macchiavello, and Mosca [CEMM98] which requires O(1/ε) queries to obtain an ε-approximation with a constant probability. Following the derivation of the lower bound in our framework, we give a new and intuitive interpretation of the phase estimation algorithm of [CEMM98], which might be of independent interest.
Phase estimation, due to Kitaev \cite{Kit95}, is one of the most fundamental subroutines in quantum computing. In the basic scenario, one is given black-box access to a unitary U , and an eigenstate | ψ ⟩ of U with unknown eigenvalue e i θ , and the task is to estimate the eigenphase θ within ± δ , with high probability. The cost of an algorithm for us is the number of applications of U and U − 1 .We tightly characterize the cost of several variants of phase estimation where we are no longer given an eigenstate, but are required to estimate the maximum eigenphase of U , aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap γ with the top eigenspace.We give algorithms and nearly matching lower bounds for all ranges of parameters.We show that a small number of copies of the advice state (or of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having lots of advice (applications of the advice-preparing unitary) does not significantly reduce cost, and neither does knowledge of the eigenbasis of U . We immediately obtain a lower bound on the complexity of the Unitary recurrence time problem, resolving an open question of She and Yuen \cite{SY23}.Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that a phase-estimation algorithm with precision δ and error probability ϵ has cost Ω ( 1 δ log 1 ϵ ) , matching an easy upper bound. This contrasts with some other scenarios in quantum computing (e.g., search) where error-probability reduction costs only a factor O ( log ( 1 / ϵ ) ) . Our lower bound uses a variant of the polynomial method with trigonometric polynomials.
The oracle identification problem (OIP) was introduced by Ambainis et al. \cite{AIKMRY04}. It is given as a set S of M oracles and a blackbox oracle f. Our task is to figure out which oracle in S is equal to the blackbox f by making queries to f. OIP includes several problems such as the Grover Search as special cases. In this paper, we improve the algorithms in \cite{AIKMRY04} by providing a mostly optimal upper bound of query complexity for this problem: (i) For any oracle set S such that ∣S∣≤2Nd (d<1), we design an algorithm whose query complexity is O(NlogM/logN), matching the lower bound proved in \cite{AIKMRY04}. (ii) Our algorithm also works for the range between 2Nd and 2N/logN (where the bound becomes O(N)), but the gap between the upper and lower bounds worsens gradually. (iii) Our algorithm is robust, namely, it exhibits the same performance (up to a constant factor) against the noisy oracles as also shown in the literatures \cite{AC02,BNRW03,HMW03} for special cases of OIP.
Abstract. The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998; [ J. ACM, 48 (2001), pp. 778–797]) and the adversary method by Ambainis (STOC 2000; [ J. Comput. System Sci., 64 (2002), pp. 750–767]) and the compressed oracle method by Zhandry [ Proceedings of the 39 th CRYPTO, 2019, pp. 239–268] have been shown to be powerful in proving quantum query lower bounds for a wide variety of problems. In this paper, we propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem, which is from an information theory perspective. Using this method, we obtain the following new results: (1) A quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Here, the sample complexity is measured given sample access to the quantum state to be tested, while the query complexity is measured given query access to an oracle that block-encodes the quantum state. (2) A matching lower bound [Formula: see text] for quantum Gibbs sampling at inverse temperature [Formula: see text] ([Formula: see text] suppresses logarithmic factors), showing that the quantum Gibbs sampler by Gilyén, Su, Low, and Wiebe [ Proceedings of the 51 st STOC, 2019, pp. 193–204] is optimal. (3) A new lower bound [Formula: see text] for the entanglement entropy problem with gap [Formula: see text], which was recently studied by She and Yuen [ Proceedings of the 14 th ITCS, 2023, pp. 96:1–96:17]. (4) A series of quantum query lower bounds for matrix spectrum testing, based on the sample lower bounds for quantum state spectrum testing by O’Donnell and Wright (STOC 2015; [ Comm. Math. Phys., 387 (2021), pp. 1–95]). In addition, we provide unified proofs for some known lower bounds that have been proven previously via different techniques, including those for phase/amplitude estimation and Hamiltonian simulation.