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arXiv:2304.02241

A Note on Quantum Phase Estimation

Yao-Ting Lin

Published Apr 5, 2023• 0 citations
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Abstract

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 $ε> 0, p \geq 0$, 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.

DOI
10.48550/arxiv.2304.02241

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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≥0ε> 0, p \geq 0ε>0,p≥0, every algorithm requires at least Ω(p/ε)Ω(p/ε)Ω(p/ε) queries to obtain an εεε-approximation for the phase with probability at least p. However, the existing bounds hold only when p>1/2p > 1/2p>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)O(1/ε)O(1/ε) queries to obtain an εεε-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.
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.
We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of log⁡∗\log^*log∗ of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.
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.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk PPP and a single marked state, the quantum walk corresponding to PPP can find the solution using a number of queries that is quadratically smaller than the classical hitting time of PPP. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.
Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary UφU_φUφ​ happen simultaneously) without using more applications of UφU_φUφ​, albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of UφU_φUφ​. We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.